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 zerointhespectrum
conjecture obtained in a recent joint work with S.
Weinberger. We constructed for any n>5
a closed smooth ndimensional manifold M
such that the LaplaceBeltrami operator acting on L^{2}
forms of all degrees on the universal covering M^{~}
of M is invertible. Our work was inspired by the
wellknown paper of M.A. Kervaire who
constructed homology spheres with prescribed fundamental groups. We
exploit the method of the extended L^{2}cohomology
which treats the NovikovShubin 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 RP^{n},
CP^{n} and HP^{n}.
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 RP^{3},
is diffeomorphic to the product RP^{3}xRP^{2}.
This nonoriented 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 threedimensional sphere coupled to O(3) nonlinear
sigma model through the Yukawatype interaction. The
nonperturbative (global) quantum anomaly of this model results in a
Hopf term for the effective nonlinear sigma model. We obtain
this term using the "embedding'' of the CP^{1}^{ }model
into the CP^{M}
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 ChernSimons 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)
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 vectorvalued 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 realRolle inequality is very
important for applications, and it would be highly desirable to have it
also for complexvalued 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 "complexRolle" inequality
allows for very accurate counting of complex zeros for
exponential sums.
 December 1, 2000,
3:30PM (
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^{q1}u=0,
q =
const 1,
we will
discuss properties of solutions in unbounded domains (such as cone,
cylinder etc.), in particular, existence or nonexistence of positive
solutions, and asymptotic behavior of solutions.
February 2, 2001
Speaker: Andrzej
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 CrandallRabinowizt bifurcation theorem,
the finitedimensional LyapunovSchmidt 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
Speaker: Bruno De
Oliveira (Harvard
University
and University of
Pennsylvania)
Title: Complex
cobordisms and the nonembeddability of CRmanifolds
Abstract: We
give results on complex cobordisms whose ends are strictly pseudoconvex
CauchyRiemannmanifolds. Suppose the complex cobordism is given by a
complex 2manifold X with one pseudoconvex and one
pseudoconcave end. We introduce two methods to construct pseudoconcave
surfaces that show that the complex 2manifold X
giving a complex cobordism is not determined by the pseudoconvex end.
These two constructions give new methods to construct nonembeddable
CauchyRiemann 3manifolds and prove that embeddability of a strictly
pseudoconvex CauchyRiemann 3manifold is not a complexcobordism
invariant. We show that a new phenomenon occurs: there are CRfunctions
on the pseudoconvex end that do not extend to holomorphic functions on X.
We also show that the extendability of the CRfunctions 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
Speaker: Alexander
Turbiner (Institute
for Nuclear Sciences  National
University of Mexico)
Title: Hydrogenic
chains in magnetic field (stateoftheart variational calculations)
Abstract: It
was anticipated by KadomtsevKudriavtsev and Ruderman at 70es, that
unusual chemical compounds can appear in a strong magnetic field, which
do not exist without magnetic field. Oneelectron exotic molecular
systems are of main content of the talk. Stateofart choosing
variational trial functions is used. In particular, it admits to carry
out the most accurate study of H_{2}^{+}
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 H_{3}^{++} and H_{4}^{+++}.
In the contrary, H_{2}^{+ }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 L^{2}version
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 AtiyahSinger index
theorem. It can also be proved by computing the variations of eta
invariants, in using a result of BoossWojciechowski 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 Jholomorphic curves in contact geometry.
Abstract:
We will describe some basic properties of Jholomorphic
curves in the symplectization of a contact manifold, which will lay
down an analytic foundation for the applications of Jholomorphic
curves in contact geometry.

Fall
2001 
 September 28, 2001
Speaker: Andrzej
Borisovich (Gdansk
University)
Title: Nielsen
fixed points theory, symplectic maps and PoincareBirkhoff
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 noncompressible 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
number.
