Analysis-Geometry Seminar

Archive talks.

Spring 2009, Fall 2008, Spring 2008, Fall 2007  Spring 2006, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2001, Winter 2001, Fall 2000, 1999-2000, 1998-9, 1997-8    or  1996-7.

Fall 2000
  • October 20, 2000
    Speaker: Michael Farber  (Tel Aviv University)
    Title: On the zero in the spectrum conjecture
    Abstract:  In the talk I will describe a negative solution to the zero-in-the-spectrum conjecture obtained in a recent joint work with S. Weinberger. We constructed for any n>5 a closed smooth n-dimensional manifold M such that the Laplace-Beltrami operator acting on L2 forms of all degrees on the universal covering M~ of M is invertible. Our work was inspired by the well-known paper of M.A. Kervaire who constructed homology spheres with prescribed fundamental groups. We exploit the method of the extended L2-cohomology which treats the Novikov-Shubin phenomenon ( "zero in the continuous spectrum").
  • November 3, 2000
    Speaker: Burkhard Wilking (UPenn)
    Title: New Examples of Manifolds with Positive Sectional Curvature Almost Everywhere
    Abstract: There are only very few examples of Riemannian manifolds with positive sectional curvature known.  In fact ,in dimensions larger than 24, the known examples are diffeomorphic to locally rank 1 symmetric spaces.
        We will construct metrics with positive sectional curvature on a open and dense set of points on the projective tangent bundles of RPn, CPn and HPn.
        The so called deformation conjecture says that these kind of metrics can be deformed into metrics with positive sectional curvature everywhere.
        However, the simplest new example within our class,  the projective tangent bundle of RP3, is diffeomorphic to the product RP3xRP2.  This non-oriented manifold is known not to admit a metric with positive sectional curvature. Thus the construction provides a counterexample to the deformation conjecture.
  • November 10, 2000
    Speaker: Alexander G. Abanov (Department of Physics, SUNY at Stony Brook
    Title: Topological terms in effective action induced by Dirac fermions
    Abstract: We derive an effective action for Dirac fermions on three-dimensional sphere coupled to O(3) non-linear sigma model through the Yukawa-type interaction.  The non-perturbative (global) quantum anomaly of this model results in a Hopf term for the effective non-linear sigma model.  We obtain this term using the "embedding'' of the CP1 model into the CPM generalization of the model which makes the quantum anomaly perturbative.  This perturbative anomaly is calculated by means of a gradient expansion of a fermionic determinant and is given by the Chern-Simons term for an auxiliary gauge field.
  • November 17, 2000
    Speaker: Yulij S. Ilyashenko  (Cornell University)
    Title: Some bounds of the number of limit cycles for Abel and Lienard equations
  •          (partly based on a joint work with A.Panov).
    Abstract: We estimate the number of limit cycles of planar vector field through the size of the domain of the Poincaré map,  the increment of this map and the width of the complex domain to which the  Poincaré map  may be analytically extended. The estimate is based on the relation between growth and zeros of holomorphic functions. This estimate is then applied to getting the upper bound of the number of limit cycles of Lienard equation dx/dt  = y - F(x),  d y/dt = -y through the (odd) power of the monic polynomial F and magnitudes of its coefficients. In the same way, an upper bound of the number of limit cycles of the Abel equation is obtained.
  • November 24, 2000
    Speaker: Mathai Varghese (University of Adelaide and MIT Cancelled
       Professor Mathai will give a talk sometime during the winter quarter. 
  • December 1, 2000
    Speaker: Sergei  Yakovenko (Weizmann Institute
    Title: Rolle theorem for complex- and vector-valued functions (joint work with A. Khovansjkii) 
    Abstract: The classical Rolle theorem implies an inequality between the number of zeros of a smooth function of one real variable, and that of its derivative. This real-Rolle inequality is very important for applications, and it would be highly desirable to have it also for complex-valued functions.
        Alas, the simplest examples show that no such  inequality may exist. Instead we establish a geometric inequality
    between curvature and spherical length of spatial curves, which translates into an inequality for analytic functions very
    similar to the Rolle inequality - except that it does not concern zeros...
       Suprisingly, despite this drawback the "complex-Rolle" inequality allows for very accurate counting of complex zeros  for exponential sums.
  • December 1, 2000,   3:30PM (Note the special time)
    Speaker: Stanislav Molchanov (University of North Carolina, Charlotte)
    Title Schroedinger operators with the mixed spectrum
    Abstract: The central part of the famous (and still open) Anderson conjecture tells about possible coexistence of dense point and
    absolutely continuous spectra for multidimensional Schroedinger operators with "weak disorder".
       The talk will describe several recent results on the models with mixed (dense point plus absolutely continuous) spectra. These models include Schroedinger operators with random sparse potentials and the surface Anderson model.
  • December 8, 2000
    Speaker:   Steven Rosenberg (Boston University)
    Title Gauge theory techniques for flat connections in quantum cohomology
    Abstract: Quantum cohomology gives a finite dimensional integrable system via the Dubrovin connection.  We use gauge theory techniques to help find flat sections for the Dubrovin connection, which are key ingredients in Givental's approach to mirror symmetry.
Winter 2001
  • January 26, 2001
    Speaker: Vladimir Kondratiev  (Moscow State University)
    Title: Qualitative properties of solutions for elliptic semilinear equations
    Abstract: For the equations of the form
            Du + |u|q-1u=0,         q = const 1,
    we will discuss properties of solutions in unbounded domains (such as cone, cylinder etc.), in particular, existence or non-existence of positive solutions, and asymptotic behavior of solutions.
  • February 2, 2001
    SpeakerAndrzej Borisovich (Gdansk University)
    Title Bifurcations in Plateau problem
    Abstract: The talk is devoted  to a general method of the study of  bifurcations in the Plateau problem with parameters  in the boundary  conditions. It is based on the Crandall-Rabinowizt bifurcation theorem, the finite-dimensional  Lyapunov-Schmidt type reduction for Fredholm  maps of index 0, the key function by Sapronov and some others  constructions. Many new bifurcations of the minimal surfaces were found and  studed.
      All the results found by the author's method fit the results of physical experiments. In particular,  the experiments by J.Plateau with minimal films was described mathematically.
  • February 9, 2001
    SpeakerBruno De Oliveira (Harvard University and University of Pennsylvania)
    Title Complex cobordisms and the non-embeddability of CR-manifolds
    Abstract: We give results on complex cobordisms whose ends are strictly pseudoconvex Cauchy-Riemann-manifolds. Suppose the complex cobordism is given by a complex 2-manifold X with one pseudoconvex and one pseudoconcave end. We introduce two methods to construct pseudoconcave surfaces that show that the complex 2-manifold X giving a complex cobordism is not determined by the pseudoconvex end. These two constructions give new methods to construct non-embeddable Cauchy-Riemann 3-manifolds and prove that embeddability of a strictly pseudoconvex Cauchy-Riemann 3-manifold is not a complex-cobordism invariant. We show that a new phenomenon occurs: there are CR-functions on the pseudoconvex end that do not extend to holomorphic functions on X. We also show that the extendability of the CR-functions from the pseudoconvex end is necessary but not sufficient for embeddability to be preserved under complex cobordisms.
       In the talk we will use differential/complex/algebraic geometric arguments so any geometer is welcome.
  • February 16, 2001
    SpeakerAlexander Turbiner (Institute for Nuclear Sciences - National University of Mexico)
    Title Hydrogenic chains in magnetic field (state-of-the-art variational calculations)
    Abstract: It was anticipated by Kadomtsev-Kudriavtsev and Ruderman at 70es, that unusual chemical compounds can appear in a strong magnetic field, which do not exist without magnetic field. One-electron exotic molecular systems are of main content of the talk. State-of-art choosing variational trial functions is used. In particular, it admits to carry out the most accurate study of H2+ molecular ion. It is shown that in a strong magnetic field the systems (pppe) and (ppppe) form bound states giving rise to exotic molecular ions H3++ and H4+++. In the contrary, H2+ ion becomes unstable for some orientations of molecular axis towards a magnetic field line. It leads to a conceptual question about a content of neutron star atmosphere.
  • March 2, 2001
    Speaker: Mikhail Shubin (Northeastern University)
    Title:  Semiclassical asymptotics and gaps in the spectra of magnetic Schrödinger operators
    Abstract:  I will discuss an L2version of the semiclassical approximation for  a magnetic Schrödinger operator with periodic electric
    and magnetic fields and a Morse type electric potential. In particular, the existence of arbitrarily large number of gaps in the spectrum can be established for a  small coupling constant.
         This is a joint work with V. Mathai.
  • March 9, 2001
    Speaker: Mathai Varghese (University of Adelaide and MIT)
    Title:   On some aspects of noncommutative Bloch theory
    Abstract:  I  will discuss the noncommutative Bloch theory of Hamiltonians appearing in the quantum Hall effect and also the mathematics of a model of the quantum Hall effect.

    Spring 2001
  • March 30, 2001
    Speaker: Yuri A. Kordyukov (Ufa State Aviation Technical University)
    Title:   Adiabatic limits and spectral theory for Riemannian foliations.
  • April 6, 2001
    Speaker: Mikhail Shubin (Northeastern University)
    Title:  A discreteness of spectrum criterion for the magnetic Schrödinger operators
    Abstract:  A necessary and sufficient condition for the discreteness of spectrum of a magnetic Schr\"odinger operator will be explained. In case when the magnetic field vanishes it becomes the Molchanov criterion (1953).

    (This is joint work with V.Kondratiev.)

