Departmental Colloquium
Spring 2016
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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January 12 *Tuesday* |
Arul Shankar
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It is a classical result of Mordell that the rational points on an elliptic curve form a finitely generated abelian group. The rank of an elliptic curve is the rank of the corresponding finitely generated abelian group. A conjecture, due to Goldfeld and Katz-Sarnak, asserts that the average rank of elliptic curves over Q is equal to 1/2. However, it was previously not known that the average rank was even finite. We will prove that the average rank is finite, and in fact less than 1. This is joint work with Manjul Bhargava.
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January 14 |
Giulio Tiozzo
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Given a group of isometries of a metric space, one can draw a random sequence of group elements, and look at its action on the space. What are the asymptotic properties of such a random walk? The answer depends on the geometry of the space. Starting from Furstenberg, people considered random walks of this type, and in particular they focused on the case of spaces of negative curvature. On the other hand, several groups of interest in geometry and topology act on spaces which are not quite negatively curved (e.g., Teichmuller space) or on spaces which are hyperbolic, but not proper (such as the complex of curves). We shall explore some results on the geometric properties of such random walks. For instance, we shall see a multiplicative ergodic theorem for mapping classes (which proves a conjecture of Kaimanovich), as well as convergence and positive drift for random walks on general Gromov hyperbolic spaces. This also yields the identification of the measure-theoretic boundary with the topological boundary.
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January 21 |
Wenjia Jing
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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.
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January 26 *Tuesday* |
Giovanni Motta
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High-dimensional time series are the most common type of dataset in the "big data" revolution. They arise in many areas, including neuroscience and econometrics. If the number of series is large, Principal Components Analysis is a powerful tool to reduce the dimensionality of the series. In the traditional (stationary) framework the latent factors can be recovered by the principal components of the (time-invariant) spectral-density matrix of the multivariate time series. If the parameters of the process are time-varying, the process becomes non-stationary in time. Our previous approach for fitting dynamic non-stationary factor models to multivariate time series is based on the principal components of the estimated time-varying spectral-density matrix. This approach allows the spectral matrix to be smoothly time-varying, which imposes very little structure on the moments of the underlying process. However, the estimation delivers time-varying filters that are two-sided and thus unsuitable for prediction. Moreover, the estimation of the spectral matrix strongly depends on the chosen bandwidths for smoothing over frequency and time. As an alternative, we propose a new semi-parametric approach in which only part of the model is allowed to be time-varying. More precisely, the latent factors admit a dynamic representation with time-varying auto-regressive coefficients while the loadings are constant over time. Estimation of the model parameters is accomplished by application of the EM algorithm and the Kalman filter. The time-varying parameters are modeled locally by polynomials and estimated by maximizing the likelihood locally. Compared to estimation of the factors by principal components, our new approach produces superior results in particular for small cross-sectional dimension.
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February 2 *Tuesday* |
Sean Lawley
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Mathematics and biology enjoy a mutually beneficial relationship. On one side, mathematical tools yield deep insight into biological systems. On the other side, biological problems create fundamentally new questions in mathematics. In this talk, I will walk through a sampling of my own work to demonstrate both sides of this relationship, with an emphasis on how biology has led to new mathematical machinery and theorems. Biologically, I will discuss problems in neuroscience, biochemical reactions, and intracellular virus trafficking. Mathematically, I will describe my work in probability theory and stochastic processes, partial differential equations, asymptotic analysis, and homogenization.
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February 4 |
Francois Monard
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Given an unknown physical quantity to be imaged, the extent to which an imaging approach best exploits the mathematics and physics of the problem highly impacts (i) what can be reconstructed, and (ii) the available resolution of the resulting images. Improvements in both directions are of tremendous interest for medical applications, for example, and can be achieved by changing the underlying inverse problem into another one displaying better invertibility and conditioning. We will illustrate this strategy by discussing the mathematical analysis of inverse problems associated with two models. The first is the Boltzmann transport equation, a model for Optical Tomography and Single Photon Emission Computerized Tomography. The second is a class of elliptic partial differential equations with internal measurements, motivated by coupled-physics (or hybrid) medical imaging. In this context, the analysis of some models concerned with the reconstruction of conductivity and elasticity tensors has shown manifest improvements in both resolution and the capability of accessing previously unavailable anisotropic features.
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February 9 *Tuesday* |
Lizhen Lin
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While theoretically justified and computationally efficient point estimators were developed in robust estimation for many problems, robust Bayesian analogues are not sufficiently well-understood. We propose a novel approach to Bayesian analysis that is provably robust to the presence of outliers and contaminations in the data, and is computationally scalable to big data. Our approach is based on the idea of splitting the data into several non-overlapping subsets, evaluating the posterior distribution given each subset data, and then combining the resulting subset posterior measures by taking the geometric medians. The resulting final measure is called the median posterior which is the ultimate object used for inference. We show several strong theoretical results for the median posterior, including concentration rates and provable robustness. We illustrate and validate the method through experiments on simulated and real data. [Joint work with Stas Minker, Sanvesh Srivastava and David Dunson]
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February 16 Tuesday |
Ignacio Rodriguez-Brenes
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Normal somatic cells lose the ability to divide after a limited number of cell divisions. This phenomenon, known as replicative senescence, is an important barrier to tumor progression. Essentially all human cancers acquire mechanisms that allow them to escape replicative senescence, most often through high levels of expression of the enzyme telomerase. In this talk we will discuss mathematical models that quantify the protection offered by replicative senescence as a tumor suppressor mechanism. We will also discuss the emergence of mutants that escape replicative senescence and present results on the mean, variance, distribution, and asymptotic behavior of the mutant population. These results introduce the concept of incorporating replicative limits as part of the Luria-Delbrück mutational framework. We end by discussing applications to anti-telomerase cancer therapies.