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 (TechnionIsrael
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
3regular 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 longrange
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 nontrapping 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 D_{t}_{ }
of Dirac operators on a closed manifold M decomposed
along a hypersurface in terms of the spectral flow of D_{t}_{
}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 2dimensional conformal invariant of Riemannian
metrics on 4manifolds, similar to conformal length for Riemann
surfaces. The approach makes use of two
ingredients. The first is the ConwayThompson 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 4manifolds
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 higherdimensional
analog of the logarithmic upper and lower bounds for conformal length
on Riemann surfaces, due to P. Buser and P. Sarnak.
January 18, 2002
Speaker: Leonid
Friedlander (The
University of Arizona)
Title: On the density of states
of periodic media in the large coupling limit
Abstract:
Let W_{0 }be
a domain in the cube
(0,p )^{n},
and let c_{t}(c)
be a function that equals 1 inside W_{0},
equals t in
(0,p )^{n}
\ W_{0}, and that is extended periodically to R^{n}.
It is known that, in the limit t®„, the spectrum of the operator c_{t}
exhibits the bandgap 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
nonexistence of positive solutions for second order nonlinear
ellitpic equations in unbounded domains
Abstract:
We consider a nonlinear elliptic secondorder 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 nonexistence conditions are obtained. They depend
on the type of domain and the nonlinear terms in the
equation.
February 1, 2002
Speaker: Ognjen
Milatovich (Northeastern
University)
Title: Essential
selfadjointness of Schrödinger type operators.
Abstract:
Several essential selfadjointness 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
boundaryvalue 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 boundaryvalue problems
are isomorphisms (Fredholm operators) between Sobolevtype
spaces of functions ''with s
and '' sd  derivatives'', where d
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 boundaryvalue problems.
Due to the fact, that for elliptic and parabolic
pseudodifferential initial boundaryvalue 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 (BenGurion
University)
Title: 1d Periodic Differential
Operator of Order 4
Abstract:
We consider a differential operator
L = d^{4}/dx^{4}+ d/dx
p(x) d/dx + q(x),
x Ī R^{1}
with 1periodic
functions p(x) and q(x).
We prove that characteristic equation for its Floquet multipliers is
inverse and hence its spectrum in L^{2}(R^{1})
may be described using some hyperelliptic 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 L_{0}=d^{4}/dx^{4}. Then p(x)ŗq(x)ŗ0.
April 26, 2002
Speaker: Maxim Braverman
(Northeastern University)
Title: New proof of the
CheegerMuller 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 MayerVietoris 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 zetafunction
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 longrange order. One of the most
striking features of quasicrystals is their symmetry groups:
these may include fivefold 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 Xray diffraction spectrum. The talk
will conclude with a description of current work on this question,
using spaces of almostperiodic 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 "BatalinVilokovisky Doperators")
and odd Poisson brackets. I will show how this relation between
differential operators and brackets can be made 11 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_{∞ }≤ c_{1}λ^{(n}^{1)/4}φ_{2 },
where n is the dimension
of M and c_{1}
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
φ_{k}_{∞ }≥ c_{2}λ _{k}^{(n}^{1)/8}
φ_{k} ^{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
originpreserving 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" differentialgeometric structure.
October 18, 2002
Speaker: Marina
Ville (CNRS
and Boston University)
Title:
Milnor numbers of minimal
surfaces in 4manifolds
Abstract: When
a sequence of smooth embedded complex curves (C_{n})
in CP^2 degenerates to a
branched curve C_{0}, we lose
topology ( g(C_{n})>g(C_{0}) ) 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_{0 }
which tells us how much topology we have lost going from C_{n}
to C_{0}.
So we ask: is there anything even
remotely similar if the C_{n}'s
and C_{0}
are more general surfaces (e.g. minimal surfaces) in a 4manifold? 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 semibounded 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 WienerHopf 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 WienerHopf
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
3manifolds play an important role in calibrated geometry of G_{2}manifolds.