  • April 13, 2001
    Speaker: Weiping Zhang  (Nankai Institute of Mathematics, visiting  MIT)
    Title:   Toeplitz operators and index theorems on odd dimensional manifolds.
    Abstract:  We discuss various index theorems for Topelitz operators on odd dimensional manifolds. On closed manifolds, classically, the corresponding index theorem can be derived from the Atiyah-Singer index theorem. It can also be proved by computing the variations of eta invariants, in using a result of Booss-Wojciechowski which expresses the Toeplitz index via spectral flows. We also descibe a recent result with Xianzhe Dai on an extension of the above index theorem to the case of manifolds with boundary.
  • April 27, 2001
    Speaker: Gang Liu  (UCLA)
    Title:   Moduli space of J-holomorphic curves in contact geometry.
    Abstract:  We will describe some basic properties of J-holomorphic curves in the symplectization of a contact manifold, which will lay down an analytic foundation for the applications of J-holomorphic curves in  contact geometry.
  • Fall 2001
    • September 28, 2001
      Speaker:   Andrzej Borisovich (Gdansk University)
      Title:   Nielsen fixed  points theory, symplectic maps and Poincare-Birkhoff theorem
      Abstract:  The talk is devoted to the study of the fixed points set of the area preserving  (simplectic) selfmaps of the plane annulus.
      The study of such maps, which describe the motions of  a non-compressible  fluid, began with  works of  Poincare and Birkhoff. The fist result says that, if the map is homotopic to  identity, satisfies the twist condition and the rotation number on the boundary of annulus is nonzero, then it has at least two distinct fixed points. More general results in this direction were obtained by Franks,
      who poved that, under more general assumptions, number  of fixed points is more then  2n, where n is the rotation
          The author's approach is based on the Nielsen fixed points theory and the notion  of special degree for symplectic maps. It allows to prove analogous results in more general situation  and without the twist condition.
    • October 12, 2001
      Speaker:   Patrick Iglesias (CNRS)
      Title:   Extension and examples of the moment map for singular and infinite differentiable spaces
    • October 19, 2001
      Speaker:   Tatiana Toro (University of Washington)
      Title: What happens when the Poisson kernel equals 1 a.e.?
      Abstract:  In this talk we would like to convey the idea that Poisson kernel of a domain W being 1 almost everywhere is a very rigid condition.  If  W is bounded Lewis and Vogel showed it must be a ball.  We will discuss what happens in the unbounded case.
    • October 26, 2001
      Speaker:   Patrick McDonald (New College of USF)
      Title:   Exit time moments for Brownian motion and spectral geometry
      Abstract:  Given a compact Riemannian manifold with boundary, we use exit time moments for Brownian motion to construct a sequence of geometric invariants for the underlying manifold.  While these invariants are not spectral, they determine much of the spectral and Riemannian geometry of the manifold.  In particular, we discuss a number of comparison theorems for the invariants, we prove that the invariants always determine the heat content asymptotics associated to the manifold, and, for generic domains in Euclidean space, we prove that the invariants determine the Dirichlet spectrum.
    • November 2, 2001
      Speaker: Robert Brooks (Technion-Israel Institute of Technology)
      Title:   A Statistical Model for Riemann Surfaces
      Abstract:  (joint work with Eran Makover) We study the question: What does a typical Riemann surface look like geometrically? We model the problem of picking a Riemann surface at random on the problem of picking a 3-regular graph at random, and show that this gives an interesting picture of a typical Riemann surface.
    • November 16, 2001
      Speaker: Andrįs Vasy (MIT)
      Title:   Semiclassical estimates in asymptotically Euclidean scattering
      Abstract:  In this joint work with Maciej Zworski, we study long-range perturbations of the Laplacian on an asymptotically Euclidean space. We show how positive commutator estimates obtained via the symbol calculus can be used to show the limiting absorption principle, and give estimates for the resolvent, under non-trapping assumptions, as Planck's constant h goes to 0.
    • November 30, 2001
      Speaker: Paul Kirk  (Indiana University)
      Title:   A splitting theorem for the spectral flow of a path of Dirac operators
      Abstract:  We explain how to calculate the spectral flow of  a path Dt  of Dirac operators on a closed manifold M decomposed along a hypersurface in terms of the spectral flow of Dt on the pieces with respect to elliptic boundary conditions, Maslov indices of the Boundary conditions, and terms coming from an adiabatic stretching procedure. The method is elementary and we show how other similar theorems in the literature follow easily.
    • December 7, 2001
      Speaker: Victor Nistor (Penn State University)
      Title:   Geometric operators on manifolds with cylindrical ends and generalizations
      Abstract:  I will begin by recalling a few classical results on the analysis on manifolds with cylindrical ends, including Fredholm conditions and a determination of the spectrum of the Laplace operator. Then I will describe a class of manifolds for which one can obtain similar results. This class of manifolds is described in terms of Lie algebras of vector fields, as in geometric scattering theory. These results are based, in part, on joint works with B. Ammann, R. Lauter, R. Melrose, and B. Monthubert.
    Spring 2002
  • January 4, 2002
    Speaker:   Mikhail Katz  (Bar Ilan University)
    Title: Cup length, systolic geometry, and surjectivity of period map
    Abstract:  The result concerns a 2-dimensional conformal invariant of Riemannian metrics on 4-manifolds, similar to conformal length for Riemann surfaces.  The approach makes use of two ingredients.  The first is the Conway-Thompson unimodular lattices of high density, known to exist through probabilistic or averaging procedures, but difficult to pin down explicitly.  The second ingredient is the current work in gauge theory, which targets the surjectivity of the period map for the class of 4-manifolds with B+ = 1.  The outcome is a polynomial asymptotic lower bound for the conformal invariant as the Betti number increases.  On the other hand, a polynomial upper bound is the result of joint work with V. Bangert.  The polynomial upper and lower bounds can be viewed as a higher-dimensional analog of the logarithmic upper and lower bounds for conformal length on Riemann surfaces, due to P. Buser and P. Sarnak.
  • January 18, 2002  Cancelled
    Speaker:   Leonid Friedlander  (The University of Arizona)
    Title: On the density of states of periodic media in the large coupling limit
    Abstract:  Let W0 be a domain in the cube (0,p )n, and let ct(c) be a function that equals 1 inside W0, equals t in
    (0,p )n \ W0, and that is extended periodically to Rn. It is known that, in the limit  t®„, the spectrum of the operator -ct
    exhibits the band-gap structure. We establish the asymptotic behavior of the density of states function in the bands.
  • January 25, 2002
    Speaker:    Vladimir Kondratiev  (Moscow State University)
    Title: Existence and non-existence of positive solutions for second order non-linear ellitpic equations in unbounded domains
    Abstract:  We consider a non-linear elliptic second-order equation, which has a linear divergence type ellpitic operator as its  principal part. It is considered in domains like cone, cylinder, paraboloid or strip. Existence and non-existence conditions are obtained. They depend on the type of domain and  the non-linear terms in the equation.
  • February 1, 2002
    Speaker:   Ognjen Milatovich (Northeastern University)
    Title: Essential self-adjointness of  Schrödinger type operators.
    Abstract:  Several essential self-adjointness conditions for the Schrödinger  type operators on manifodls and in sections of vector bundles will be explained in the talk. These conditions are expressed in terms of completeness of certain metrics on the manifold. These metrics are  naturally associated with the operator.
                    (Joint work with M.Braverman and M.Shubin.)
  • February 8, 2002
    Speaker:   Alexander Kozhevnikov (University of Haifa)
    Title: On a Complete scale of isomorphisms for elliptic and parabolic pseudodifferential boundary-value problems.
    Abstract:  In monographs by J.-L. Lions and E. Magenes (1968) and  Ya. Roitberg (1996) a theorem on complete scale of isomorphisms has been established which, roughly speaking, means that  operators generated by elliptic  differential boundary-value problems are  isomorphisms (Fredholm operators) between Sobolev-type spaces of functions ''with  and '' s-d - derivatives'', where  is the order of the elliptic operator. The completeness of the scale means that  s  can be an arbitrary  real number.
          In a monograph by S. Eidelman and N. Zhitarashu (1990) a theorem on a complete scale of isomorphisms has been obtained for parabolic differential initial boundary-value problems.
          Due to the fact, that for  elliptic and parabolic pseudodifferential initial boundary-value problems there exist  parametrices belonging to the Boutet de Monvel algebra, a much shorter proof has been found as well as some applications of the results.
  • February 22, 2002
    Speaker:   Mikhail Shubin (Northeastern University)
    Title: New criteria of discreteness of spectrum for Schrödinger  operators
    Abstract:  New necessary and sufficient conditions for the discreteness of spectrum  for (magnetic)  Schrödinger operators will be explained. The fact that  all these conditions are equivalent to the discreteness of spectrum leads to the equivalence of these conditions between each other. This leads to new properties  of the Wiener capacity, which at the moment have no direct proofs.
  • April 5, 2002
    Speaker:   Vadim Tkachenko (Ben-Gurion University)
    Title: 1d Periodic Differential Operator of Order 4
    Abstract:  We consider a differential operator
                    L = d4/dx4+ d/dx p(x) d/dx + q(x),           x Ī R1
    with 1-periodic functions  p(x)  and q(x). We prove that characteristic equation for its Floquet multipliers is inverse and hence its spectrum in  L2(R1)   may be described using some hyper-elliptic Riemann surface.
         We prove the following uniqueness theorem:   Let  U(l) be the monodromy matrix of operator  L  and let its characteristic determinant   det(U(l)-r I)  be the same as of operator L0=d4/dx4.  Then  p(x)ŗq(x)ŗ0.
  • April 26, 2002
    Speaker:   Maxim Braverman (Northeastern University)
    Title: New proof of the Cheeger-Muller Theorem
    Abstract:  We present a short analytic proof of the equality between the analytic and combinatorial torsion. We use the same approach as in the proof given by Burghelea, Friedlander and Kappeler, but avoid using the difficult Mayer-Vietoris type formula for the determinants of elliptic operators. Instead, we provide a direct way of analyzing the behaviour of the determinant of the Witten deformation of the Laplacian. In particular, we show that this determinant can be written as a sum of two terms, one of which has an asymptotic expansion with computable coefficients and the other is very simple (no zeta-function regularization is involved in its definition).
  • May 10, 2002
    Speaker: Benji Fisher  (Boston College)
    Title: Quasicrystals:  Algebra, Geometry, Number Theory, and Physics
    Abstract:  TIn the 1980's, almost simultaneously, the first mathematical and physical quasicrystals were discovered; the first mathematical one was the Penrose tiling of the plane.  These structures are not periodic, but still have long-range order.  One of the most striking features of quasicrystals is their symmetry groups:  these may include five-fold rotations and other symmetries that are "crystallographically forbidden'' for ordinary  (periodic) crystals in two and three dimensions.
           This talk will describe some recent work on classifying the symmetry types of quasicrystals in two and three dimensions.  Another, related question,  is how to detect the symmetry type of a physical quasicrystal.  For example, certain symmetry types reveal themselves in the X-ray diffraction spectrum.  The talk will conclude with a description of current work on this question, using spaces of almost-periodic functions to study the quantum mechanics of quasicrystals.
  • May 17, 2002
    Speaker: Jacek Szmigielski    (University of Saskatchewan and Yale)
    Title: Non smooth water waves and continued fractions
    Abstract:  One model of nonlinear, strongly dispersive water waves, called the CH equation (Camassa, Holm) admits non smooth travelling waves which have corners in their profiles, and yet they behave in many respects like the smooth solitons of other integrable models of nolinear waves.  Those special non smooth solitons are dubbed peakons and the intention of this talk is to give a gentle overview of their collision properties.  The mathematical locale within which all their properties become transparent is an inverse spectral problem for the  discrete Dirichlet string which in turn is an implicit part of the famous treatise by Stieltjes on continued fractions.  An appropriate adaptation of Stieltjes's work provides a decisive insight into some mechanical questions regarding collisions of peakons.  This is a part of joint work with R.Beals, D. Sattinger.

