Northeastern University         Department of Mathematics

Applied and Interdisciplinary Mathematics Seminar  

Meets 11am to 12pm Tuesdays in 511 LA 
(Unless otherwise noted)

Organizers: Chris King, Robert McOwen, Adam Ding

The seminar features talks in Applied and Interdisciplinary Mathematics (AIM) by our own faculty as well as faculty from other departments at Northeastern and other Universities in the Boston Area. For additional information or to be added to the mailing list, please contact Robert McOwen.
 

  Future Talks

Date: April 24, 2018
     Speaker:   Heikki Haario (Lappeenranta University of Technology, Department of Mathematics and Physics)
     Title: Statistical invariance in chaos and random patterns
Abstract:  We present a recent algorithm for parameter estimation of chaotic dynamical sysems. The idea is to consider simulated state vectors (or real measurements) as samples from the underlying attractor in the state space, and create a statistical feature vector to characterize the variability of it. With slight modifications, the approach can be applied to SDE systems, as well as to identify model parameters of reaction-diffusion systems by ensembles of Turing patterns, such as produced by, e.g, the Fitzhugh-Nagumo model, a classical model of excitable media.
 
 

  Past Talks

Date: April 10, 2018
     Speaker:   Daniel Wichs (CCIS, Northeastern University)
     Title: Encrypted Computation
Abstract:  This talk will explore recent progress in cryptosystems that enable computation over encrypted data. We will discuss fully homomorphic encryption and signatures, as well as extensions to the multi-party setting.
 
Date: March 20, 2018
     Speaker:   Emanuele Viola (CCIS, Northeastern University)
     Title: Interleaved group products
Abstract:  Let G be the special linear group SL(2,q). We show that if (a1,a2) and (b1,b2) are sampled uniformly from large subsets A and B of G^2 then their interleaved product a1 b1 a2 b2 is nearly uniform over G. This extends a result of Gowers (2008) which corresponds to the independent case where A and B are product sets. We obtain a number of other results. For example, we show that if X is a probability distribution on G^m such that any two coordinates are uniform in G^2, then a pointwise product of s independent copies of X is nearly uniform in G^m, where s depends on m only. Similar statements can be made for other groups as well. (Results obtained with Timothy Gowers.) These results have applications in computer science, which is the area where they were first sought by Miles and Viola (2013).
 
Date: February 27, 2018
     Speaker:   Ehsan Elhamifar (CCIS and ECE, Northeastern University)
     Title: Subset Selection and Summarization in Sequential Data
Abstract:  The increasing amount of sequential data, including video, speech and text, across different areas of science requires robust and scalable techniques to reduce data redundancy. In this talk, I will discuss the problem of subset selection and summarization for sequential data and focus on the development of efficient optimization algorithms to tackle the problem. While best subset selection is, in general, an NP-hard problem, I will present efficient and robust algorithms based on convex and submodular optimization, that come with theoretical performance guarantees and can be applied to large-scale data and online observations. These methods, in contrast to classical approaches, can deal with general nonlinear data models, arbitrary similarity measures and data nuisances. I will discuss successful applications of these methods to summarization of instructional videos and to learning the grammar of complex tasks using the vast amount of data on the Internet.
 
Date: February 20, 2018
     Speaker:   Nicolai Panikov (Biotechnology Program, Northeastern University)
     Title: Mathematical Modeling of Complex Microbial Dynamics in Bioreactors and in Nature
Abstract:  The talk deals with using non-linear ODEs to simulate complex dynamic behavior of microbial populations, such as growth in batch and continuous culture, well-mixed homogeneous and spatially organized growth (biofilms, colonies), steady-state and transient processes, switching from one nutrient sources to another, adaptation to starvation and other stresses, super-fast and near-zero growth, etc. My review will start from simple unstructured models with 1-2 variables (e.g. cell mass and limiting nutrient concentrations) and then proceed step-by-step to more structured models including the genome-scale simulations operating with up to thousand variables. As expected, the best results were obtained with "fundamentally correct" models of intermediate complexity. I will discuss the strategy how to find the desired level of model complexity as dependent on goal/objectives of particular study.
 
Date: February 13, 2018
     Speaker:   Elina Robeva (Math Dept, MIT)
     Title: Maximum Likelihood Density Estimation under Total Positivity
Abstract:  Nonparametric density estimation is a challenging problem in theoretical statistics: in general the maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints allows a path forward. This talk offers an invitation to non-parametric density estimation under total positivity (i.e. log-supermodularity) and log-concavity. Totally positive random variables are ubiquitous in real world data and possess appealing mathematical properties. Given i.i.d. samples from such a distribution, we prove that the maximum likelihood estimator under these shape constraints exists with probability one. We characterize the domain of the MLE and show that it is in general larger than the convex hull of the observations. If the observations are 2-dimensional or binary, we show that the logarithm of the MLE is a tent function (i.e. a piecewise linear function) with "poles" at the observations, and we show that a certain convex program can find it. In the general case the MLE is more complicated. We give necessary and sufficient conditions for a tent function to be concave and supermodular, which characterizes all the possible candidates for the MLE in the general case.
 
Date: November 28, 2017
     Speaker:   Anand Osa (Courant Institute, NYU)
     Title: Coarse-grained models for interacting flapping swimmers
Abstract:  I will present the results of a theoretical investigation into the dynamics of interacting flapping swimmers. Our study is motivated by ongoing experiments in the Applied Math Lab at the Courant Institute, in which freely-translating, heaving hydrofoils interact hydrodynamically to choose their relative positions and velocities. We develop a discrete dynamical system in which flapping swimmers shed point vortices during each flapping cycle, which in turn exert forces on the swimmers. We present a framework for finding exact solutions to the evolution equations and for assessing their stability, giving physical insight into the preference for certain observed "schooling states". The model may be extended to arrays of flapping swimmers, and to configurations in which the swimmers' flapping frequencies are incommensurate. Generally, our results indicate how hydrodynamics may mediate schooling and flocking behavior in biological contexts.
 
Date: 4:00-5:00 pm, November 14, 2017 (Note unusual time!)
Place: 315 Behrakis (Note unusual place!)
     Speaker:   Stuart Brorson (Adjunct, Math Dept, Northeastern University)
     Title: A new calculation of Feigenbaum's constant alpha
Abstract:  The logistic map is a widely-studied example of a dynamical system which behaves chaotically for certain parameter ranges. It is of particular interest because it displays a "period doubling" transition from order to chaos. In the late 1970s, Mitchell Feigenbaum showed that the period doubling behavior could be characterized by two numbers, delta and alpha. He also showed these numbers were universal in the sense that any map obeying fairly broad conditions would evidence the same values of delta and alpha. Therefore, delta and alpha are mathematical constants on the same footing as pi and e.
In my talk I will discuss where delta and alpha come from, and present a high-precision calculation (over 1000 digits) of alpha using the new computer language Julia. This represents the most digits of alpha ever calculated. I will talk about features in Julia which enable this computation. My talk will be of interest to all, and specifically accessible to undergraduates.
Note:  This talk is jointly sponsored with the NU Math Club.
 