We study associative cones in R^{7}
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
3Manifolds with Yamabe Invariant Greater Than RP^{3}
Abstract: In
a joint work with Andre Neves, we classify prime 3manifolds with
Yamabe invariant greater than Y_{2},
the Yamabe invariant of RP^{3}.
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
3manifolds had been 0 and
Y_{1} = 6 (2p^{2})^{2/3},
the Yamabe invariant of S^{3}.
We increase the number of known values by 50% by proving that the
Yamabe invariants of RP^{3} and RP^{2}xS^{1}
are equal to Y_{2} = Y_{1}/2^{2/3}.
We also prove that any 3manifold with Yamabe invariant Y
> Y_{2} must either be S^{3}
or a connect sum with an S^{2}
bundle over S^{2} (S^{2}xS^{1}
or the 3dimensional Klein bottle). Hence, the Poincare conjecture for
manifolds with Y > Y_{2}
(which includes an infinite number of 3manifolds) follows. Also, since
the classification of 3manifolds reduces to the classification of
prime 3manifolds, it is natural to try to make a list of prime
3manifolds. Ordered by their Yamabe invariants, we conclude that the
first five prime 3manifolds are S^{3}, S^{2}xS^{1},
K (the 3dimensional Klein bottle), RP^{3},
and RP^{2} xS^{1}.

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
K_{0}(A)
x K^{0}(A)→ K
where K_{0}(A)
is defined by projective Amodules, K^{0}(A)
is defined by finitedimensional Amodules, 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
nonlinear boundary conditions on a noncompact part of the boundary
Abstract: We
will describe new results about the following problems, concerning
solutions of second order linear elliptic equations with nonlinear
boundary conditions in unbounded domains: Phragmén 
Lindelöf type theorems, existence or nonexistence of positive
solutions.
February 7, 2003
Speaker: Ilya
Zakharevich )
Title:
Geometry of biPoisson
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 finitedimensional Poisson pairs. While
answers to some flavors of the classification problem are known (for
example, biPoisson 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 HardyLittlewood 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,formboundedness, 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 S_{1}
into another, S_{2}, one means
a function which defines a bounded linear mapping of S_{1}
into S_{2} 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 pseudodifferential 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 arXiv.org. 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 = H_{0} + V,
where H_{0} = D
is the Laplacian on the Euclidean space, with an arbitrary real or
complexvalued distributional potential V.
Analogous results for the relativistic Schrödinger operator
where
H_{0} = (D
+m^{2})^{1/2} m
will be discussed. This is joint work with Vladimir Maz'ya.
April 11, 2003
Speaker: Semyon
Alesker (Tel
Aviv University)
Title:
Noncommutative
determinants and MongeAmpere equations.
Abstract: There
are various constructions of noncommutative determinants (super,
quantum, GelfandRetakh...). 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 MongeAmpere 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 (BarIlan
University)
Title:
Projective geometry of
quantum mechanics, EinsteinPodolskyRosen 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 foursphere
Abstract: We
construct sequences of Willmore surfaces in S^{4}
by using a Baecklund transformation of Willmore surfaces.
For Willmore tori with nonzero 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 nontrivial 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 3manifolds with first Betti
number 2, a sufficient condition for our systolic inequality to hold is
the nonvanishing of a certain Massey product.
May 23, 2003
Speaker: Maxim
Braverman (Northeastern
University)
Title:
Topological calculation
of the phase of the zetaregularized determinants
Abstract: We
show that for a large class of elliptic operators the phase of the
zetaregularized 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 semiinfinite 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 oneparticle
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 HelmholtzVolterra 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 HelmholtzVolterra criterion
using the de Rham complex on an infinitedimensional space of fields.
September 12, 2003, 3:30
PM (
the special time)
Speaker: Fabio
Podesta (University
of Florence, Italy)
Title:
Cohomogeneity One Kaehler
manifolds and new examples of KaehlerEinstein Cohomogeneity One
Kaehler manifolds and new examples of KaehlerEinstein 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 KaehlerEinstein metrics on some cohomogeneity
one compact Kaehler manifolds, when the principal orbits are Levi
nondegenerate.