    Fall 2002
  • September 20, 2002
    Speaker:  Theodore Voronov (University of Manchester (UMIST))
    Title: Differential operators, brackets and connections
    Abstract: I am going to describe remarkable relations between differential operators and bracket structures. It is known that an operator of the second order acting on functions defines a "bracket", i.e. a symmetric bilinear operation on functions satisfying the Leibniz rule (basically, the polarized principal symbol). This works on ordinary manifolds as well as on supermanifolds. In the super context, this gives a relation between odd Laplacians (or "Batalin-Vilokovisky D-operators") and odd Poisson brackets. I will show how this relation between differential operators and brackets can be made 1-1 if one considers the algebra of densities instead of the algebra of functions. This construction implicitly involves "generalized" connections: notice that every differential operator of the second order acting on functions encodes in its coefficients an "upper connection" in the bundle of volume forms (basically, the subprincipal symbol). An extension of these ideas to operators of higher order would lead to homotopy algebras.
       This is a joint work with Hovhannes Khudaverdian.

  • October 4, 2002
    Speaker:  Harold G. Donnelly (Purdue University)
    Title: Bounds for eigenfunctions of the Laplacian on compact Riemannian manifolds
    Abstract: Suppose that φ is an eigenfunction of  -Δ  with eigenvalue  λ≠0. It is proved that

    ||φ||∞  ≤   c1λ(n-1)/4||φ||2 ,

    where n is the dimension of  M and  c1 depends only upon a bound for the absolute value of the sectional curvature of  M and a lower bound for the injectivity radius of  M. It is then shown that if  M admits an isometric circle action, and the metric is generic, one has exceptional sequences of eigenfunctions satisfying the complementary bounds

    ||φk||∞  ≥   c2λ k(n-1)/8 ||φk|| 2 .


  • October 11, 2002
    Speaker:  Stanislav Dubrovskiy (Northeastern University)
    Title: Moduli space of symmetric connections
    Abstract: We are interested in local differential invariants of a symmetric connection, under smooth coordinate changes. We consider the action of origin-preserving diffeomorphisms on a space of jets of connections and calculate dimensions of moduli spaces in generic case. We show that the corresponding Poincarč series is a rational function. This confirms one more time the 1894' finitness claim of Tresse, stated for any "natural" differential-geometric structure.
  • October 18, 2002
    Speaker:  Marina Ville (CNRS and Boston University)
    Title: Milnor numbers of minimal surfaces in 4-manifolds
    Abstract: When a sequence of smooth embedded complex curves (Cn)  in  CP^2  degenerates to a branched curve C0, we lose topology  ( g(Cn)>g(C0) ) and gain singularity. Milnor gave a precise meaning to this assertion. There is a quantity - now called the Milnor number - we can compute on the branch points of  C which tells us how much topology we have lost going from  Cn  to  C0
       So we ask: is there anything even remotely similar if the  Cn's and  C0  are more general surfaces (e.g. minimal surfaces) in a 4-manifold? Can we define a Milnor number for a sequence of minimal surfaces? It turns out that we have to define not one, but two Milnor numbers (the "tangent" and the "normal" one); these numbers coincide in the complex case.
       We will define these Milnor numbers, give explain geometric and topological interpretations and show how they give a partial answer to our question above.

  • October 25, 2002
    Speaker:  Mikhail Shubin (Northeastern University)
    Title: A new family of necessary and sufficient discreteness of spectrum conditions for Schrödinger operators
    Abstract: I will explain new criteria of discreteness of spectrum for the Schrödinger operators with semi-bounded below potentials. They extend a well known result by A.Molchanov (1953) who was the first to formulate a necessary and sufficient condition for the discreteness of spectrum in terms of the Wiener capacity.
       We provide a new family of such conditions which depend on a functional parameter describing "negligible" sets.
       (This is a joint work with V.Maz'ya.)

  • November 1, 2002
    Speaker:  Roland Duduchava (Razmadze Mathematical Institute (Tbilisi, Georgia))
    Title: Mathematical theory of cracks: the Wiener-Hopf method
    Abstract:  A crack in elastic media is modelled by the Neumann boundary value problem for a homogeneous second order partial differential equation with constant coefficients. We use potentials and the Wiener-Hopf method to obtain a full asymptotic expansion of the solution. We show that these asymptotics do not contain logarithmic terms.
  • November 8, 2002
    Speaker:  Mihaela Iftime (Northeastern University)
    Title: On cylindrically symmetric solutions of Einstein's field equation
    Abstract: I will present a stationary cylindrically symmetric solution of Einstein's equation with dust and positive cosmological constant. The solution approaches Einstein static universe on the axis of rotation.
  • November 15, 2002
    Speaker:  Eric Wang (Northeastern University)
    Title: Associative cones and integrable system
    Abstract: Associative 3-manifolds play an important role in calibrated geometry of G2manifolds. We study associative cones in R7 with isolated singularity. In particular, we give an integrable system formulation using moving frame. We will discuss some applications.
                       (joint work with Shengli Kong)

  • November 22, 2002
    Speaker:  Jeff Viaclovsky (MIT)
    Title: Fully nonlinear equations on Riemannian manifolds
    Abstract: We define a conformal invariant using maximal volumes, and use this to prove existence of solutions to a class of conformally invariant fully nonlinear second order PDEs.
  • December 6, 2002
    Speaker:  Hubert Bray (MIT)
    Title: Classification of Prime 3-Manifolds with Yamabe Invariant Greater Than RP3
    Abstract: In a joint work with Andre Neves, we classify prime 3-manifolds with Yamabe invariant greater than Y2, the Yamabe invariant of RP3. The Yamabe invariant of a closed manifold is a smooth topological invariant which is defined geometrically and reduces to the Euler characteristic in dimension 2 by the Gauss Bonnet formula. However, prior to this work, the only known values of the Yamabe invariant for 3-manifolds had been 0 and
                        Y1 = 6 (2p2)2/3,
    the Yamabe invariant of S3. We increase the number of known values by 50% by proving that the Yamabe invariants of RP3 and RP2xS1 are equal to Y2 = Y1/22/3. We also prove that any 3-manifold with Yamabe invariant Y > Y2  must either be S3 or a connect sum with an S2 bundle over S2 (S2xS1 or the 3-dimensional Klein bottle). Hence, the Poincare conjecture for manifolds with Y > Y2 (which includes an infinite number of 3-manifolds) follows. Also, since the classification of 3-manifolds reduces to the classification of prime 3-manifolds, it is natural to try to make a list of prime 3-manifolds. Ordered by their Yamabe invariants, we conclude that the first five prime 3-manifolds are S3, S2xS1, K (the 3-dimensional Klein bottle), RP3, and RP2 xS1.

  • Spring 2003
  • January 10, 2003
    Speaker:  Dimitri Gourevitch (Université de Valenciennes)
    Title: Noncommutative index on the quantum sphere
    Abstract: Noncommutative (NC) index was introduced by A.Connes.
    Given a NC algebra A, NC index is defined as a pairing
                K0(A) x K0(A)→ K
    where K0(A) is defined by projective A-modules, K0(A) is defined by finite-dimensional A-modules, and K is the basic field. In the talk I will introduce quantum orbits whose particular case is a quantum sphere, define a version of the NC index well adapted to the algebras in question and compute NC index on this sphere.

  • January 24, 2003
    Speaker:  Alexander Kozhevnikov (University of Haifa)
    Title: Random fields estimation and elliptic boundary value problems
    Abstract: A basic equation in the random fields estimation theory is solved by reducing it to an elliptic boundary value problem in an external domain. The notions of random fields and estimation theory will be explained during the talk.
  • January 31, 2003
    Speaker:  Vladimir Kondratiev (Moscow State University)
    Title: Elliptic problems with non-linear boundary conditions on a non-compact part of the boundary
    Abstract: We will describe new results about the following problems, concerning solutions of second order linear elliptic equations with non-linear boundary conditions in unbounded domains: Phragmén - Lindelöf type theorems, existence or non-existence of positive solutions.
  • February 7, 2003
    Speaker:  Ilya Zakharevich )
    Title: Geometry of bi-Poisson structures
    Abstract: Nowadays, the theory of "compatible" pairs of Poisson brackets permeates seemingly unrelated domains of math, from the classical method of separation of variables, to quantum cohomology. We discuss the recent progress in the geometry of finite-dimensional Poisson pairs. While answers to some flavors of the classification problem are known (for example, bi-Poisson structures are related to the "most general" settings of the twistor transform), others lead to complicated questions in geometry, complex analysis, and the theory of PDE of the principal type.
  • February 14, 2003
    Speaker:  Tatyana Shaposhnikova (Linköping University - Sweden)
    Title: Pointwise interpolation inequalities for derivatives and their applications
    Abstract: I overview recent results, obtained together with V.Maz'ya, concerning interpolation inequalities for functional and fractional derivatives.
       A typical example is the Landau type inequality on the real line
                   |u'(x)|2    ≤  8/3 Mu(x) Mu''(x),
       where the constant 8/3 is best possible and M is the Hardy-Littlewood maximal operator.
       Similar inequalities are used in an elementary proof of a theorem by H.Brezis and P.Mironescu on the continuity of the composition operator in fractional Sobolev spaces.
       New limiting properties of fractional Sobolev spaces initiated recently by Bourgain, Brezis, and Mironescu will be discussed as well.

  • February 21, 2003
    Speaker:  Vladimir Mazya (Linköping University - Sweden)
    Title: Maximum principles for solutions of elliptic and parabolic systems
    Abstract: Maximum principles for solutions of elliptic and parabolic equations of the second order are classical and very important facts of the theory of partial differential equations. Recently Kresin and Maz'ya found a complete algebraic description of elliptic and parabolic systems satisfying the maximum principle. Unexpected phenomena occur when the boundary of a domain has edges and vertices.
       The talk is a survey of these and related results. No advanced knowledge of PDEs is required.