Date: November 7, 2017
     Speaker:   Christopher Riedl (CCIS and the D'Amore-McKim School of Business, Northeastern University)
     Title: Optimal Experimental Design in Online Experiments
Abstract:  We illustrate and extend a dynamic method to design and execute Bayesian optimal experiments. The method extends seminal work in statistics and computer science on Bayesian optimal design that have not taken hold due to computational limitations. Our procedure takes as primitives a class of parametric models of strategic behavior, a class of experimental designs, and priors on the behavioral parameters. The method then selects the experimental design that maximizes the information from the experiment and thus increases statistical power. We propose and evaluate two computational improvements to make the application of our approach more tractable. First, we develop a probabilistic bootstrapping procedure to reduce the number of hypothetical experimental outcomes that need to be considered. Second, we apply modern Artificial Intelligence methods to learn a multi-dimensional function to efficiently optimize the space of possible experiments. Together, these two improvements reduce computational complexity by several orders of magnitude over the naive approach while simultaneously increasing precision and reducing regret. This allows for a completely automated approach to designing experiments using only the space of possible experiments as input, which is critically important for the large scale experimentation performed by internet companies that run thousands of experiments daily.
 
Date: November 2, 2017.
     Speaker:   Suncica Canic (Department of Mathematics, University of Houston)
     Special Talk: 12:00-1:30 pm, Interaction between fluids and structures motivated by real-life problems: micro-swimming soft robots, vascular stents, and heart valves.
      Colloquium: 4:30-5:30 pm, Capturing nonlinear interaction between fluids and fiber-reinforced composite materials: a unified mathematical framework. More Info.
 
Date: 12:00-1:00 pm, October 17, 2017 (Note unusual time!)
     Speaker:   Michelle Borkin (CCIS, Northeastern University)
     Title: Data Visualization Across Disciplines
Abstract:  What can help enable both the treatment of heart disease and the discovery of newborn stars? Visualization. Specifically interdisciplinary data visualization, the sharing and co-development of tools and techniques across domains. In this talk I will share sample results from my own research and experience crossing disciplines and bringing together the knowledge and experts of computational physics, computer science, astrophysics, radiology and medicine. I will present new visualization techniques and tools inspired by this work for the astronomical and medical communities.
 
Date: April 18, 2017
     Speaker:   Michael Allshouse (MIE, Northeastern University)
     Title: Application of Lagrangian coherent structures to oceanic transport
Abstract:  Trajectory based analysis uncovers insight into the transport of fluid systems not apparent from Elarian data. Structures that dominate transport are revealed by these trajectories and are referred to as Lagrangian coherent structures. These structures have been used to elucidate transport in systems ranging in scale from blood flow to the dynamics of Jupiter's Giant Red Spot. Coherent structures are used here to address problems of near-coastal ocean transport. We first present the theory behind these coherent structures. Then apply coherent structures to the surface transport of oil from spills and show that the fate of the oil can be dramatically affected by surface wind, which has not been included in previous studies. Another problem we analyze with coherent structures is the subsurface transport of cold, salty water from the abyss to the coastline. Much like waves on the ocean surface, density disturbances propagate through the stratified ocean as internal gravity waves with amplitudes of ~100m. As these internal waves approach the continental slope, they steepen and break resulting in foreign material being transported with the wave up the slope. We will show how biota and sediments can be trapped in the coherent structures formed by the nonlinear shoaling internal waves.
 
Date: April 11, 2017
     Speaker:   Chun-An (Joe) Chou (MIE, Northeastern University)
     Title: Interpretable Medical Decision Support via Combinatorial Optimization
Abstract:  Accurate medical diagnosis and prediction tools are particularly important to help one make better decisions on personalized treatment and intervention. Various predictive models/approaches were developed, and successfully achieved very high accuracy. However, most current tools built using machine learning techniques as a black box are not preferable because they fail to provide transparent information, e.g., what-if decision rules. In this talk, we present a combinatorial optimization approach to building accurate and interpretable decision models, driven by data, and demonstrate several practical cases compared to state-of-the-art machine learning methods.
 
Date: April 4, 2017
     Speaker:   Carina Curto (Penn State)
     Title: What can topology tell us about the neural code?
Abstract:  Cracking the neural code is one of the central challenges of neuroscience. Neural codes allow the brain to represent, process, and store information about the outside world. Unlike other types of codes, they must also reflect relationships between stimuli, such as proximity between locations in an environment. In this talk, I will explain why algebraic topology provides natural tools for understanding the structure and function of neural codes.
 
Date: March 28, 2017
     Speaker:   Mehdi Behroozi (MIE, Northeastern University)
     Title: Asymptotic Analysis in Large-scale Optimization of Logistics and Network Problems: A Geometric Perspective
Abstract:  Most problems in logistics, transportation, and network optimization are solved via linear and integer programming. However, most of these approaches are computational inefficient for large-scale problems. A geometric perspective can provide computational efficiency and at the same time bring new insights by looking at the input data differently and exploiting intrinsic geographic characteristics of these problems. In this talk we apply this new perspective to two practical transportation problems.
In the first problem, we study a fundamental trade-off in determining the net carbon footprint that results from widespread adoption of centralized delivery services. On the one hand, such services are efficient because they aggregate demand together and can plan efficient routes; on the other hand, a household that already undertakes a large amount of driving may have an "economy of scale" of its own because it is likely to be near a grocery store at some point anyway. We model this problem as a Generalized Travelling Salesman Problem (GTSP) and we perform a probabilistic analysis of the asymptotic behavior of the GTSP in order to showcase this trade-off and the efficiency of such services.
In the second problem, we consider a data-driven distributionally robust version of the Euclidean travelling salesman problem in which we compute the worst-case spatial distribution of demand against all distributions whose Wasserstein distance to an observed demand distribution is bounded from above. This constraint allows us to circumvent common overestimation that arises when other procedures are used, such as fixing the center of mass and the covariance matrix of the distribution. Numerical experiments confirm that our new approach is useful as a decision support tool for dividing a territory into service districts for a fleet of vehicle when limited data is available.
 
Date: March 21, 2017
     Speaker:   Benjamin Allen (Emmanuel College)
     Title: Evolutionary dynamics on any population structure
Abstract:  Evolution occurs in populations of reproducing individuals. The structure of a population can affect which traits evolve. Understanding evolutionary game dynamics in structured populations is difficult. Mathematical results have recently emerged for special structures where all individuals have the same number of neighbors. But the general case, where the number of neighbors can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class which suggests there is no efficient algorithm. Whether there exists a simple solution for weak selection was unanswered. Here we provide a solution for weak selection that applies to any graph or social network. Our method relies on calculating the coalescence times of random walks. We evaluate large numbers of diverse and heterogeneous population structures for their propensity to favor cooperation. We study how small changes in population structure---graph surgery---affect evolutionary outcomes. We find that cooperation flourishes most in societies that are based on strong pairwise ties.
 