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 KacMoody
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 KacMoody group and thus of a
symmetric space, the quotient of an affine KacMoody 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 KacMoody algebras. We show
in particular that it can be reduced to well known problems in finite
dimensions.
October 1, 2003, 2PM ( the special
date and time)
Speaker: Thomas
Kappeler (University
of
Zurich)
Title:
Wellposedness 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
a 1.
October 17, 2003
Speaker: Maxim
Braverman (Northeastern
University)
Title: The
L2torsion without
the determinant class condition
Abstract:
We define the combinatorial and the analytic L2torsions 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 L2torsions 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
CheegerMuller type theorem stating the equality between the analytic
and the combinatorial L2torsions.
(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 CP^{2}
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 CP^{2}
and show that any nonsuperminimal Hamiltonian stationary Lagrangian
torus in CP^{2}
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:
LowDimensional
Homogeneous Einstein Manifolds
Abstract:
I will describe joint work with Christoph Böhm, investigating
the Einstein equation for Ginvariant 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 casebycase
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 12dimensional simply connected homogeneous torus bundles which do
not admit Ginvariant Einstein metrics. These are the first
nonexistence 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:
Timelike isothermic
surfaces associated to Grassmannian Systems
Abstract:
C.L. Terng defined the U/K system for a symmetric
space based on a semisimple U in 1997. This is a
nonlinear first order system of partial differential equations defined
using U/K. This system gives rise to a
oneparameter family of flat connections called the Lax connection of
the U/K system. I will show that timelike isothermic surfaces in
pseudoriemannian space R^{nj,j }are
associated to the Grassmannian O(nj+1,j+1)/O(nj,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 FeynmanKac
like integral, combined with a simple Markov property and estimates on
the solution to a initialboundary problem on a ball in Euclidean space
we prove Morse inequalities. Based on this argument we will discuss
also the case of a BottMorse function where the idea is to compare the
associated Laplacians with respect to Bismut connection and LeviCivita
connection around the critical submanifolds.
February 6, 2004
Speaker: Victor Roitburd (Rensselaer
Polytechnic Institute)
Title:
Asymptotic dynamics of
nonequilibrium freeboundary problems
Abstract: Freeboundary
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 freeboundary problems
for the heat equation, and explain how some types of them relate to
reactiondiffusion systems. The talk will be mostly concerned with
asymptotic dynamics of solutions of a freeboundary problem arising
from the socalled solidstate 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 UniversityPurdue 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 pseudodifferential
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 MxT^{4},
where T^{4} 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 nonlinear map r > r'+r^{2 }on
functions of one real variable. The importance
of this transform lies in its relations with some nonlinear 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 ( the special
date and time)
Speaker: Alexander
Turbiner (Institute
for Nuclear Sciences  National University of Mexico)
Title: Perturbations
of
integrable systems and DysonMehta integrals
Abstract: OlshanetskyPerelomov
quantum Hamiltonians are unique both completely integrable and
exactlysolvable 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 Liealgebraic 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 manybody anharmonic
oscillators turned out to be among these perturbed problems.
Corrections to eigenvalues are given by ratios of generalized
DysonMehta integrals, hence they can be found by algebraic means. They
are interesting by themselves.
December 3, 2004
Speaker: Gideon
Maschler (University
of Toronto)
Title: ConformallyEinstein
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 nonempty zero set of the conformal factor, giving rise to
examples of asymptotically hyperbolic Einstein metrics. Locally, these
metrics are given by a 3parameter family on the total space of a line
bundle over a KählerEinstein 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 3dimensional manifolds. In particular, we will
reformulate the Poincare' Conjecture in terms of the new convexity.