  • February 28, 2003
    Speaker:  Vladimir Mazya (Linköping University - Sweden)
    Title: Old and new spectral criteria for the Schroedinger operator
    Abstract:  The lecture is a survey of the conditions on the potential responsible for various spectral properties of the Schroedinger operator:positivity and strict positivity, semiboundedness, descreteness of the spectrum,form-boundedness, finiteness and descreteness of the negative spectrum, etc.
  • March 7, 2003
    Speaker:  Vladimir Mazya (Linköping University - Sweden)
    Title: Theory of Sobolev multipliers
    Abstract:  By a multiplier acting from one function space S1 into another, S2, one means a function which defines a bounded linear mapping of S1 into S2 by pointwise multiplication. In particular, multipliers in spaces of differentiable functions arise in various problems of analysis and the theory of differential and integral equations. For example, coefficients of differential operators can be naturally considered as multipliers. The same is true for symbols of pseudo-differential operators. Multipliers also appear in the theory of differential mappings preserving Sobolev spaces. Solutions of boundary value problems can be sought in classes of multipliers. Because of their algebraic properties, multipliers are suitable objects for generalization of basic facts of calculus (theorem on superposition, implicit function theorem etc.). Regardless of the substantiality and the usefulness of multipliers in Sobolev spaces, until recently they attracted relatively little attention. In the present talk I give a survey of principal known results in this area.
  • March 14, 2003
    Speaker:  Matvei Libine (University of Massachusetts at Amherst)
    Title: Equivariant Forms and Character Formulas
    Abstract: This talk is based on my article math.RT/0208024 available on I will talk about interplay between geometry and representation theory. Namely between the integral localization formula for equivariant forms and the Weyl and Kirillov's character formulas. I will explain the compact group case and then I will move on to recent developments for NONcompact groups.
  • April 4, 2003
    Speaker:  Igor Verbitsky (University of Missouri)
    Title: The form boundedness problem for the Schrödinger operator and its relativistic countepart  
    Abstract: We present necessary and sufficient conditions for the relative form boundedness and compactness of the Schrödinger operator  H = H0 + V,  where  H0 = -D   is the Laplacian on the Euclidean space, with an arbitrary real- or complex-valued distributional potential V. Analogous results for the relativistic Schrödinger operator where                         
                                H0 = (-D +m2)1/2 -m 
    will be discussed. This is joint work with Vladimir Maz'ya.

  • April 11, 2003
    Speaker:  Semyon Alesker (Tel Aviv University)
    Title: Non-commutative determinants and Monge-Ampere equations.
    Abstract: There are various constructions of non-commutative determinants (super, quantum, Gelfand-Retakh...). First we discuss old and new properties of the Diedonne and Moore determinants. Based on these constructions, we introduce a class of plurisubharmonic functions of quaternionic variables and quaternionic Monge-Ampere equations. They are analogous to the classical real and complex cases. Then we discuss the solvability of the Dirichlet problem for them. Some connections to geometry will be mentioned.
  • April 18, 2003
    Speaker:  Ari Belenkiy (Bar-Ilan University)
    Title: Projective geometry of quantum mechanics, Einstein-Podolsky-Rosen experiment and Bell's correlation
    Abstract: Applications of newly developed state vector stochastic reduction on Kahler manifolds (following Lane Hughston) are considered. Looking at the EPR experiment from the point of view of finite dimensional projective geometry might lend support one of the alternatives considered by John Bell.
  • April 25, 2003
    Speaker:  Katrin Leschke (Technische Universität Berlin and University of Massachusetts at Amherst)
    Title: Sequences of Willmore surfaces in the four-sphere
    Abstract: We construct sequences of Willmore surfaces in S4 by using a Baecklund transformation of Willmore surfaces.
       For Willmore tori with non-zero normal bundle degree the sequence has to be finite, and we obtain a classification result.
       (This is joint work with Franz Pedit)

  • May 9, 2003
    Speaker:  Nadja Kurt (University of Massachusetts at Amherst)
    Title: Discrete curves and the Toda Lattice
    Abstract: A novel interpretation of the one dimensional Toda lattice hierarchy (and reductions thereof) will be given in terms of flows on discrete curves. Among others the three Poisson structures of the Toda lattice (trihamiltonian structure) will be derived from a canonical structure on closed curves.
  • May 16, 2003
    Speaker:  Alex Suciu (Northeastern University)
    Title: Free abelian covers and systolic inequalities
    Abstract:  I will describe recent work with M. Katz and M. Kreck, on systolic inequalities satisfied by arbitrary Riemannian metrics on a compact, orientable, smooth manifold X. Applying Gromov's filling inequality to the typical fiber of the map from X to its Jacobi torus, we prove an interpolating inequality for two flavors of shortest length invariants of loops. The inequality is a lower bound for the total volume of the manifold. The procedure works, provided X is aspherical, and the lift of the typical fiber of the Jacobi map is non-trivial in the homology of the maximal free abelian cover of X. For nilmanifolds, our ``fiberwise'' inequality typically gives stronger information than the filling inequality for X itself. For 3-manifolds with first Betti number 2, a sufficient condition for our systolic inequality to hold is the non-vanishing of a certain Massey product.
  • May 23, 2003
    Speaker:  Maxim Braverman (Northeastern University)
    Title: Topological calculation of the phase of the zeta-regularized determinants
    Abstract: We show that for a large class of elliptic operators the phase of the zeta-regularized determinant is a topological invariant which can be explicitly calculated. We consider some examples where the phase is related to such classical topological invariants as the degree of the map and the Hopf invariant. Some of our examples were known to physicists. But not only the proofs but the very formulations of the results were not rigorous even by the standards usual for the physical literature.
       (Joint project with A. Abanov)

  • Fall 2003
  • September 5, 2003
    Speaker:  Boris Pavlov (V.A. Fock Institute of Physics, St.Petersburg, Russia and University of Auckland, New Zealand)
    Title: Modelling of quantum networks
    Abstract: A mathematical design of a quantum network is equivalent to the Inverse Scattering Problem for the Schrodinger equation on a composite domain consisting of quantum wells and finite or semi-infinite quantum wires attached to them. We suggest an alternative approach based on using of a solvable model to this difficult problem and to the relevant problems of choice and optimization of the construction and working parameters of the quantum network. We suggest a general principle of construction of quantitatively consistent solvable models for one-particle scattering processes in the network assuming that the transmission of an electron across the wells from one quantum wire to the other happens due to excitation of oscillatory modes in the well. This approach permits to obtain an explicit approximate formula for transmission coefficients based on numerical results on the discrete spectrum of the Schrodinger operator on the quantum wells.
  • September 12, 2003
    Speaker:  Theodore Voronov (University of Manchester (UMIST))
    Title: Inverse problem of calculus of variations and forms on field space
    Abstract: The inverse problem of calculus of variations (in its simplest form) is to find out whether given functions depending on fields and their derivatives are the variational derivatives of some functional. There is a classical Helmholtz--Volterra condition, which is necessary and locally sufficient. We give an alternative criterion in terms of the identical vanishing of the variation of a certain functional on an extended space where the number of independent variables is increased by one, and explain its relation with the Helmholtz--Volterra criterion using the de Rham complex on an infinite-dimensional space of fields.
  • September 12, 2003,  3:30 PM (Note the special time)
    Speaker:  Fabio Podesta (University of Florence, Italy)
    Title: Cohomogeneity One Kaehler manifolds and new examples of Kaehler-Einstein Cohomogeneity One Kaehler manifolds and new examples of Kaehler-Einstein metrics
    Abstract: I will discuss the geometry of compact Kaehler manifolds with vanishing first Betti number and which admit an isometric action of a compact Lie group with codimension one principal orbits. I will also show how to construct new invariant Kaehler-Einstein metrics on some cohomogeneity one compact Kaehler manifolds, when the principal orbits are Levi non-degenerate.
  • September 19, 2003
    Speaker:  Gudlaugur Thorbergsson (University of Cologne, Germany)
    Title: Isometric actions on symmetric spaces
    Abstract:  In the first part of the talk I will review the definitions and basic properties of variationally complete and polar actions on Riemannian manifolds. In the second part I will explain my joint work with Gorodski in which we prove that a variationally complete action on a compact symmetric space is hyperpolar. The converse was already proved by Conlon in 1971.
  • September 26, 2003
    Speaker:  Ernst Heintze (University of Augsburg, Germany)
    Title: Involutions of Kac-Moody algebras and infinite dimensional symmetric spaces
    Abstract: In finite dimensions, compact Lie groups with a biinvariant metric are important examples of Riemannian manifolds. They are in turn special examples of the so called symmetric spaces G/K where K is the fixed point set of an involution on G.
       The closest analogue of a compact Lie group in infinite dimensions is an affine Kac-Moody group and thus of a symmetric space, the quotient of an affine Kac-Moody group by the fixed point set of an involution.
       The purpose of this talks is to outline a new classification of these infinite dimensional symmetric spaces or equivalently of the involutions of affine Kac-Moody algebras. We show in particular that it can be reduced to well known problems in finite dimensions.

  • October 1, 2003,  2PM (Note the special date and time)
    Speaker:  Thomas Kappeler (University of Zurich)
    Title: Well-posedness of KdV in H-1(T)
    Abstract: In this talk I present recent results on the normal form for the KdV equation on the circle. They are used to show that KdV is well posed on the Sobolev spaces H -a(T)   for 0 1.
  • October 17, 2003
    Speaker:  Maxim Braverman (Northeastern University)
    Title: The L2-torsion without the determinant class condition
    Abstract:  We define the combinatorial and the analytic L2-torsions of a flat Hilbertian bundle as an element of the determinant line of its extended cohomology. In the case when the bundle is of determinant class, our definitions reduces to the construction of Carey, Farber, and Mathai. In the general case, we show that the ratio of the analytic and the combinatorial L2-torsions is equal to the relative torsion introduced by Carey, Mathai, and Mishchenko. In particular, applying the recent result of Burghelea, Friedlander, and Kappeler we obtain a Cheeger-Muller type theorem stating the equality between the analytic and the combinatorial L2-torsions.
       (Joint work with A. Carey, M. Farber, and V. Mathai)

  • October 24, 2003
    Speaker:  Hui Ma (University of Massachusetts at Amherst)
    Title: Hamiltonian stationary Lagrangian surfaces in CP2
    Abstract: A Lagrangian submanifold in a Kähler manifold is called Hamiltonian stationary if its area is critical with respect to all Hamiltonian deformation. We present a (new) equivalent condition of Hamiltonian stationary Lagrangian surfaces in CP2 and show that any nonsuperminimal Hamiltonian stationary Lagrangian torus in CP2 can be constructed from a pair of commuting Hamiltonian ODE on a finite dimensional subspace of a certain loop Lie algebra.
       (joint work with Weihuan Chen and Franz Pedit)

  • October 31, 2003
    Speaker:  Megan M. Kerr (Wellesley College)
    Title: Low-Dimensional Homogeneous Einstein Manifolds
    Abstract:  I will describe joint work with Christoph Böhm, investigating the Einstein equation for G-invariant metrics on compact homogeneous spaces. We prove that every compact, simply connected homogeneous space of dimension less or equal than 11 admits a homogeneous Einstein metric. The result is sharp: Wang and Ziller showed that in dimension 12 the compact, simply connected homogeneous space SU(4)/SU(2) does not admit any homogeneous Einstein metrics. (Here SU(2) < Sp(2) < SU(4) and SU(2) is maximal in Sp(2).) Classification results up to dimension seven have been published, but the case-by-case classification gets significantly more difficult as the dimension increases, since the number of spaces to be considered rapidly increases with the dimension. I will also describe an infinite family of 12-dimensional simply connected homogeneous torus bundles which do not admit G-invariant Einstein metrics. These are the first non-existence examples where the isotropy representation has four summands.
  • November 7, 2003
    Speaker:  Mikhail Shubin (Northeastern University)
    Title: Semiclassical asymptotics and vanishing of quantum Hall conductivity
    Abstract: I will explain a new method of obtaining semiclassical asymptotics for magnetic Schroedinger operators with invariant Morse type potentials on covering spaces of compact manifolds. It provides a new existence proof for spectral gaps and also gives an information about the spectral projections, implying vanishing of classes of these projections in $K$-theory for small coupling constant. An important corollary is vanishing of the corresponding higher traces in cyclic cohomology, which in turn implies vanishing of the quantum Hall conductivity for weak magnetic fields.
       This is a joint work with Yu.Kordyukov and V.Mathai