Date: February 7, 2017 (CANCELLED)
     Speaker:   Michael Malioutov (Northeastern University)
     Title: SCOT approximation and asymptotic inference
Abstract:  Approximation of stationary strongly mixing processes by m-Markov models with sparse memory structure (SCOT) and the Le Cam-Hajek-Ibragimov-Khasminsky locally minimax theory of statistical inference for them is outlined. In our previous papers we proved SCOT equivalence to 1-MC with state space-alphabet consisting of the SCOT contexts. For a fixed alphabet size and growing sample size, the Local Asymptotic Normality is proved and applied for establishing asymptotically optimal inference. We outline what obstacles arise for a large SCOT alphabet size and not necessarily vast sample size.
 
Date: 12:00-1:00 pm, January 17, 2017 (Note unusual time!)
     Speaker:   Dmitri Krioukov (Northeastern University)
     Title: Clustering Implies Geometry in Networks
Abstract:  Two common features of many large real networks are that they are sparse and that they have strong clustering, i.e., large number of triangles homogeneously distributed across all nodes. In many growing real networks for which historical data is available, the average degree and clustering are roughly independent of the growing network size. Recently, (soft) random geometric graphs, also known as latent-space network models, with hyperbolic and de Sitter latent geometries have been used successfully to model these features of real networks, to predict missing and future links in them, and to study their navigability, with applications ranging from designing optimal routing in the Internet, to identification of the information-transmission skeleton in the human brain. Yet it remains unclear if latent-space models are indeed adequate models of real networks, as random graphs in these models may have structural properties that real networks do not have, or vice versa.
We show that the canonical maximum-entropy ensemble of random graphs in which the expected numbers of edges and triangles at every node are fixed to constants, are approximately soft random geometric graphs on the real line. The approximation is exact in the limit of standard random geometric graphs with a sharp connectivity threshold and strongest clustering. This result implies that a large number of triangles homogeneously distributed across all vertices is not only a necessary but also sufficient condition for the presence of a latent/effective metric space in large sparse networks. Strong clustering, ubiquitously observed in real networks, is thus a reflection of their latent geometry.
 
Date: December 6, 2016
     Speaker:   Jose Angel Martinez-Lorenzo (ECE, Northeastern University)
     Title: Next Generation Multi-Coded Compressive Systems for High-Capacity Sensing and Imaging Applications
Abstract:  Millimeter-wave sensing and imaging systems are used ubiquitously for a wide range of applications, such as atmospheric sounding of the earth to forecast the weather, non-destructive-testing to assess the condition of civil infrastructure, deep space observation to explore the composition of the galaxies, security monitoring to detect potential threats in airport checkpoints, and biological imaging of superficial tissues for wound diagnosis and healing. These systems typically operate well when the scene does not change rapidly. Unfortunately, this is not the case in emerging societally-important applications like swarms of drones in rescue missions, smart self-driving cars on roadways, or cyber-physical systems searching for suicide bombers when they are on the move. These new applications require 4D (space + time) sensing and imaging at high video frame, as well as adapting the sensing process based on the evolution of the scene. The primary challenges of implementing 4D adaptable sensing and imaging system can be summarized as follows: 1) the system must be capable of handling variable dynamics, i.e. objects may be moving with different velocities and be located at different focal ranges; 2) the system must be capable of sampling sufficient signal to noise ratio data during the limited period of time available by the scene dynamics; and 3) the system must sample data at massive rates, of the order of several gigabytes per second, in order to perform fast 4D video reconstruction with high volumetric frame rate. With these challenges in mind, one of the key features of these 4D sensing and imaging systems will be the ability to extract the maximum amount of information (measured in bits), which describes properties of an object located in the imaging domain, from the data collected by the sensing system in a given period of time (seconds); this is the information rate (measured in bits per seconds). This information rate has an upper bound, which is the maximum amount of information that can be transferred by an electromagnetic wave from one region of space into another (the system’s physical capacity). This maximum information rate can only be approached when the mutual information of successive measurements is minimized. One way to achieve this is to merge traditional 1D temporal modulation with 3D dynamical modulation of the wavefield in space – a novel technique known as 3D spatial wave-field coding or spatial light modulation. 1D- temporal codes enable one to reach a 1D-information rate close to the upper bound imposed by the 1D-physical capacity, also known as Shannon capacity. Combining 3D-volumetric wavefield coding and 1-D-temporal codes should enable one to reach an information rate close to that imposed by the 4D-physical capacity.
This talk will cover the theoretical principles and fundamental limitations of adaptable compressive sensing and imaging systems using 4D coding. This coding can be implemented by novel Multi-Coded Compressive System using the following physical structures: spatial light modulators, vortex-meta-lenses, and compressive reflectors. Preliminary results will be presented in four domains: 1) multi-scale computational modeling; 2) system design and optimization; 3) real-time distributed imaging; and 4) hardware design integration and validation.
 
Date: November 8, 2016
     Speaker:   Zheyang Wu (Worcester Polytechnic Institute)
     Title: Methods for Detecting Weak and Sparse Signals Among Correlated Features
Abstract:  A group of optimal tests, such as the Higher Criticism test (HC), Berk-Jones test (B-J), $\phi$-divergence tests, etc., have been shown asymptotically optimal under the independence assumption. They were also shown theoretical advantages under dependence cases. However, due to the difficulty of p-value calculation for these tests under the dependence case, current applications are either based on de-correlation of input tests or empirical testing methods. We demonstrate that de-correlation is not an appropriate strategy; properly incorporated correlation information can help to improve statistical power. Meanwhile, we provide a solution to calculate the p-values under a broad range of correlation structures. Under stronger correlations our method is more accurate than the recently proposed GHC (the generalized Higher Criticism) method, which also targets at the correlated data problem. Moreover, our method applies to a wider family of goodness-of-fit (GOF) tests. This family covers the above-mentioned optimal tests, some of which are more powerful than GHC under various correlations.
 
Date: October 18, 2016
     Speaker:   Jian Zou (Worcester Polytechnic)
     Title: Conquering Big Data in Volatility Inference and Risk Management
Abstract:  The field of high-frequency finance has experienced a rapid evolvement over the past few decades. One focus point is volatility modeling and analysis for big data setting. It plays a major role in finance and economics. In this talk, we focus on the statistical inference problem on large volatility matrix using high-frequency financial data, and propose a methodology to tackle this problem under various settings. We illustrate the methodology with the high-frequency price data on stocks traded in New York Stock Exchange in 2013. The theory and numerical results show that our approach perform well while pooling together the strengths of regularization and estimation from a high-frequency finance perspective.
 
