Seminars and Colloquia by Series

The Complexity of Random Functions of Many Variables II

Series
School of Mathematics Colloquium
Time
Thursday, September 1, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gérard Ben Arous – Courant Institute, NYU

Please Note: Link to the Stelson Lecture announcement http://www.math.gatech.edu/news/stelson-lecture-dr-g-rard-ben-arous

This Colloquium will be Part II of the Stelson Lecture. A function of many variables, when chosen at random, is typically very complex. It has an exponentially large number of local minima or maxima, or critical points. It defines a very complex landscape, the topology of its level lines (for instance their Euler characteristic) is surprisingly complex. This complex picture is valid even in very simple cases, for random homogeneous polynomials of degree p larger than 2. This has important consequences. For instance trying to find the minimum value of such a function may thus be very difficult. The mathematical tool suited to understand this complexity is the spectral theory of large random matrices. The classification of the different types of complexity has been understood for a few decades in the statistical physics of disordered media, and in particular spin-glasses, where the random functions may define the energy landscapes. It is also relevant in many other fields, including computer science and Machine learning. I will review recent work with collaborators in mathematics (A. Auffinger, J. Cerny) , statistical physics (C. Cammarota, G. Biroli, Y. Fyodorov, B. Khoruzenko), and computer science (Y. LeCun and his team at Facebook, A. Choromanska, L. Sagun among others), as well as recent work of E. Subag and E.Subag and O.Zeitouni.

Finding hyperbolic-like behavior in non-hyperbolic spaces

Series
School of Mathematics Colloquium
Time
Wednesday, June 8, 2016 - 15:30 for 1 hour (actually 50 minutes)
Location
Clary theater
Speaker
Ruth Charney – Brandeis University –
In the early '90s, Gromov introduced a notion of hyperbolicity for geodesic metric spaces. The study of groups of isometries of such spaces has been an underlying theme in much of the work in geometric group theory since that time. Many geodesic metric spaces, while not hyperbolic in the sense of Gromov, nonetheless display some hyperbolic-like behavior. I will discuss a new invariant, the Morse boundary of a space, which captures this behavior. (Joint work with Harold Sultan and Matt Cordes.)

The slicing problems for sections of proportional dimensions

Series
School of Mathematics Colloquium
Time
Thursday, April 14, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alexander Koldobskiy – University of Missouri, Columbia –
We consider the following problem. Does there exist an absolute constant C such that for every natural number n, every integer 1 \leq k \leq n, every origin-symmetric convex body L in R^n, and every measure \mu with non-negative even continuous density in R^n, \mu(L) \leq C^k \max_{H \in Gr_{n-k}} \mu(L \cap H}/|L|^{k/n}, where Gr_{n-k} is the Grassmannian of (n-k)-dimensional subspaces of R^n, and |L| stands for volume? This question is an extension to arbitrary measures (in place of volume) and to sections of arbitrary codimension k of the hyperplace conjecture of Bourgain, a major open problem in convex geometry. We show that the above inequality holds for arbitrary origin-symmetric convex bodies, all k and all \mu with C \sim \sqrt{n}, and with an absolute constant C for some special class of bodies, including unconditional bodies, unit balls of subspaces of L_p, and others. We also prove that for every \lambda \in (0,1) there exists a constant C = C(\lambda) so that the above inequality holds for every natural number, every origin-symmetric convex body L in R^n, every measure \mu with continuous density and the codimension of sections k \geq \lambda n. The latter result is new even in the case of volume. The proofs are based on a stability result for generalized intersections bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies.

On two dimensional gravity water waves with angled crests

Series
School of Mathematics Colloquium
Time
Thursday, March 17, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sijue Wu – University of Michigan –
In this talk, I will survey the recent understandings on the motion of water waves obtained via rigorous mathematical tools, this includes the evolution of smooth initial data and some typical singular behaviors. In particular, I will present our recently results on gravity water waves with angled crests.

Almost orthogonality in Fourier analysis: From singular integrals, to function spaces, to the structural coloration of biological tissues

Series
School of Mathematics Colloquium
Time
Thursday, March 10, 2016 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rodolfo Torres – University of Kansas
Decomposition techniques such as atomic, molecular, wavelet and wave-packet expansions provide a multi-scale refinement of Fourier analysis and exploit a rather simple concept: “waves with very different frequencies are almost invisible to each other”. Starting with the classical Calderon-Zygmund and Littlewood-Paley decompositions, many of these useful techniques have been developed around the study of singular integral operators. By breaking an operator or splitting the functions on which it acts into non-interacting almost orthogonal pieces, these tools capture subtle cancelations and quantify properties of an operator in terms of norm estimates in function spaces. This type of analysis has been used to study linear operators with tremendous success. More recently, similar decomposition techniques have been pushed to the analysis of new multilinear operators that arise in the study of (para) product-like operations, commutators, null-forms and other nonlinear functional expressions. In this talk we will present some of our contributions in the study of multilinear singular integrals, function spaces, and the analysis of nanostructure in biological tissues, not all immediately connected topics, yet all centered on some notion of almost orthogonality.