  • December 5, 2003
    Speaker:  Martin Magid (Wellesley College)
    Title: Time-like isothermic surfaces associated to Grassmannian Systems
    Abstract:  C.-L. Terng defined the U/K system for a symmetric space based on a semi-simple U in 1997. This is a non-linear first order system of partial differential equations defined using U/K. This system gives rise to a one-parameter family of flat connections called the Lax connection of the U/K system. I will show that time-like isothermic surfaces in pseudo-riemannian space Rn-j,j are associated to the Grassmannian O(n-j+1,j+1)/O(n-j,j) x O(1,1)-system.
  • Spring 2004
  • January 16, 2004
    Speaker:  Ionel Popescu (MIT)
    Title: A Probabilistic Approach to Morse Inequalitites
    Abstract: Starting with the heat kernel of the Witten Laplacian in terms of Feynman-Kac like integral, combined with a simple Markov property and estimates on the solution to a initial-boundary problem on a ball in Euclidean space we prove Morse inequalities. Based on this argument we will discuss also the case of a Bott-Morse function where the idea is to compare the associated Laplacians with respect to Bismut connection and Levi-Civita connection around the critical submanifolds.
  • February 6, 2004
    Speaker:  Victor Roitburd (Rensselaer Polytechnic Institute)
    Title: Asymptotic dynamics of non-equilibrium free-boundary problems
    Abstract: Free-boundary models provide a convenient and rather accurate description for many phase transition type phenomena, such as freezing/melting or burning. I'll give a brief and elementary introduction to free-boundary problems for the heat equation, and explain how some types of them relate to reaction-diffusion systems. The talk will be mostly concerned with asymptotic dynamics of solutions of a free-boundary problem arising from the so-called solid-state combustion. Numerical experiments reveal a huge variety of dynamical scenarios (some animations will be shown). Nonetheless, it turns out that the possible asymptotic regimes (a global attractor) occupy a compact set in the space of all the regimes, and that its Hausdorff dimension is finite. In some sense the PDE system behaves like a fancy nonlinear oscillator. The proofs are based on classical potential theory estimates. An elementary description of the Hausdorff dimension and of its computation will be given. Results of the talk are obtained in a joint work with Michael Frankel of Indiana University-Purdue University Indianapolis
  • February 20, 2004
    Speaker:  Dimitri Yafaev (Université de Rennes 1)
    Title: Scattering by magnetic fields
    Abstract:  For the magnetic Schrödinger operator we will discuss the definition and spectral properties of the scattering matrix. In particular, the essential spectrum of the scattering matrix can be found in terms of the decay of the magnetic field at infinity. Under appropriate conditions we can also describe singularities of the scattering amplitude (the Schwartz kernel of the scattering matrix).
       An important point of our approach is that we consider the scattering matrix as a pseudo-differential operator on the unit sphere and find an explicit expression of its principal symbol in terms of the vector potential. Another ingredient is an extensive use of a special gauge adapted to a given magnetic field.

  • March 19, 2004
    Speaker:  Dan Mangoubi (Technion - Israel Institute of Technology)
    Title: Symplectic Aspects of the First Eigenvalue of the Laplacian
    Abstract: Let (M, w) be a compact symplectic manifold of  dim > 2. We are interested to know whether we can find a Riemannian metric on M compatible with w and with arbitrary large first positive eigenvalue. L. Polterovich proved that it is the case under some technical condition on M, which is fullfiled for manifolds of the form MxT4, where T4 is the torus.
       We will discuss ideas which hopefully will let us prove it for any compact (M, w)  of dim>2.

  • Fall 2004
  • October 15, 2004
    Speaker:  Maxim Braverman (Northeastern University)
    Title: The phase of the determinant of a Dirac type operator and the degree of a map.
    Abstract: I will present an an example of a Dirac type operator depending on a map V from a manifold to a sphere, the phase of whose determinant can be calculated in terms of the topological degree of V. An important part of our calculation is the study of the imaginary part of the spectrum of this operator. In this study a new version of the Witten deformation technique is used.
       A special case of our result was suggested by physicists. But not only the proofs but the very formulations of the results were not rigorous even by the standards usual for the physical literature.
       (Joint project with A. Abanov)

  • October 29, 2004
    Speaker:  Mikhail Shubin (Northeastern University)
    Title: The Miura transform
    Abstract: The Miura transform is a non-linear map r ---> r'+r2 on functions of one real variable. The importance of this transform lies in its relations with some non-linear partial differential equations, e.g. the famous Korteweg - de Vries equation. I will describe properties of this transform, in particular, recently found description of the image of this transform in some spaces of functions and distributions on R. This description is related with the spectra of the Schr\"odinger operators on R.
       The talk will be based on joint results of T.Kappeler, P.Perry, P.Topalov and the speaker.

  • November 18, 2004,  12:00 (Note the special date and time)
    Speaker:  Alexander Turbiner (Institute for Nuclear Sciences - National University of Mexico)
    Title: Perturbations of integrable systems and Dyson-Mehta integrals
    Abstract: Olshanetsky-Perelomov quantum Hamiltonians are unique both completely integrable and exactly-solvable multidimensional Hamiltonians related to root systems. We will show that they admit algebraic forms being represented as linear differential operators with polynomial coefficients, which allows a Lie-algebraic interpretation of these Hamiltonians. The existence of algebraic form also allows to present a quite general class of perturbations for which one can develop a constructive, `algebraic perturbation theory', where all corrections are found by pure algebraic means. These perturbations can be classified in terms of representation theory. Physically relevant many-body anharmonic oscillators turned out to be among these perturbed problems. Corrections to eigenvalues are given by ratios of generalized Dyson-Mehta integrals, hence they can be found by algebraic means. They are interesting by themselves.
  • December 3, 2004
    Speaker:  Gideon Maschler (University of Toronto)
    Title: Conformally-Einstein Kähler metrics
    Abstract: We describe the classification of Kähler metrics which are conformal to Einstein metrics on manifolds of complex dimension m>2. Included is the case where the Einstein metric is defined only away from the non-empty zero set of the conformal factor, giving rise to examples of asymptotically hyperbolic Einstein metrics. Locally, these metrics are given by a 3-parameter family on the total space of a line bundle over a Kähler-Einstein base. In the global classification, the metrics are extended to an associated projectivized bundle. The allowed Chern numbers for these bundles are parameterized via a discrete subset lying on a family of plane plane algebraic curves. This work is joint with A. Derdzinski.
  • Spring 2005
  • January 21, 2005
    Speaker:  Gabriel Katz (Bennington College)
    Title: Morse theory on manifolds with boundary and convexity
    Abstract: Classical Morse Theory links singularities of Morse functions with the topology of a closed manifold. The singularities cause an interruption of the gradient flow; and the homology or even the topological type of the manifold can be expressed in terms of such interruptions (i.e. in terms of descending manifolds, attaching maps, spaces of the flow trajectories which connect the singularities).
       On manifolds with boundary an additional source of the flow interruption occurs: it comes from a particular geometry of the boundary, or rather from the failure of the boundary to be convex with respect to the flow. In fact, one can trade the singularities in the interior of the manifold for these new non convexity effects. In our approach, these boundary effects take a central stage, while the singularities remain in the background.
       We will discuss some applications of this philosophy to 3-dimensional manifolds. In particular, we will reformulate the Poincare' Conjecture in terms of the new convexity.

  • February 3, 2005,  1:30PM (Note the special date and time)
    Speaker:  Mikhail Agranovich (Moscow Institute of Electronics and Mathematics, Russia)
    Title: Spectral problems for second order strongly elliptic systems with spectral parameter in boundary or transmission conditions
    Abstract: We consider spectral problems in Rn, n і 3, for a second order strongly elliptic system satisfying some additional conditions. The spectral parameter is contained in the boundary or transmission conditions on a Lipschitz surface S. It is either closed or bounded and non-closed. The aim is to describe spectral properties of the corresponding operators of the Neumann-to-Dirichlet type on S in the simplest Sobolev spaces. In the second case, the Dirichlet and Neumann problems with boundary data on S are also considered.
  • February 11, 2005
    Speaker:  Joseph Coffey (New York University)
    Title: Failure of parametric H-principle for maps with prescribed Jacobian
    Abstract: Let M and N be closed n-dimensional manifolds, and equip N with a volume form σ. Let μ be an exact n-form on M. Arnold then asked the question: When can one find a map  f:M→N  such that  f*σ=μ. In 1973 Eliashberg and Gromov showed that this problem is, in a deep sense, trivial: It satisfies an h-principle, and whenever one can find a bundle map fbdl:T M→T N which is degree 0 on the base and such that  fbdl*(σ)=μ  one can homotop this map to a solution f. That is if the naive topological conditions are satisfied on can find a solution. There is no further interesting geometry in the problem.
    We show the corresponding parametric h-principle fails- if one considers families of maps inducing μ from σ, one can find interesting topology in the space of solutions which is not predicted by an h-principle. Moreover the homotopy type of such maps is "quantized": for certain families of forms homotopy type remains constant, jumping only at discrete values.