Date: September 27, 2016
     Speaker:   Ting Zhang (Boston University)
     Title: Semiparametric Model Building for Regression Models with Time-Varying Parameters
Abstract:  I consider the problem of semiparametric model building for linear regression models with potentially time-varying coefficients. By allowing the response variable and explanatory variables be jointly a nonstationary process, the proposed meth- ods are widely applicable to nonstationary and dependent observations, for example time-varying autoregressive processes with heteroscedastic errors. We propose a local linear shrinkage method that is capable of achieving variable selection and param- eter estimation simultaneously in a computationally efficient manner. Its selection consistency along with the favorable oracle property is established. Due to the fear of losing efficiency, an information criterion is further proposed for distinguishing time-varying and time-invariant components. Numerical examples are presented to illustrate the proposed methods.
 
Date: April 12 , 2016
     Speaker:   Edmund Yeh (ECE, Northeastern University)
     Title: Throughput and Delay Scaling in Content-centric Wireless Networks
Abstract:  We study the throughput and delay scaling of wireless networks based on a content-centric network architecture, where users are mainly interested in retrieving content stored in the network, rather than in maintaining source-destination communication. Nodes are assumed to be uniformly distributed in the network area. Each node has a limited-capacity content store, which it uses to cache contents according to a given caching scheme. Requested content follows a general popularity distribution, and users employ multihop communication to retrieve the requested content from the closest cache. We derive the throughput-delay tradeoff of the content-centric wireless network model. It is shown that in contrast to the source-destination-based communication model, the use of caching can simultaneously increase network throughput and decrease delay in the content-centric model. For the Zipf content popularity distribution, we explicitly find the optimal cache allocation to maximize throughput and minimize delay.
Next, we investigate heterogeneous wireless networks where, in addition to wireless nodes, there are a number of base stations uniformly distributed at random in the network area. Here, users retrieve the requested content from the closest wireless caching point, or from the closest base station (assumed to have access to all contents), whichever is nearer. We show that in order for a heterogeneous network to achieve better performance than an ad hoc network in the order sense, the number of base stations needs to be greater than the ratio of the number of nodes to the number of content types. Furthermore, we show that the heterogeneous network does not yield performance advantages in the order sense if the Zipf content popularity distribution exponent exceeds 3/2.
Joint work with Milad Mahdian
 
Date: March 15, 2016  
     Speaker:   Usama Kadri (MIT)
     Title: Acoustic-gravity waves, theory & applications
Abstract:  The classical water-wave theory ignores the effects of water compressibility, on the grounds that acoustic waves are virtually decoupled from free-surface waves. In the linear theory, this assumption is well justified for many applications, as acoustic propagation modes possess vastly different spatial and/or temporal scales from free-surface waves due to the fact that the sound speed in water far exceeds the maximum surface wave phase speed. In keeping with the incompressible wave equation formulation, for describing waves in a fluid layer with a rigid bottom and a free-surface, any prescribed frequency corresponds to a single propagating gravity wave. However, accounting for both gravity and compressibility, then reveals a countable infinity of propagation modes, some of which are non-evanescent compression modes (do not decay with distance) that are called acoustic-gravity waves.
Acoustic-gravity waves are generated as a response to a sudden change of the water surface, e.g. due to tsunami or other sudden severe sea state. They travel at near the speed of sound in water (ca. 1500 m/s), which turns them into perfect precursors. Acoustic-gravity waves is an emerging field that is rapidly gaining popularity among the scientific community, as it finds broad utility in physical oceanography, marine biology, geophysics, and multiphase flows, to name a few. Acoustic-gravity waves are compression-type waves that can be generated by wind-wave and wave-wave interactions, movements of the tectonic lithosphere plates, landslides, and submarine explosions. They might be found in the oceans, lakes, rivers, atmosphere, as well as in hydraulic systems and flow pipelines.
This talk is an overview on acoustic-gravity waves, confined to three major projects I am currently conducting, and looking to study further. In particular, I will briefly discuss the application for early detection of tsunami, transportation of water in deep ocean, and then focus on the nonlinear triad resonance theory of acoustic-gravity waves, concluding with various implications.
 
Date: 12:00-1:00 pm, January 26, 2016   (Note unusual time)
     Speaker:   Wenjia Jing (Univ of Chicago)
     Title: Stochastic homogenization and random fluctuation theory for partial differential equations
Abstract:  Partial differential equations with oscillatory coefficients arise in many applications, such as the modeling of composite materials and flame propagation. The fine scale variations of the physical media are often unknown and are typically modeled as random. The theory of stochastic homogenization amounts to exploring the self-averaging mechanism of the PDEs, and it leads to mean field approximations of the large scale behavior of the solution. The theory of random fluctuations aims at studying the differences between the solution and its homogenization limit, leading to higher order, and often Gaussian, corrections to the approximation. In this talk, I will present some results on homogenization of the Hamilton-Jacobi equations and on the fluctuation theory for elliptic equations with random potentials.
 
Date: 12:00-1:00 pm, January 19, 2016   (Note unusual time)
     Speaker:  Francois Monard (University of Michigan)
     Title: Toward imaging modalities with high(er) resolution
Abstract:  Given an unknown physical quantity to be imaged, the extent to which an imaging approach (or the corresponding inverse problem) exploits at best the mathematics and the physics of the problem highly impacts (i) what can be reconstructed of this quantity, and (ii) the resolution available on the resulting images. Improvements in both directions (i) and (ii), of tremendous interest for, e.g., medical applications, can then be achieved by changing the underlying inverse problem into another one displaying better invertibility and conditioning.
This strategy for obtaining improvements will be illustrated in two ways, first by discussing some inverse problems for the Boltzmann transport equation, a model for optical tomography and SPECT. Second, we will review some coupled-physics (or hybrid) medical imaging modalities. Such modalities involve a coupling between a typically poorly-resolved soft tissue imaging modality (e.g., EIT, Elastography) and a highly-resolved one (e.g., Ultrasound, MRI), in order to derive imaging models displaying both high contrast and high resolution. In this context, the mathematical analysis of some models concerned with the reconstruction of conductivity and elasticity tensors has shown manifest improvements in both resolution and in the capability of accessing previously unavailable anisotropic features.
 
Date: 12:00-1:00 pm, January 12, 2016   (Note unusual time)
     Speaker:  Emanuel Lazar (Univ Penn)
     Title: Topological Data Analysis: Structure in Spatial Point Sets
Abstract:  Many physical systems can be abstracted as large sets of point-like particles. Understanding how such particles are arranged is thus a very natural problem, though describing this “structure” in an insightful yet tractable manner can be difficult. We consider a configuration space of local arrangements of neighboring points, and consider a stratification of this space via Voronoi topology. Theoretical results help explain limitations of purely metric solutions to this problem, and computational examples illustrate the unique effectiveness of topological ones. Applications to computational materials science are considered.
 