February 3, 2005, 1:30PM ( 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 R^{n},
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
nonclosed. The aim is to describe spectral properties of the
corresponding operators of the NeumanntoDirichlet 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
Hprinciple for maps with prescribed Jacobian
Abstract: Let M and N
be closed
ndimensional manifolds, and equip N with a volume
form σ. Let μ be an exact nform 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 hprinciple, and whenever one can find a
bundle map f_{bdl}:T
M→T N which is degree 0 on the base and such that f_{bdl}*(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 hprinciple 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 hprinciple. 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 WessZuminoNovikovWitten
model) and to gauge theory in three dimensions. We show that these
measures are positive, and that the WessZuminoNovikovWitten measure
where the Riemann surface is P^{1}
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
KontsevichVishik. 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 BransonGilkey 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 EinsteinHilbert 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 odddimensional manifold, which is close enough to a unitary
representation, we define a refinement of the RaySinger 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 RaySinger torsion, while its phase is
determined by the etainvariant. The fact that the RaySinger torsion
and the etainvariant 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 blockdiagonal
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 P_{k}(Λ)
of matrices of rank exactly k;
(c) tangent plane, normal subspace and
shape operator for a given point S Ī
P_{k}(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 П_{i=1}^{k} P_{i}(Λ)
is equal to 2^{nk}.
 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 algebroanalytic
mechanism which is similar to the Lax LA 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 timedependent 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 T^{3}
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 S^{n }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 evendimensional 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.
http://www.math.toronto.edu/ivrii/Research/preprints/Talk_5.php
 September 29, 2006
Speaker: Steven
Rosenberg (Boston
University)
Title: ChernSimons
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 LeviCivita connection takes values in pseudodifferential
operators. Using the Wodzicki residue, we can define Pontrjagin forms,
but these all vanish. The corresponding secondary ChernSimons classes
are nontrivial in general. (Joint work with F. TorresArdilo.)
 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 pseudodifferential
operators, which extend the usual trace on traceclass operators and
satisfy the trace condition Tr(AB)= Tr(BA).
In the talk I will consider the algebra
of odd class logarithmic pseudodifferential 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 pseudodifferential
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 pseudodifferential operators. This determinant is
multiplicative, i.e. satisfies det(AB)= det(A) det(B), whenever the
KontsevichVishik 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/λ^{d}
< 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 ( the special
date and time)
Speaker: Alexander
Turbiner (Institute
for Nuclear Sciences  National University of Mexico)
Title: Anharmonic
oscillator
and doublewell potential: how to approximate eigenfunctions
Abstract: A
simple uniform approximation of the logarithmic derivative of the
lowest eigenfunction for both the quantummechanical quartic anharmonic
oscillator and the doublewell potential given by V= m^{2}
x^{2}+g x^{4} at
arbitrary g ≥ 0 for m^{2}>0
and m^{2}<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 D_{X}
module M and an Rconstructible
sheaf F such that the intersection ofthe
characterictic variety of M and the microsupport of F
is contained in the zerosection of T*X.
If this intersection is compact, then thecohomology of the complex of
solutions RHom_{D}( MÄ F,O_{X}
) is
finite dimensional over C and its index χ(X; M,F)
is given by the formula (SchapiraSchneiders):
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 RiemannRoch and AtiyahSinger theorems.
 December 11, 2006, 3PM ( 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: Largetime
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 H^{d}.
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 SturmLiouville equation with general potential,
and elements of the harmonic anlysis on H^{d}.
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 generaltype manifold.
 September
28, 2007
Speaker: Vadim
Tkachenko (BenGurion
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
onedimensional 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
rightinvariant (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 Fréchetanalytic 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 CamassaHolm 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 noncompact
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 secondorder 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 noncompact weighted manifold. We will also
observe that if the Cauchy boundary ¶_{C}
M : = MM of M,
where M
is the completion of M, is almost polar, then the weighted Laplacian is
essential selfadjoint. 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 finitedimensional 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 1tomany
operators, etc. In finitedimensional theory one must, in addition to
Jordan blocks, consider Kronecker blocks; they have only discrete
parameters. In infinitedimensional 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
4manifolds; although highly nongeneric 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
HelfferNier (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
highlyexcited 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=
m^{2} x^{2}+ x^{4}
with single well (m^{2} ≥ 0)
and for the doublewell potential (m^{2}
< 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 EremenkoGabrielovShapiro about complex zeroes of AHO
eigenfunctions is mentioned.