  • February 25, 2005
    Speaker:  Michael E. Taylor (University of North Carolina at Chapel Hill)
    Title: Scattering length and spectral theory of Schrodinger operators
    Abstract: The theory of capacities has played an important role in potential theory for a long time. About 30 years ago, M. Kac began to explore the use of the notion of scattering length of a positive potential, as an analogue of capacity. We will discuss some basic properties of this scattering length, and apply it to study the spectrum of Schrodinger operators with positive potentials. We will obtain variants of some results of Molchanov and Maz'ya and Shubin.
  • March 18, 2005
    Speaker:  Jonathan Weitsman (University of California Santa Cruz)
    Title: Measures on Banach manifolds and supersymmetric quantum field theory
    Abstract: We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. As examples of our construction we produce measures corresponding to spaces of maps from a Riemann surface to a semisimple Lie group (the Wess-Zumino-Novikov-Witten model) and to gauge theory in three dimensions. We show that these measures are positive, and that the Wess-Zumino-Novikov-Witten measure where the Riemann surface is P1 has finite mass. As an application we show that formulas arising from expectations in this measure reproduce the results of Frenkel and Zhu from vertex operator algebras.
  • March 25, 2005
    Speaker:  Raphaėl Ponge (Ohio State University)
    Title: Spectral asymmetry, zeta functions and the noncommutative residue
    Abstract:  Motivated by an approach developed by Wodzicki, we look at the spectral asymmetry of (possibly nonselfadjoint) elliptic PsiDO's in terms of theirs zeta functions. Using formulas of Wodzicki we look at the spectral asymmetry of elliptic PsiDO's which are odd in the sense of Kontsevich-Vishik. Our main result implies that the eta function of a selfadjoint elliptic odd PsiDO is regular at every integer point when the dimension and the order have opposite parities (this generalizes a well known result of Branson-Gilkey for Dirac operators) and allows us to relate the spectral asymmetry of a Dirac operator on a Clifford bundle to the Riemannian geometric data. As a consequence, we can express the Einstein-Hilbert action of a Riemanian metric in terms of the difference of two zeta functions of a Dirac operator, hence obtain a new spectral interpretation of this action.
  • April 8, 2005
    Speaker:  Oleg Gleizer (UCLA)
    Title: TBA
    Abstract: TBA
  • Spring 2006
    • September 16, 2005
      Speaker:  Maxim Braverman (Northeastern University)
      Title: Refined Analytic Torsion
      Abstract: For an acyclic representation of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to a unitary representation, we define a refinement of the Ray-Singer torsion associated to this representation. This new invariant can be viewed as an analytic counterpart of the refined combinatorial torsion introduced by Turaev.
         The refined analytic torsion is a holomorphic function of the representation of the fundamental group. When the representation is unitary, the absolute value of the refined analytic torsion is equal to the Ray-Singer torsion, while its phase is determined by the eta-invariant. The fact that the Ray-Singer torsion and the eta-invariant can be combined into one holomorphic function allows to use methods of complex analysis to study both invariants. I will present several applications of this method.
         (Joint work with Thomas Kappeler)

    • January 13, 2006
      Speaker:  Alexander Kushkuley (Clearing Corporation)
      Title: On positive semidefinite approximation of matrces with prescribed block-diagonal structure
      Abstract: Let  sym(Λ)  be an affine plane of symmetric n x n  matrices with fixed positive definite block diagonal submatrix  Λ  and let  P(Λ) Ģ  sym(Λ)  be the set of all positive semidefinite matrices in   sym(Λ). Consider the following optimization problem: given  A Ī sym(Λ)   find the closest matrix to  A  in  P(Λ) .  The problem belongs to a class of "positive semidefinite completion" problems that are usually solved by methods of convex programming. A related rank reduction problem is to find a positive semidefinite approximation to matrix   A Ī sym(Λ)   of rank less or equal than some  0 < k < n . In this paper both problems are studied in a rather straightforward manner, as problems of finding critical points of Euclidian distance function. Besides presenting some algorithms, we observe, that
        (a)    the number of critical points of distance function on P(Λ) is allways finite;
        (b)   P(Λ)  is stratified by connected open manifolds Pk(Λ) of matrices of rank exactly  k;
        (c)   tangent plane, normal subspace and shape operator for a given point  S Ī Pk(Λ)  Ģ sym(Λ)  can be characterized algebraically in terms of  S;
        (d)   if  Λ  is diagonal then total Betti number of the stratified space  Пi=1k Pi(Λ)    is equal to  2n-k.

    • January 20, 2006
      Speaker:  Peter Topalov (Northeastern University)
      Title: Solutions of mKdV in classes of functions unbounded at infinity
      Abstract: Investigation of relation between the Korteweg - de Vries and modified Korteweg - de Vries equations (KdV and mKdV) leads to a new algebro-analytic mechanism which is similar to the Lax L-A pair but includes a first order operator Q instead of the 3rd order operator A. This allows an explicit control of eigenfunctions of the Schr\"odinger operator L when its time-dependent potential satisfies KdV. In particular, we establish global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity.
         (joint work with T. Kappeler, P. Perry, and M. Shubin)

    • March 24, 2006
      Speaker:  Mihai Stoiciu (Williams College)
      Title: The Distribution of the Eigenvalues of Random CMV Matrices
      Abstract: Recent developments in the theory of orthogonal polynomials on the unit circle have emphasized the importance of CMV matrices; they are the unitary analog of Jacobi matrices. We prove that the asymptotic local statistical distribution of various classes of random CMV matrices is Poisson. This means that, as in the case of random Schrodinger operators, there is no local correlation between the eigenvalues.
    • March 31, 2006
      Speaker:  Mikhail Shubin (Northeastern University)
      Title: Crystal lattice vibrations and specific heat at low temperatures
      Abstract: Behavior of the specific heat of a solid at low temperature is a classical subject in solid state physics which dates back to a pioneering work by Einstein (1907) and its refinement by Debye (1912). Using a special quantization of crystal lattices and calculating the asymptotic of the integrated density of states at the bottom of the spectrum, we obtain a rigorous derivation of the classical Debye  T3  law.
         The talk is based on joint work by the speaker and T.Sunada.

    Fall 2006
    • September 8, 2006
      Speaker:  Boris Botvinnik (University of Oregon)
      Title: Moduli spaces of metrics/(conformal classes) and analytical torsion.
      Abstract: We study the homotopy type of the moduli spaces M+(M) of metrics with positive scalar curvature on a smooth compact manifold M. We use analytical torsion to describe rational homotopy type of the moduli space M+(M) for the sphere Sn with n≥ 5.
    • September 22, 2006
      Speaker:  Victor Ivrii (University of Toronto)
      Title: Magnetic Schrödinger Operator: Geometry, Classical and Quantum Dynamics and Spectral Asymptotics
      Abstract: I consider even-dimensional Scrödinger operator with the small Planck parameter and a large coupling parameter &mu, and discuss connections between the geometry of magnetic field, classical and quantum dynamics of the corresponding movements and the remainder estimate in the spectral asymptotics.

    • September 29, 2006
      Speaker:  Steven Rosenberg (Boston University)
      Title: Chern-Simons classes on loop spaces
      Abstract:  The loop space of a Riemannian manifold has a family of canonical Riemannian metrics indexed by a Sobolev space parameter. The curvature of the Levi-Civita connection takes values in pseudodifferential operators. Using the Wodzicki residue, we can define Pontrjagin forms, but these all vanish. The corresponding secondary Chern-Simons classes are nontrivial in general. (Joint work with F. Torres-Ardilo.)
    • October 13, 2006
      Speaker:  Peter Topalov (Northeastern University)
      Title: Liouville billiard tables
      Abstract: We will discuss the dynamical and the spectral properties of a special class of billiard tables with completely integrable billiard ball map. Using a simple idea that goes back to Beltrami we will construct such billiard tables on surfaces of constant curvature.
    • October 27, 2006
      Speaker:  Jonathan Weitsman (University of California Santa Cruz)
      Title: Equivariant Morse theory, old and new: Hamiltonian loop group spaces and hyperkahlerian group actions.
      Abstract: In 1980 Raoul Bott gave his Poincare Symposium Lectures on Morse Theory, Old and New, at the end of which he previewed his work with M. Atiyah on Yang Mills theory in two dimensions. These ideas have since given rise to an explosion in the understanding of equivariant Morse theory. In the talk I will discuss more recent developments involving the study of loop group actions and the extension of the ideas of Atiyah and Bott to the case of hyperkahler manifolds. This is joint work with Bott and Tolman and with Daskalopoulos and Wilkin.
    • November 3, 2006
      Speaker:  Maxim Braverman (Northeastern University)
      Title: Symmetrized trace and symmetrized determinant of elliptic operators
      Abstract: Determinant and trace of (pseudo-)differential operators on a closed manifold M are defined using certain renormalization procedure. As usual, such a renormalization leads to anomalies. Namely,
         1. the defined objects depend on the choices made during the renormalization;  
         2. det(AB)≠ det(A) det(B) and Tr(AB)≠ Tr(BA)
         The above anomalies is not a bug of the procedure, but are coursed by the nature of the problem. It is known that there is no trace on the algebra of all pseudo-differential operators, which extend the usual trace on trace-class operators and satisfy the trace condition Tr(AB)= Tr(BA).
         In the talk I will consider the algebra of odd class logarithmic pseudo-differential operators on a manifold of odd dimension. This algebra contains, in particular, all differential operators and their logarithms. For operators in this algebra I suggest a new, more symmetric, renormalization of the trace. The obtained trace "almost" don't have anomalies. In particular, Tr(AB)= Tr(BA). When restricted to the algebra of odd classical pseudo-differential operators this trace coincides with the canonical trace of Kontsevich and Vishik. Using the new trace I construct a new determinant of odd classical elliptic pseudo-differential operators. This determinant is multiplicative, i.e. satisfies det(AB)= det(A) det(B), whenever the Kontsevich-Vishik multiplicative anomaly formula for usual determinants holds. In particular, it is multiplicative for operators whose leading symbols commute. When restricted to operators of Dirac type the new determinant provides a sign refined version of the determinant constructed by Kontsevich and Vishik.

    • November 17, 2006
      Speaker:  Dan Mangoubi (University of Montreal)
      Title: On the Inner Radius of Nodal Domains
      Abstract: Let M be a closed Riemannian manifold of dimension d. We consider the inner radius R of a nodal domain for a large eigenvalue λ of the Laplacian. We prove that  A/λd < R < B/√λ.   For d=2 we prove a sharp bound: A/√λ < R. Our proof is based on estimation of the volume of positivity of a harmonic function and a Poincare type inequality by Maz'ya.
    • December 4, 2006,  3:30 (Note the special date and time)
      Speaker:  Alexander Turbiner (Institute for Nuclear Sciences - National University of Mexico)
      Title: Anharmonic oscillator and double-well potential: how to approximate eigenfunctions
      Abstract: A simple uniform approximation of the logarithmic derivative of the lowest eigenfunction for both the quantum-mechanical quartic anharmonic
      oscillator and the double-well potential given by V= m2 x2+g x4 at arbitrary  g ≥ 0 for m2>0 and m2<0, respectively, is presented. It is shown that if this approximation is taken as unperturbed problem it leads to an extremely fast convergent perturbation theory. The case of sextic oscillator is briefly mentioned. A connection to WKB approximation is discussed.

    • December 8, 2006
      Speaker:  Pierre Schapira (Université Pierre et Marie Curie)
      Title: Index theorem for elliptic pairs
      Abstract:  An elliptic pair on a complex manifold X is the data of a coherent DX -module M and an R-constructible sheaf F such that the intersection ofthe characterictic variety of M and the microsupport of F is contained in the zero-section of T*X. If this intersection is compact, then thecohomology of the complex of solutions RHomD( MÄ F,OX ) is finite dimensional over C and its index χ(X; M,F) is given by the formula (Schapira-Schneiders):
                              χ(X; M,F)= T*X  µeu(M ) Č  µeu(F).
      Here µeu(M ) is the microlocal Euler class of M and µeu(F) is Kashiwaras microlocal Euler class of F. In this talk, we shall explain the meaning of this formula and its links with classical Riemann-Roch and Atiyah-Singer theorems. 