Date: November 17, 2015
     Speaker:  Francois Monard (Mathematics, University of Michigan)
     Title: X-ray transforms on surfaces and tensor tomography
Abstract:  A family of integral geometric problems on surfaces consists of assessing what is reconstructible of a given integrand from knowledge of its integrals along all non-trapped geodesics, and how to reconstruct it. Such a class of problems includes the problem of reconstructing a function defined on the surface (with application to, e.g., computerized tomography in media with variable index of refraction), or a symmetric tensor field (in connection to other imaging methods, or arising as the linearized boundary rigidity problem on a manifold). In this talk we will discuss some inversion approaches to reconstruct functions on "simple" Riemannian surfaces, from knowledge of certain weighted geodesic ray transforms. These formulas will then be connected with recent explicit inversions of the tensor tomography problem derived by the author in the case of the Euclidean disk. In this setting, we will discuss in passing a full range characterization of the ray transform over tensors of any order, and in the case of functions, a new connection between a range characterization by L. Pestov and G. Uhlmann and the so-called Helgason-Ludwig consistency conditions for the Radon transform. Time allowing, we will go over recent results on inversion formulas for attenuated geodesic X-ray transforms on surfaces, and for unattenuated transforms on sufaces with hyperbolic trapped set (the latter in joint work with Colin Guillarmou), presenting examples where functions can be reconstructed from their ray transform even when the geometry contains trapped geodesics. Numerical illustrations will be presented throughout the talk.
 
Time & Date: 12:00-1:00 pm November 3, 2015   (Note unusual time!)
     Speaker:  Cris Moore (Santa Fe Institute)
     Title: The hunt for a quantum algorithm for Graph Isomorphism
Abstract:  After Shor discovered his quantum algorithm for factoring, many of us were hopeful that a similar algorithm could work for Graph Isomorphism. Our hopes were dashed --- but dashed beautifully, by the representation theory of the symmetric group (to which I will give a friendly introduction). As a result, we now know that no quantum algorithm remotely similar to Shor's will solve this problem.
This is joint work with several people, but I'll focus on joint work with Alex Russell and Leonard Schulman.
 
Time & Date: Noon, Friday, October 30, 2015   (Note unusual date and time!)
     Speaker:   Ken Duffy (MIT Research Laboratory of Electronics)
     Title: Guesswork, quantifying computational security
Abstract:  The talk explores the connection among computational security, probability and information theory. The security of many systems is predicated on the following logic: a user selects a string, for example a password, from a list; an inquisitor who knows the list can query each string in turn until gaining access by chancing upon the user's chosen string; the resulting system is deemed to be computationally secure so long as the list of strings is large.
Implicit in that definition is the assumption that the inquisitor knows nothing about the likely nature of the selected string. If instead one assumes that the inquisitor knows the probabilities with which strings are selected, then the random variable of interest is dubbed Guesswork, the number of queries required to identify a stochastically selected string, and the quantification of computational security becomes substantially more involved.
In this talk we review the seminal work of J. Massey (1994) and E. Arikan (1996) on the moments of Guesswork before describing some of our contributions, which start by establishing a Large Deviation Principle (LDP) for Guesswork, which enables direct estimates of the Guesswork distribution. Moreover, as the LDP is a covariant property, it facilitates a significant broadening of the remit of Guesswork to include the computational security of multi-user systems. These developments, as well as others related to information theoretic security, will be discussed. No prior knowledge of the subject will be assumed.
This talk is based on work with M. Christiansen (NUIM), F. du Pin Calmon (MIT) and M. Medard (MIT).
 
Date: October 27, 2015   - POSTPONED!
     Speaker:   Richard Jordan (MIT Lincoln Laboratory)
     Title: Clusters and Communities in Air Traffic Delay Networks
Abstract:  The air transportation system is a network of many interacting, capacity-constrained elements. When the demand for airport and airspace resources exceed the available capacities of these resources, delays occur. The state of the air transportation system at any time can be represented as a weighted directed graph in which the nodes correspond to airports, and the weight on each arc is the delay experienced by departures on that origin-destination pair. Over the course of any day, the state of the system progresses through a time series, where the state at any time-step is the weighted directed graph described above. This talk will describe approaches for the clustering of air traffic delay network data from the US National Airspace System, in order to identify characteristic delay states, as well as characteristic types of days. The similarity of delay states during clustering are evaluated on the basis of not only the in- and out-degrees of the nodes, but also network-theoretic properties such as the eigenvector centralities, and the hub and authority scores of different nodes. Finally, we’ll look at community detection, that is, the grouping of nodes (airports) based on their similarities within a system delay state. The type of day is found to have an impact on the observed community structures.
Joint work with Hamsa Balakrishnan, Karthik Gopalakrishnan and Jacob Avery, Department of Aeronautics & Astronautics, MIT
 
Date: October 13, 2015
     Speaker:  Sayan Mukherjee (Statistical Science, Duke University)
     Title: Stochastic topology: random walks and percolation
Abstract:  The graph Laplacian and random walks on graphs has impacted statistics, computer science, and mathematics. I will motivate why it is of interest to extend these graph based algorithms to simplicial complexes, which capture higher-order relations. I will describe recent efforts to define random walks on simplicial complexes with stationary distributions related to the combinatorial (Hodge) Laplacian. This work will touch on higher-order Cheeger inequalities, an extension of label propagation to edges or higher-order complexes, and a generalization of results for near linear time solutions for linear systems. Given n points down from a point process on a manifold, consider the random set which consists of the union of balls of radius r around the points. As n goes to infinity, r is sent to zero at varying rates. For this stochastic process, I will provide scaling limits and phase transitions on the counts of Betti numbers and critical points.
 
Date: October 6, 2015
     Speaker:  David and Yakir Reshef (Harvard University and the Broad Institute)
     Title: Equitability and the maximal information coefficient
Abstract:  As data sets continue to grow in dimensionality, how can we use them not just to exploit the relationships we already know about but also to detect unanticipated, new relationships between features? This is commonly done by using a statistic to rank relationships in the data set and then manually examining the top of the resulting list. For such a strategy to succeed though, the statistic must give similar scores to equally noisy relationships of different types (e.g., linear, parabolic, non-functional, etc.), given some notion of noise. In this talk we will formalize this property, called equitability, and show how it is related to a variety of traditional statistical concepts. Next, we will introduce the maximal information coefficient (MIC), a statistic with state-of-the-art equitability in a wide range of practical settings. Finally, we will examine the performance, in terms of equitability, power against independence, and runtime, of a number of state-of-the-art measures of dependence and will conclude with a discussion of tradeoffs to consider in choosing an appropriate measure of dependence for various settings.
 