A generalization to different
onedimensional 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 3manifold groups?
Abstract: Every
finitely presented group G can be realized as the
fundamental group of a (smooth, compact, connected, orientable)
4dimensional manifold. Requiring that G be the
fundamental group of a Kaehler manifold, or that of a 3manifold, 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 http://arxiv.org/abs/0709.4350

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 200304, 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 odddimensional 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 FarberTuraev torsion. In fact,
the torsion quadratic form can be viewed as an analytic analogue of the
PoincareReidemeister 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 BurgheleaHaller
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 FarberTuraev torsion and prove some weak version
of this conjecture. As an application we establish a relationship
between the CappellMiller 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 odddimensional 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 FarberTuraev torsion. In fact,
the torsion quadratic form can be viewed as an analytic analogue of the
PoincareReidemeister 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 BurgheleaHaller
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 FarberTuraev torsion and prove some weak version
of this conjecture. As an application we establish a relationship
between the CappellMiller and the combinatorial torsions.
(joint work with T. Kappeler)
 February 22, 2008
Speaker: Yaron
Ostrover (MIT)
Title: BrunnMinkowskitype
inequality in symplectic geometry
Abstract: The
BrunnMinkowski 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 HoferEkelandZehnder capacity of
convex domains. We discuss the BrunnMinkowskitype 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
ArtsteinAvidan).

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 KortewegDe Vries Equation
Abstract: We prove local
existence and
uniqueness of solutions of the focusing modified Korteweg  de Vries
equation u_{t} + u^{2}u_{x} + u_{xxx}
= 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 (ga_{k})_{k
≥ 1} of the periodic spectrum of Hill's operator d^{2}/dx^{2}+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 pseudodifferential 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 Ktheoretic 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 Diractype operator
with a nonselfadjoint 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 zetaregularized
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 zetaregularized determinant.
In
the second half of the talk I will present a new regularization for the
determinant of the elliptic differential operator on odddimensional
manifolds, called the symmetrized determinant, and explain that it is
adequate for the study of the nonlinear sigma model. The symmetrized
determinant has very nice features. In particular, it is
multiplicative, whenever the KonsevichVishik 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 originpreserving
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 kjets 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" differentialgeometric 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 wellknown 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 "oscillationcontrol" 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 onetoone on the space of
continuous functions K on the boundary of the billiard. This
allows us to obtain spectral rigidity of the corresponding
LaplaceBeltrami 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 singlevalued
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 (
RoseHulman
Institute
of Technology )
Title: TBA
Abstract:
 March
24, 2009 (Note special date)
Speaker: Alexander Turbiner (
UNAM,
Mexico )
Title: A
new continuous family of twodimensional exactlysolvable and
(super)integrable Schroedinger equations
Abstract: It
is shown that the SmorodinskyWinternitz potential, BC_{2}
rational model, 3body Calogero model, Wolves potential (G_{2}rational
model in Hamiltonian Reduction nomenclature) are the members of a
continuous family of twodimensional exactlysolvable 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 nonsemisimple Lie algebra gl(2) x R^{k+1}
realized as vector fields on line bundles over kHirzebruch surface.
 April 3,
2009
Speaker: Justin
Holmer ( Brown
University )
Title: Motion
of mKdV 2solitons 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
fourdimensional Yang Mills theory, and explore connections with the
invariants of threemanifolds.
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 finitedimensional vector spaces, and may be
indicative of the surprising ways in which integrals on
infinitedimensional spaces differ fromand are much simpler
thantheir finitedimensional analogs.
In this talk I will survey the basic ideas of
integration on function spaces and then discuss these new developments.