    • December 11, 2006,  3PM (Note the special date and time)
      Speaker:  Frédéric Klopp (Université Paris 13)
      Title: Renormalization of certain exponential sums
      Abstract: The talk is devoted to a simple renormalization formula for Gaussian exponential sums. We apply it to study the behavior of these sums ; we obtain new results on the curlicues seen on graphs of such sums and well as recover some known results on their growth. (joint work with A. Fedotov)
    Spring 2007
    Fall 2007
    • September 21, 2007
      Speaker:  Mark Kelbert (University of Swansea, UK)
      Title: Large-time behaviour of a branching diffusion on a hyperbolic space
      Abstract: This is a joint work with Y Suhov (University of Cambridge, UK). We consider a general hyperbolic branching diffusion on a Lobachevsky space Hd. The question is to evaluate the Hausdorff dimension of the limiting set on the absolute. In the case of a homogeneous branching diffusion, an elegant formula for the Hausdorff dimension was obtained by Lalley and Sellke (1997) for d=2 and by Karpelevich, Pechersky and Suhov (1998) for a general d. Later on, Kelbert and Suhov (2006) extended the formula to the case where the branching diffusion was in a sense asymptotically homogeneous (i.e. its main relevant parameter, the fission potential, approached a constant limiting value near the absolute). In this talk I show that the Hausdorff dimension of the limiting set is determined by maximum points of the fission potential. The method is based on properties of the minimal solution to a Sturm--Liouville equation with general potential, and elements of the harmonic anlysis on Hd. We also relate the Hausdorff dimension with properties of recurrence and transience of a branching diffusion, as was defined by Grigoryan and Kelbert (2003) on a general-type manifold.
    • September 28, 2007
      Speaker:  Vadim Tkachenko (Ben-Gurion University, Israel)
      Title: A Criterion for Hill Operators to be Spectral Operators of Scalar Type
      Abstract: We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schroedinger operator) to be a spectral operator of scalar type in the sense of Danford. The conditions show the remarkable fact that this property is independent of smoothness (or even analyticity) of the potential. In the course of our analysis we also establish a functional model for Hill operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion.
         The problem of deciding which Hill operators are spectral operators of scalar type appears to have been open for about 40 years.
         This is a joint work with F.Gesztezy.

    • October 5, 2007
      Speaker:  Peter Topalov (Northeastern University)
      Title: Analyticity of Riemannian exponential maps on Diff(T)
      Abstract:  We will discuss the exponential maps induced by Sobolev type right-invariant (weak) Riemannian metrics of order k≥ 1 on the Lie group D = Diff(T) of smooth, orientation preserving diffeomorphism of the circle. In particular, it will be shown that each of these Riemannian exponential maps defines a Fréchet-analytic chart of the identity. The Lie group D and its algebra come up in hydrodynamics, playing the role of a configuration space for the Burgers and Camassa-Holm equation. The latter equation is a model for one -dimensional wave propagation in shallow water.
    • October 12, 2007
      Speaker:  Graeme Smith (University of Bristol and Boston University)
      Title: The quantum channel capacity and the (super)additivity of coherent information
      Abstract:  I will describe what is currently known about the capacity of a quantum channel for high fidelity quantum communication. In contrast to the simple formula for the capacity of a classical channel, which was found by Shannon in 1948, there is no known characterization of the quantum capacity as a finite optimization problem. There are a handful of channels, known as degradable, for which the capacity can be found. I will discuss the structure of these channels, as well as some channels that are less well behaved.
         No previous knowledge of quantum information will be assumed.

    • October 19, 2007
      Speaker:  Jun Masamune (Worcester Polytechnic Institute)
      Title: Self adjointness, Liouville property, and stochastic completeness of a non-compact weighted manifold
      Abstract: A weighted manifold M is a manifold furnished with a Riemann tensor and a measure which has a smooth density against the Riemann measure. It carries a second-order elliptic operator called the weighted Laplacian. A weighted manifold M is said to be stochastic complete if the Brownian motion associated to the weighted Laplacian can be found in M for any positive time. In this talk we will discuss a Liouville type property which implies the stochastic completeness and observe that the stochastic completeness implies the essential self adjointness of the weighted Laplacian of a non-compact weighted manifold. We will also observe that if the Cauchy boundary C M : = M-M of M, where M  is the completion of M, is almost polar, then the weighted Laplacian is essential self-adjoint. The main results of the talk are obtained in the joint work with A. Grigorfyan.
    • October 26, 2007
      Speaker:  Ilya Zakharevich 
      Title: On spectral theory of operator pencils A + t B: V --> W
      Abstract: A spectral theory of an operator in a finite-dimensional vector space is completely determined by its Jordan decomposition. A pencil A + tB is a generalization of an operator; it may encode more complicated data of linear algebra, such as partially defined operators, a 1-to-many operators, etc. In finite-dimensional theory one must, in addition to Jordan blocks, consider Kronecker blocks; they have only discrete parameters. In infinite-dimensional case, such blocks acquire parameters, which carry a semantic of "fuzzy eigenvalues".
    • November 2, 2007
      Speaker:  Marina Ville (Northeastern University)
      Title: Branch points, braids and minimal surfaces.
      Abstract:  Branch points are a type of singularity of immersed surfaces in 4-manifolds; although highly non-generic they are quite interesting because
          1) they generalize the branch points of complex algebraic curves
          2) they occur as singularities of minimal surfaces.
         As in the complex case, a branch point p of a surface S can be studied through its link L(p), where S intersects a small sphere centred at p. This link comes naturally as a braid and we will discuss the connection between this braid and the differential topology/geometry of the branched surface S.
         If S is just a disk, L(p) is a knot. One question is still very much open: what knot types correspond to minimal disks? We will report some recent progress on this (joint work with Marc Soret), where crucial help came from a computer program.

    • November 9, 2007
      Speaker:  Nilufer Koldan (Northeastern University)
      Title: Semiclassical Asymptotics of Witten's Laplacian on Manifolds with Boundary
      Abstract: In 1982 E. Witten introduced a deformation of the de Rham complex of differential forms on a compact closed manifold M using a Morse function f and a small parameter h. Witten's Laplacian can be defined in the same way as the usual Laplacian but by using Witten's deformed differential instead of the standard de Rham differential. In the semiclassical asymptotics of the eigenvalues of Witten's Laplacian, only small neighborhoods of the critical points of f play a role.
         On a manifold with boundary, Witten's Laplacian can be defined in the same way, but we need to specify its domain. I will define a specific domain and will show that for this particular operator, all the interior and some of the boundary critical points play a role. I will write a model operator by considering the operator only around these points and this will lead us to the semiclassical asymptotics of Witten's Laplacian.
         This is an improvement of the results of Helffer-Nier (2005). We use a new variational method based on a paper by Kordyukov, Mathai, Shubin (2005) but with more extended use of quadratic forms instead of the operators.

    • November 16, 2007
      Speaker:  Alexander Turbiner (Institute for Nuclear Sciences - National University of Mexico)
      Title: Can semiclassical approximation be modified to study the ground state?
      Abstract:  Semiclassical (WKB) approximation is one of main methods to study the Schroedinger equation. Usually, their applicability is limited to highly-excited states. It is proposed a simple approach based on a combination of WKB approximation at large distances with perturbation theory at small distances. It allows to construct uniform approximation of the ground state eigenfunction for the anharmonic oscillator (AHO) V= m2 x2+ x4 with single well (m2 ≥ 0) and for the double-well potential (m2 < 0). It is shown that if this approximation is treated as unperturbed problem it leads to an extremely fast convergent perturbation theory. A possible connection to recent remarkable results by Eremenko-Gabrielov-Shapiro about complex zeroes of AHO eigenfunctions is mentioned.
         A generalization to different one-dimensional and multidimensional AHO as well as to the problem of hydrogen in a magnetic field is discussed.

    • November 30, 2007
      Speaker:  Alex Suciu (Northeastern University)
      Title: Which Kaehler groups are 3-manifold groups?
      Abstract: Every finitely presented group G can be realized as the fundamental group of a (smooth, compact, connected, orientable) 4-dimensional manifold. Requiring that G be the fundamental group of a Kaehler manifold, or that of a 3-manifold, is very restrictive. A natural question (raised by Donaldson, Goldman, and Reznikov in the 1990s) is then: What if both conditions are required to hold? I will address this question in my talk.
         This is joint work with Alex Dimca, see

    Spring 2008
    • January 18, 2008
      Speaker:  Jayant Shah (Northeastern University)
      Title: Riemannian Geometry of infinite dimensional spaces of planar shapes
      Abstract: One of the problems in Computer Vision is how to quantify similarity and variation in object shapes within a category and between categories. An approach to this problem is to define a Riemannian metric on a manifold of shapes and understand its geometry in terms of its geodesics and curvature. In 2003-04, Peter Michor and David Mumford formulated a general framework to carry out such an analysis. I will describe this framework and some recent results in the case of a relatively simple shape space of closed curves in the plane.
    • February 1, 2008
      Speaker:  Robert McOwen (Northeastern University)
      Title: The Fundamental Solution of an Elliptic Problem in Nondivergent Form
      Abstract: TBA
    • February 8, 2008
      Speaker:  Maxim Braverman (Northeastern University)
      Title: A Canonical Quadratic Form on the Determinant Line of a Flat Vector Bundle
      Abstract: We introduce and study a canonical quadratic form, called the torsion quadratic form, on the determinant line of the cohomology of a flat vector bundle over a closed oriented odd-dimensional manifold. This quadratic form caries less information than the refined analytic torsion, introduced in our previous work, but is easier to construct and closer related to the combinatorial Farber-Turaev torsion. In fact, the torsion quadratic form can be viewed as an analytic analogue of the Poincare-Reidemeister scalar product, introduced by Farber and Turaev. Moreover, it is also closely related to the complex analytic torsion defined by Cappell and Miller and we establish the precise relationship between the two. In addition, we show that up to an explicit factor, which depends on the Euler structure, and a sign the Burghelea-Haller complex analytic torsion, whenever it is defined, is equal to our quadratic form. We conjecture a formula for the value of the torsion quadratic form at the Farber-Turaev torsion and prove some weak version of this conjecture. As an application we establish a relationship between the Cappell-Miller and the combinatorial torsions.
         (joint work with T. Kappeler)