Date: April 28, 2015
     Speaker:  Christian Borgs (Microsoft Research)
     Title: Privacy and Estimation for Non-Parametric Graph Models
Abstract:  When analyzing large networks, statisticians often assume a generative model in which the observed graph is assumed to come from a stochastic block model, i.e., a random graph with inhomogeneous edge probabilities given in terms of a small block matrix. A non-parametric version of these stochastic block models are so-called W-random graphs, given in terms of an integrable, symmetric function W on the unit square. In this talk I discuss the question on how to recover a good approximation to W from just a single sample of a W-random graph, and how to do this in a way which is differentially private. I will also discuss the relationship to the theory of graph convergence, in particular graph convergence for sparse graphs.
This is joint work with J. Chayes, H. Cohn, S. Ganguly and A. Smith.
 
Date: 10:30-11:30 am, April 21, 2015 (Note unusual time!)
     Speaker:  Daniel Wichs (College of Computer and Information Science, NU)
     Title: Tamper Detection and Non-malleable Codes
Abstract: We consider a public and keyless code which is used to encode a message and derive a codeword. The codeword can be adversarially tampered via a function from some "tampering-function family". We study the different types of security guarantees that can be achieved in this scenario for different families of tampering functions.
Firstly, we will study tamper-detection codes, which must detect that a tampered codeword is invalid. We show that such codes exist for any tampering-function family over n-bit codewords, as long as (1) the family is sufficiently small, of size < 2^{2^n}, (2) the functions can only have a few fixed points x such that f(x)=x, (3) the functions must have high entropy of f(x) over a random x. Such codes can also be made efficient when the family is of size 2^{poly(n)}. For example, this holds for the family of all low-degree polynomials excluding constant and identity polynomials. Such tamper-detection codes generalize the algebraic manipulation detection (AMD) codes of Cramer et al. (EUROCRYPT '08).
Next, we will study non-malleable codes, which require that a tampered codeword either decodes to the original message m, or to some unrelated value that doesn't provide any information about m. We achieve such codes for any sufficiently small function family. We give a modular construction of non-malleable codes by combining tamper-detection codes and leakage-resilient codes.
 
Date: April 14, 2015
     Speaker:  Abhyudai Singh (Electrical and Computer Engineering, University of Delaware)
     Title: Stochastic inference of regulatory networks inside living cells
Abstract: In the noisy cellular environment many mRNA and protein species occur at low integer molecular counts, and hence are subject to large stochastic fluctuations in copy numbers over time. Far from being a hindrance, signatures of protein/mRNA noise levels can be informative about the underlying gene network topology. In this talk, I will present recently developed mathematical techniques that harness fluctuations in the levels of biochemical species for systems identification of gene regulatory networks. I will describe current efforts in using these techniques for reverse engineering the genetic circuitry of the Human immunodeficiency virus (HIV). Our results show that HIV encodes a noisy promoter and stochastic expression of a key viral regulatory protein can drive HIV into latency, a drug-resistant state of the virus.
 
Date: April 7, 2015
     Speaker:  Chris Rycroft (School of Engineering and Applied Sciences, Harvard University)
     Title: Computation of three-dimensional standing water waves
Abstract:  We develop a method for computing three-dimensional gravity-driven water waves, which we use to search for time-periodic standing wave solutions. We simulate an inviscid, irrotational, incompressible fluid bounded below by a flat wall, and above by an evolving free surface. The simulations require computing a velocity potential in the bulk of the fluid, which we carry out using a fourth-order finite element method; this computationally expensive step is solved using a parallel multigrid algorithm. Several families of large-amplitude three-dimensional standing waves are found in both shallow and deep regimes, and their physical characteristics are examined and compared to previously known two-dimensional solutions.
 
Date: March 31, 2015
     Speaker:  Lydia Bourouiba (Department of Civil and Environmental Engineering, MIT)
     Title: Fluid dynamics of contact and disease transmission
Abstract:  The mechanisms governing the transfer of pathogens between infected and non-infected members of a population are critical in shaping the outcome of an epidemic. Despite major efforts aimed at the mathematical modeling and mitigation of infectious diseases, the fundamental mechanisms of pathogen spreading for most infectious diseases remain poorly understood. Drawing upon fluid experiments and mathematical modeling, I will discuss the dynamics of contact in mathematical epidemiology and revisit the discussion of transmission of various pathogens through the lens of fluid fragmentation.
 
Date: March 24, 2015
     Speaker:  L. Mahadevan (School of Engineering and Applied Sciences, Harvard University)
     Title: Shape: biology, physics and mathematics
Abstract:  The range of shapes in the plant (and animal) world is "enough to drive even the sanest man mad", wrote Darwin. Motivated by qualitative and quantitative biological observations, I will show that there is a "method in the madness" - using examples of growth and form in tissues and organs such as the undulating fringes on a leaf, the looping of your gut, and the convolutions in your brain. In each case, we will see how a combination of biological and physical experiments, mathematical models and computations allow us to unravel the quantitative basis for the diversity and complexity of biological form, while creating new subjects of study in geometry, analysis and statistics.
 
Date: March 17, 2015
     Speaker:  Gilbert Strang (Department of Mathematics, MIT)
     Title: Banded Matrices and Fast Inverses
Abstract:  The inverse of a banded matrix A has a special form with low rank submatrices except at the main diagonal. That form comes directly from the Nullity Theorem. Then the inverse of that matrix A-1 is the original A, which can be found by a remarkable "local" inverse formula. This formula uses only the banded part of A-1 and it offers a very fast algorithm to produce A.
That fast algorithm has a potentially valuable application. Start now with a banded matrix B. (Possibly B is the positive definite beginning of a covariance matrix C, but covariances outside the band are unknown or too expensive to compute). It is a poor idea to assume that those unknown covariances are zero. Much better to complete B to C by a rule of maximum entropy; for Gaussians this rule means maximum determinant.
As statisticians and also linear algebraists discovered, the optimally completed matrix C is the inverse of a banded matrix. Best of all, the matrix actually needed in computations is that banded C-1 (which is not B !). And C-1 comes quickly and efficiently from the local inverse formula.
A very special subset of banded matrices contains those whose inverses are also banded. These arise in studying orthogonal polynomials and also in wavelet theory|the wavelet transform and its inverse are both banded ( = FIR flters). We describe a factorization for all banded matrices that have banded inverses.
 
Date: March 3, 2015
     Speaker:  Mark Kon (Department of Mathematics and Statistics, Boston University)
     Title: The Marr conjecture and uniqueness of wavelet transforms
Abstract:  The inverse question of identifying a function from the nodes (zeroes) of its wavelet transform has several formulations. These include whether the nodes of a heat or hypoelliptic solution determine its initial conditions, and in mathematical vision theory the Marr conjecture, which asks whether an image can be mathematically reconstructed from its edge information. We prove a general version of this conjecture by reducing it to the moment problem, using a basis dual to the Taylor basis xn on Rd.
 