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February 18 |
Shuyang Bai
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In time series analysis, the notion "long memory" stands for a slow hyperbolic decay of correlation as the time lag increases. Long memory has made surprisingly frequent appearances both in nature and in human life. In statistical inference, the sample sum plays a central role. Under long memory, the sample sum behaves drastically differently compared to the "short-memory" (e.g., independent) case. In this talk, we shall introduce a class of long-memory models given by nonlinear filters of white noise. We then show how their sample sums behave in terms of the probabilistic scaling limits. These results exhibit nicely how "long" the "memory" can be kept.
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February 23 |
Veniamin Morgenshtern
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This talk will consist of two parts. In the first part, I will review the simplest classical result on sparse signal recovery from incomplete data (compressed sensing). Underlying this result is the basic assumption that the columns of the measurment (design) matrix are only weakly correlated. This assumption is violated in many important applications. For example, in genetic data analysis, in super-resolution microscopy, and in radar imaging. In the second part of the talk, I will focus on super-resolution microscopy and on radar imaging and explain the conditions under which convex optimisation can solve the inverse problem in these cases. The analysis relies on novel constructions in Fourier analysis and on concentration inequalities for random matrices with specific structure. I will conclude with a brief discussion of a related open problem that is currently of great interest to me.
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March 3 |
Vitaly Bergelson
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A classical theorem due to H. Weyl states that if P is a real polynomial such that at least one of its coefficients (other than the constant term) is irrational, then the sequence P(n), n=1,2,... is uniformly distributed mod 1. After briefly reviewing various approaches to the proof of Weyl's theorem, we will discuss some recent extensions which involve "generalized polynomials", that is, functions which are obtained from the conventional polynomials by the use of the greatest integer function, addition and multiplication. We will explain the role of dynamical systems on nil-manifolds in obtaining these results and discuss the intrinsic connection between the generalized polynomials and the polynomial extensions of Szemeredi's theorem on arithmetic progressions. We will conclude with formulating and discussing some natural open problems and conjectures.
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March 10 |
Suncica Canic
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Fluid-structure interaction (FSI) problems arise in many applications. The widely known examples are aeroelasticity and biofluids. In biofluidic applications, including the interaction between blood flow and cardiovascular tissue, the coupling between the fluid and structure is highly nonlinear because the density of the structure (tissue) and the density of the fluid (blood) are roughly the same. In such problems, geometric nonlinearities of the fluid-structure interface and significant exchange in the energy between the moving fluid and structure play important roles in the physical and mathematical description of the underlying biological problem. The problems are further exacerbated by the fact that the walls of major arteries are composed of several layers, each with different mechanical characteristics. No mathematical results exist so far that analyze existence of solutions to fluid-structure interaction problems in which the structure is composed of several different layers. In this talk we summarize the main difficulties in studying the underlying problem, and present a computational scheme based on which the existence of a weak solution to this class of FSI problems with multi-layered structures was obtained. Our results reveal a new physical regularizing mechanism in FSI problems: inertia of the fluid-structure interface with mass regularizes evolution of the FSI solution. This means that in our large (muscular) arteries, the inner-most layer of arterial walls, which consists of an elastic lamiae covered with endothelial cells, smooths out the propagation of the pressure wave in the cardiovasuclar system. All theoretical results will be illustrated with numerical examples. This talk will be accessible to graduate students and to general mathematical/scientific audience. This is a joint work with Boris Muha (University of Zagreb, Croatia), and with Martina Bukac (Notre Dame University).
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March 24 |
Dick Canary
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Higher Teichmuller theory studies deformation spaces of ``geometric'' representations into higher rank Lie groups. One goal is to develop analogues of the rich geometric theory of Teichmuller spaces of Riemann surfaces. We will begin by discussing examples of higher Teichmuller spaces and attempt to give some flavor of the general theory. In the second half of the talk we will discuss a pressure metric on higher Teichmuller spaces which is a generalization of the classical Weil-Petersson metric on Teichmuller space. Its definition is dynamical in nature and is inspired by Thurston's reformulation of the Weil-Petersson metric and subsequent work by Wolpert, Bonahon, McMullen and Bridgeman. (This portion of the talk describes joint work with Bridgeman, Labourie and Sambarino.)
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March 31 |
Mike Wolf
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Some spaces of surface group representations into Lie groups may be related to geometric structures on a surface and to harmonic maps of surfaces into symmetric spaces. The latter provide for ways of imagining limits of degenerating representations as singular geometric objects such as R-trees and buildings. We focus on examples involving hyperbolic surfaces, affine spheres, convex real projective structures and 'opers', with a plan to explain all of the words. Joint work with David Dumas, John Loftin; we will also discuss work of Jorge Acosta.
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