    • February 15, 2008
      Speaker:  Maxim Braverman (Northeastern University)
      Title: A Canonical Quadratic Form on the Determinant Line of a Flat Vector Bundle
      Abstract: We introduce and study a canonical quadratic form, called the torsion quadratic form, on the determinant line of the cohomology of a flat vector bundle over a closed oriented odd-dimensional manifold. This quadratic form caries less information than the refined analytic torsion, introduced in our previous work, but is easier to construct and closer related to the combinatorial Farber-Turaev torsion. In fact, the torsion quadratic form can be viewed as an analytic analogue of the Poincare-Reidemeister scalar product, introduced by Farber and Turaev. Moreover, it is also closely related to the complex analytic torsion defined by Cappell and Miller and we establish the precise relationship between the two. In addition, we show that up to an explicit factor, which depends on the Euler structure, and a sign the Burghelea-Haller complex analytic torsion, whenever it is defined, is equal to our quadratic form. We conjecture a formula for the value of the torsion quadratic form at the Farber-Turaev torsion and prove some weak version of this conjecture. As an application we establish a relationship between the Cappell-Miller and the combinatorial torsions.
         (joint work with T. Kappeler)

    • February 22, 2008
      Speaker:  Yaron Ostrover (MIT)
      Title: Brunn-Minkowski-type inequality in symplectic geometry
      Abstract: The Brunn-Minkowski inequality for volumes of bodies is a fundamental inequality in geometry and has numerous applications. In the symplectic world, the analogue of volume is given by the notion of a symplectic capacities. In this talk we focus on a specific example which arose from Hamiltonian dynamics, namely the Hofer-Ekeland-Zehnder capacity of convex domains. We discuss the Brunn-Minkowski-type inequality for this capacity, explain its meaning in the context of symplectic geometry, and mention some of its applications.
         (This is a joint work with Shiri Artstein-Avidan).
    Fall 2008
    • September 19, 2008
      Speaker:  Siye Wu (University of Colorado)
      Title: Analytic torsion for twisted de Rham complexes
      Abstract: We define analytic torsion for twisted de Rham complexes and show that it is metric independent. We also establish several functorial properties of the torsion and compute it in some examples.
    • September 26, 2008
      Speaker:  John Gonzalez (Northeastern University)
      Title: Unbounded Solutions of the Modified Korteweg-De Vries Equation
      Abstract: We prove local existence and uniqueness of solutions of the focusing modified Korteweg - de Vries equation ut + u2ux + uxxx = 0 in classes of unbounded functions that admit an asymptotic expansion at infinity in decreasing powers of x. We show that an asymptotic solution differs from a genuine solution by a smooth function that is of Schwartz class with respect to x and that solves a generalized version of the focusing mKdV equation. The latter equation is solved by discretization methods.
    • October 17, 2008
      Speaker:  Peter Topalov (Northeastern University)
      Title: Birkhoff coordinates and spectral asymptotics
      Abstract: Using Birkhoff coordinates for the Korteweg - de Vries equation and a simple deformation argument we characterize the regularity of a distributional potential q in the Sobolev space H-1 on the circle in terms of the decay of the gap lengths (gak)k ≥ 1 of the periodic spectrum of Hill's operator d2/dx2+q on the interval [0,2]. The same method could also be used for the proof of analogous results for more general spaces. (This report is based on a joint work with F. Serier and T. Kappeler.)
    • November 14, 2008
      Speaker:  Pierre Albin (MIT)
      Title: The signature operator on a Witt space
      Abstract: A natural class of spaces to which the techniques of microlocal analysis extend is that of stratified manifolds such as Witt spaces. I will discuss joint work with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza on applying pseudo-differential techniques to study the signature operator of a Witt space. Time permitting I will discuss how this analysis extends to allow C*-algebra coefficients and allows us to define a `symmetric signature' morphism from Siegel's Witt bordism group to a K-theoretic group.
    • December 5, 2008
      Speaker:  Maxim Braverman (Northeastern University)
      Title: Topological calculation of the phase of determinant of elliptic operators
      Abstract: In the recent years several interesting examples appeared in physical literature, when the phase of the determinant of a Dirac-type operator with a non-self-adjoint potential can be computed in terms of such topological invariants of the potential as the index and the Hopf invariant. But not only the proofs but the very formulations of the results were not rigorous even by the standards usual for the physical literature.
         In the first part of the talk I will show that for a large class of elliptic operators the phase of the zeta-regularized determinant is a topological invariant which can be explicitly calculated. This covers roughly a half of the physicist's examples, and also many other interesting geometric examples. However, for the other half of the physicist's examples the answer predicted by physicists turns out to be wrong for the zeta-regularized determinant.
         In the second half of the talk I will present a new regularization for the determinant of the elliptic differential operator on odd-dimensional manifolds, called the symmetrized determinant, and explain that it is adequate for the study of the non-linear sigma model. The symmetrized determinant has very nice features. In particular, it is multiplicative, whenever the Konsevich-Vishik anomaly formula holds for the usual determinant. Finally, I will present a condition for the phase of the symmetrized determinant to be a topological invariant, and compute it in all of examples considered by physicists

    Spring 2009
    • January 23, 2009
    • Speaker: Stanislav Dubrovskiy ( Northeastern University )
      Title: Moduli space of general connections
      Abstract: Finiteness of functional moduli is a recurring theme in local differential geometry.
         In this talk we investigate the moduli space of general connections (with torsion). We consider the action of the group of origin-preserving diffeomorphisms on the space of germs of generic connections at a point. The resulting moduli space gives rise to a Poincare series. By analyzing the corresponding moduli spaces of k-jets we calculate the series and establish that it is in fact a rational function, indicating a finite number of functional invariants.
         This conforms once again the finiteness conjecture of Tresse, that algebras of invariants of "natural" differential-geometric structures are finitely generated.

    • January 30, 2009
    • Speaker: Sergei Yakovenko ( Weizmann Institute, Israel )
      Title: Oscillatory properties of Fuchsian ordinary differential equations in the real and complex domain
      Abstract: It is a well-known fact that solutions of second order linear ODE with bounded coefficients admit explicit estimate of the distance between consecutive zeros of its solutions (Sturm theory). It is virtually unknown that this fact is valid for any linear ODE away from singular points, and a complex generalization exists which allows to treat also complex zeros of (multivalued) solutions.
         Under certain assumptions on the nature of singular points it is possible to extend the above "oscillation-control" type results onto sectors with vertices at the singularities.
         Application of this simple but powerful theory allows to construct an explicit bound for the number of zeros of Abelian integrals (the Infinitesimal Hilbert 16th problem).
         The results were obtained in a joint work with Gal Binyamini and Dmitry Novikov.

    • February 13, 2009
    • Speaker: Peter Topalov ( Northeastern University )
      Title: On the Integral Geometry of Liouville Billiard Tables
      Abstract: A notion of Radon transform for completely integrable billiard tables is introduced. It will be shown that in the case of Liouville billiard tables of dimension 3 the Radon transform is one-to-one on the space of continuous functions K on the boundary of the billiard. This allows us to obtain spectral rigidity of the corresponding Laplace-Beltrami operator with Robin boundary conditions.
    • February 20, 2009
    • Speaker: Ivan Horozov ( Brandeis University )
      Title: Gravity and Regulators of Number Fields
      Abstract: We are going to describe two applications of a new tool - higher dimensional iterated integrals. One of the applications is in number theory and the other - in quantum physics.
         The number theoretic application is about Borel regulator of a number field: We express the values of the Dedekind zeta function at the positive in terms of multiple polylogarithms. Zagier has conjectured that single-valued polylogarithms are enough.
         The second application is a new approach to quantum gravity. A starting point for the gravity that we consider is general relativity in terms of a connection, using spinors. (Some people who have worked in this direction are sir Penrose, Ashtekar, Gambini, Lano, Fedosin, Agop, Buzea and Ciobanu, Mashhoon, Gronwald, and Lichtenegger, Clark and Tucker. There are many relations between the two application. For example: the loop expansion in this approach to quantum gravity is done in terms of Borel regulators of number fields.

    • February 27, 2009
    • Speaker: Dave Finn ( Rose-Hulman Institute of Technology )
      Title: TBA
    • March 24, 2009 (Note special date)
    • Speaker: Alexander Turbiner ( UNAM, Mexico )
      Title: A new continuous family of two-dimensional exactly-solvable and (super)integrable Schroedinger equations
      Abstract: It is shown that the Smorodinsky-Winternitz potential, BC2 rational model, 3-body Calogero model, Wolves potential (G2-rational model in Hamiltonian Reduction nomenclature) are the members of a continuous family of two-dimensional exactly-solvable and (super)integrable Schroedinger equations marked by some continuous parameter. Their spectra is always linear in quantum numbers. Hidden algebra of the family for integer values k of the parameter is uncovered. It is non-semi-simple Lie algebra gl(2) x Rk+1 realized as vector fields on line bundles over k-Hirzebruch surface.
    • April 3, 2009
    • Speaker: Justin Holmer ( Brown University )
      Title: Motion of mKdV 2-solitons in an external field
      Abstract: We consider the mKdV equation with a slowly varying potential term, and show that both single and double solitons remain intact but move with parameters of motion described by ODEs. These ODEs are formally predicted by symplectic projection, although the rigorous proof relies on substituting an ansatz into the equation and controlling errors using the Lyapunov functional employed in stability theory. The results are valid on a long enough time scale to observe interesting dynamics in the semiclassical limit. We confirm the results with numerical simulations. This is joint work with Maciej Zworski and Galina Perelman.
    • April 17, 2009
    • Speaker: Jonathan Weitsman ( Northeastern University )
      Title: Fermionization, convergent perturbation theory, and correlations in quantum gauge theories
      Abstract: The problem of understanding path integrals associated to quantum gauge theories is a longstanding issue in mathematical physics, and now also in differential geometry. We show that quantum gauge theories in three and four dimensions are equivalent to purely fermionic theories, where, with appropriate cutoffs, the perturbation series is convergent. Classical techniques, developed in the 1980''s, have been used in the past to understand the path measures in similar cases, and we hope that they are useful in this situation also. Meantime as a byproduct we obtain some natural conjectures about the behavior of correlations in four-dimensional Yang Mills theory, and explore connections with the invariants of three-manifolds.
         The technique of fermionization was developed in the 1970''s in the context of two dimensional quantum field theories, where the fermions are quantum solitons or vertex operators. The fact that these ideas can be used in gauge theory is unexpected. Mathematically, this is an indication that in many cases which arise naturally in geometry, measures on Banach spaces are related to much simpler algebraic objects arising from the calculation of certain determinants of operators on these spaces. This kind of relation cannot occur for finite-dimensional vector spaces, and may be indicative of the surprising ways in which integrals on infinite-dimensional spaces differ from---and are much simpler than---their finite-dimensional analogs.
         In this talk I will survey the basic ideas of integration on function spaces and then discuss these new developments.


    You can also view past talks that we have had: 1999-2000, 1998-9, 1997-8    or  1996-7.

    Back to the AG Seminar Home Page Seminars and Colloquia

    Created: January 18, 2001. Last modified: January 3, 2003.

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