Date: February 24, 2015
     Speaker:  Ariel Amir (School of Engineering and Applied Sciences, Harvard University)
     Title: Cell size control in microorganisms
Abstract:  Microorganisms in all kingdoms of life face a challenge of regulating the size and shape of their cells, control of which is essential for their viability. For decades, a popular hypothesis has been that cells can measure their absolute size, and that reaching a critical size triggers the division process. This would imply that a cell that was born smaller than average will not be smaller than average when it divides, in contrast to experiments showing that such correlations exist, and that size is partly "inherited." I will present a mathematical model inspired by statistical mechanics that sheds new light on this problem, showing that a cell does not need to know its absolute size to regulate size robustly, quantitatively explaining the experimentally measured correlations in both E. coli and budding yeast, and predicting that average cell size should depend exponentially on the growth rate.
 
Date: February 17, 2015
     Speaker:  Nayantara Bhatnagar (University of Delaware)
     Title: Lengths of Monotone Subsequences in a Mallows Permutation
Abstract:  The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied permutation statistic. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the Tracy-Widom distribution, which arises as the limiting distribution of the normalized largest eigenvalue of a random Hermitian matrix.
We study the length of the LIS and LDS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation p in S_n is proportional to q^Inv(p) where q is a real parameter and Inv(p) is the number of inversions in p. We determine the typical order of magnitude of the LIS and LDS, large deviation bounds for these lengths and a law of large numbers for the LIS for various regimes of the parameter q.
This is joint work with Ron Peled.
 
Date: February 10, 2015       POSTPONED DUE TO SNOW
     Speaker:  Edoardo M. Airoldi (Department of StatisticsHarvard University)
     Title: Statistical and machine learning challenges in the analysis of large networks
Abstract:  Network data --- i.e., collections of measurements on pairs, or tuples, of units in a population of interest --- are ubiquitous nowadays in a wide range of machine learning applications, from molecular biology to marketing on social media platforms. Surprisingly, assumptions underlying popular statistical methods are often untenable in the presence of network data. Established machine learning algorithms often break when dealing with combinatorial structure. And the classical notions of variability, sample size and ignorability take unintended connotations. These failures open to door to a number of technical challenges, and to opportunities for introducing new fundamental ideas and for developing new insights. In this talk, I will discuss open statistical and machine learning problems that arise when dealing with large networks, mostly focusing on modeling and inferential issues, and provide an overview of key technical ideas and recent results and trends.
 
DateJanuary 20, 2015
     Speaker Anand Oza  (Courant Institute)
     Title A Trajectory Equation for Walking Droplets: Hydrodynamic Pilot-Wave Theory
Abstract: Yves Couder and coworkers have demonstrated that millimetric droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic quantum realm, including single-particle diffraction, tunneling, quantized orbits, and wave-like statistics in a corral. We here develop an integro-differential trajectory equation for these walking droplets with a view to gaining insight into their subtle dynamics. We then rationalize the emergence of orbital quantization in a rotating frame by assessing the stability of the orbital solutions. The stability analysis also predicts the existence of wobbling orbital states reported in recent experiments, and the absence of stable orbits in the limit of large vibrational forcing. In this limit, the complex walker dynamics gives rise to a coherent statistical behavior with wave-like features.
 
Date: December 9, 2014
     Speaker: Mason A. Porter  (University of Oxford, U.K.)
     TitleCascades and Social Influence on Networks
Abstract: I discuss "simple" dynamical systems on networks and examine how network structure affects dynamics of processes running on top of networks. I'll give an introduction to the idea of social ("complex") contagions, and I'll present a model for multi-stage complex contagions--in which fanatics produce greater influence than mere followers.  I'll also briefly discuss the use of ideas from topics like persistent homology to examine wavefront propagation versus the appearance of new contagion clusters, and I'll present a model (without network structure) for the adoption of applications on Facebook. The last family of models illustrates how very different time-dependent dynamics can produce quantitatively similar long-time behavior, which poses both very serious challenges and exciting opportunities for the modeling of complex systems.
 
Date: December 2, 2014
     Speaker: Ramis Movassagh (Northeastern University)
     Title:   Power law violation of the area law in quantum spin chains
Abstract: The sub-volume scaling of the entanglement entropy with the system's size, n, has been a subject of vigorous study in the last decade. The area law provably holds for gapped one dimensional systems and it was believed to be violated by at most a factor of log(n) in physically reasonable models such as critical systems. In this paper, we generalize the spin -1 model of Bravyi et al to all integer spin-s chains, whereby we introduce a class of exactly solvable models that are physical and exhibit signatures of criticality, yet violate the area law by a power law. The proposed Hamiltonian is local and translationally invariant in the bulk. We prove that it is frustration free and has a unique ground state. Moreover, we prove that the energy gap scales as n^{-c}, where using the theory of Brownian excursions, we prove c >= 2. This rules out the possibility of these models being described by a conformal field theory. We analytically show that the Schmidt rank grows exponentially with n and that the half-chain entanglement entropy to the leading order scales as sqrt {n}. Geometrically, the ground state is seen as a uniform superposition of all s-colored Motzkin walks. Lastly, we introduce an external field which allows us to remove the boundary terms yet retain the desired properties of the model. Our techniques for obtaining the asymptotic form of the entanglement entropy, the gap upper bound and the self-contained expositions of the combinatorial techniques, more akin to lattice paths, may be of independent interest. (Joint work with Peter W. Shor.)
 
Date: November 25, 2014
     Speaker: Scott Aaronson  (MIT) 
     TitleWhen Exactly Do Quantum Computers Provide a Speedup?
Abstract: Twenty years after the discovery of Shor's factoring algorithm, I'll survey what we now understand about the structure of problems that admit quantum speedups.  I'll start with the basics, discussing the hidden subgroup, amplitude amplification, adiabatic, and linear systems paradigms for quantum algorithms.  Then I'll move on to some general results, obtained by Andris Ambainis and myself over the last few years, about quantum speedups in the black-box model.  These results include the impossibility of a superpolynomial quantum speedup for any problem with permutation symmetry, and the largest possible separation between classical and quantum query complexities for any problem.
 
Date: November 18, 2014
     Speaker: Francis Chung (University of Michigan)
     Title:  Electromagnetic inverse problems with partial data
Abstract:  I will begin by describing an inverse problem for Maxwell's equations, which goes roughly as follows. Suppose we can measure the electromagnetic fields on the boundary of an object -- can we use this information to recover the electrical and magnetic parameters of the interior of the object?  In this talk I will consider the partial data version of this problem, where we can measure only on part of the boundary, and describe some recent results along these lines.
 
Date: November 4, 2014
     Speaker: David R. Nelson (Lyman Laboratory of Physics, Harvard University)
     TitleDislocation-Mediated Elongation of Bacteria
Abstract: Recent experiments have revealed a remarkable growth mechanism for rod-shaped bacteria:  specialized proteins associated with cell wall elongation move at constant velocity in clockwise and counterclockwise directions on circles around the cell circumference.  We argue that this machinery attaches to dislocations in the ordered peptidoglycan cell wall, and study theoretically the dynamics of these interacting defects on the surface of a cylinder. Unlike the dislocations typical in materials science, the motion is predominantly climb (glycan strand extension) instead of glide. The activated motion of these dislocations and the resulting dynamics within a simple kinetic model show surprising effects arising from the cylindrical geometry, with important implications for bacterial growth.   Recent experiments revealing plastic deformation of bacterial cell walls bent by a hydrodynamic flow will be presented as well. 
 
DateOctober 28, 2014
     Speaker: Ken Duffy (Hamilton Institute, NUIM, Ireland)
     TitleInferring the average generation of a population of cells
Abstract: A fundamental quantity of interest in cell biology is the average number of divisions per cell since the initial progenitors, i.e. the average generation of presently living cells. It can, for example, be used to quantify dynamics and aging of the immune system, to better understand the evolution and risk of cancer, and to rank cell types in hierarchies of complex differentiation programs. Estimating average generation is a simple matter if, for example, the cells are directly observable with time-lapse microscopy or if one can stain progenitors with a division-diluting dye. For large numbers of generations and/or in vivo systems, as is necessary for several significant applications, no methodology is presently available.
In this talk we describe a scheme that we have developed to infer average generation. We shall introduce theorems regarding the average generation of an age-dependent branching process that enable us to relate its growth rate to an experimentally measurable quantity, the proportion of neutrally mutated cells. A genetic construct that enables the realization of this estimate, a version of which is presently being built by Ton Schumacher's group (Netherlands Cancer Institute), will be described. No prior biological knowledge will be assumed of the audience.
 
Date: October 21, 2014
     Speaker: Seth Lloyd (MIT)
     Title: Quantum machine learning
Abstract:  Machine learning algorithms look for patterns in data.   Frequently, that data comes in the form of large arrays of high-dimensional vectors.   Quantum computers are adept at manipulating large arrays of high-dimensional vectors.  This talk presents a series of quantum algorithms for big data analysis.   The ability of quantum computers to perform Fourier transforms, find eigenvectors and eigenvalues, and invert matrices translates into quantum algorithms for clustering, principal component analysis, and for identifying topological features such as numbers of connected components, holes and voids.  These quantum algorithms are exponentially faster than their classical counterparts: complex patterns in datasets of size N can be identified in time O( log N). 
 
Date: October 14, 2014
     Speaker: Steven G. Johnson (MIT)
     TitleThe mathematics of lasers: From nonlinear eigenproblems to linear noise
Abstract: Lasers are ubiquitous in modern technology, but their theoretical description is a complicated mixture of quantum and classical electromagnetic phenomena.   Starting from a simple toy model of a laser as an oscillator with a nonlinear gain, we will review the basic description of laser physics.  In recent years, this description has evolved into the steady-state ab-initio laser theory (SALT) of Tureci, Stone, and others: a type of nonlinear eigenproblem. Although the SALT equations could initially only be solved in simple 1d and 2d geometries, we have now demonstrated that modern numerical methods can solve SALT for the lasing modes in general 3d systems, coupling the full vectorial Maxwell equations with the nonlinear gain of the lasing medium in complex inhomogeneous media.
        This progress in computational solvers, in turn, has enabled purely analytical progress on the laser linewidth in the presence of noise.  Returning to the toy nonlinear-oscillator model of a laser, we review how the presence of noise induces a Brownian drift of the laser phase (described by a stochastic ODE) and causes its spectrum to be a finite-width peak.  The theoretical challenge is to derive the width of this peak in the context of complex laser geometries and dynamics, where the noise originates from quantum charge fluctuations (Johnson-Nyquist noise) in matter.  We show how, by linearizing the dynamics around the steady-state (noise-free) SALT solutions, we can obtain an analytical linewidth expression that encapsulates nearly all previous results (which were mostly derived in simplified 1d models; e.g. the Schawlow-Townes formula, the Petermann correction, and the Henry α correction) as special cases.
 
Date: October 7, 2014
     Speaker: Ken Kamrin (MIT)
     TitleContinuum modeling of granular flow
Abstract: Despite the ubiquity of granular matter in the world around us, the challenge of predicting the motion of a collection of flowing grains has proven to be a difficult one, from both computational and theoretical perspectives.  Grain-by-grain discrete element methods can be used, but these approaches become computationally unrealistic for large bodies of material and long times.   A broadly accurate continuum model would be ideal if it could be found, as it would provide a much more rapid means of calculating flows in real-world problems, such as those encountered in industrial design and geotechnical engineering.
With this challenge in mind, in this talk we propose a new constitutive relation for granular matter, which produces quantitatively accurate predictions for granular flow.  The model is constructed in a step-by-step fashion.  First we compose a local relation based on existing granular rheological approaches (i.e. the principle of  "inertial" rheology) and point out where the model succeeds and where it does not.  The clearest missing ingredient is shown to be the lack of an intrinsic length-scale.  To tie flow features more carefully to the characteristic grain size, we justify a nonlocal modification which takes the form of a size-dependent term in the rheology (with one new material parameter).  The nonlocal model is then numerically implemented with a custom-written User-Element in the Abaqus package, where it is shown to greatly improve flow predictions compared to the local model.  In fact, it is the first model to accurately predict all features of flows in `split-bottom cell' geometries, a decade-long open question in the field.  In total, we will show that this new model, using three material parameters, quantitatively matches the flow and stress data from over 160 experiments in several different families of geometries.  We close the talk by showing how the same model can be used to reconcile certain "strange" features of granular media that have been documented in the literature, such as the observation that thinner granular layers behave as if they are stronger, and the motion-induced "quicksand" effect wherein flow at one location reduces the yield limit everywhere.
 
Date: September 23, 2014
     Speaker: Carlos Castillo-Chavez (Arizona State University)
     Title: Dispersal, Epidemics and Disease Evolution: Challenges and Opportunities in Computational, Mathematical and Theoretical Epidemiology
Abstract: I will briefly trace the history of the field of mathematical epidemiology with emphasis on the transmission and evolution of influenza. Emphasis will be placed on the role of multiple time scales and the study of long-term versus short-term dynamics. The role of dispersal will be briefly addressed and its relation to diseases that include Leprosy and Ebola. This presentation will be based on the work of former students and collaborators including Ana Luz Vivas Barber, Edgar Diaz and others.