Closed geodesics on compact simply connected Finsler manifolds

Series
School of Mathematics Colloquium
Time
Tuesday, March 8, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Dr. Yiming Long – Nankai University
The closed geodesic problem is a classical topic of dynamical systems, differential geometry and variational analysis, which can be chased back at least to Poincar\'e. A famous conjecture claims the existence of infinitely many distinct closed geodesics on every compact Riemaniann manifold. But so far this is only proved for the 2-dimentional case. On the other hand, Riemannian metrics are quadratic reversible Finsler metrics, and the existence of at least one closed geodesic on every compact Finsler manifold is well-known because of the famous work of Lyusternik and Fet in 1951. In 1973 A. Katok constructed a family of remarkable Finsler metrics on every sphere $S^d$ which possesses precisely $2[(d+1)/2]$ distinct closed geodesics. In 2004, V. Bangert and the author proved the existence of at least $2$ distinct closed geodesics for every Finsler metric on $S^2$, and this multiplicity estimate on $S^2$ is sharp by Katok's example. Since this work, many new results on the multiplicity and stability of closed geodesics have been established. In this lecture, I shall give a survey on the study of closed geodesics on compact Finsler manifolds, including a brief history and results obtained in the last 10 years. Then I shall try to explain the most recent results we obtained for the multiplicity and stability of closed geodesics on compact simply connected Finsler manifolds, sketch the ideas of their proofs, and then propose some further open problems in this field.

Unitary representations of reductive Lie groups

Series
School of Mathematics Colloquium
Time
Thursday, March 3, 2016 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter Trapa – University of Utah
Unitary representations of Lie groups appear in many guises in mathematics: in harmonic analysis (as generalizations of classical Fourier analysis); in number theory (as spaces of modular and automorphic forms); in quantum mechanics (as "quantizations" of classical mechanical systems); and in many other places. They have been the subject of intense study for decades, but their classification has only recently emerged. Perhaps surprisingly, the classification has inspired connections with interesting geometric objects (equivariant mixed Hodge modules on flag varieties). These connections have made it possible to extend the classification scheme to other related settings. The purpose of this talk is to explain a little bit about the history and motivation behind the study of unitary representations and offer a few hints about the algebraic and geometric ideas which enter into their study. This is based on joint work with Adams, van Leeuwen, and Vogan.

Nuclear physics, random matrices and zeros of L-functions

Series
School of Mathematics Colloquium
Time
Thursday, March 3, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel Fiorilli – University of Ottawa –
While the fields named in the title seem unrelated, there is a strong link between them. This amazing connection came to life during a meeting between Freeman Dyson and Hugh Montgomery at the Institute for Advanced Study. Random matrices are now known to predict many number theoretical statistics, such as moments, low-lying zeros and correlations between zeros. The goal of this talk is to discuss this connection, focusing on number theory. We will cover both basic facts about the zeta functions and recent developments in this active area of research.

Geometric understanding and analysis of unstructured data

Series
School of Mathematics Colloquium
Time
Monday, February 29, 2016 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hongkai Zhao – University of California, Irvine
One of the simplest and most natural ways of representing geometry and information in three and higher dimensions is using point clouds, such as scanned 3D points for shape modeling and feature vectors viewed as points embedded in high dimensions for general data analysis. Geometric understanding and analysis of point cloud data poses many challenges since they are unstructured, for which a global mesh or parametrization is difficult if not impossible to obtain in practice. Moreover, the embedding is highly non-unique due to rigid and non-rigid transformations. In this talk, I will present some of our recent work on geometric understanding and analysis of point cloud data. I will first discuss a multi-scale method for non-rigid point cloud registration based on the Laplace-Beltrami eigenmap and optimal transport. The registration is defined in distribution sense which provides both generality and flexibility. If time permits I will also discuss solving geometric partial differential equations directly on point clouds and show how it can be used to “connect the dots” to extract intrinsic geometric information for the underlying manifold.

Dynamical systems and beyond

Series
School of Mathematics Colloquium
Time
Thursday, February 25, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stefan Siegmund – TU Dresden –
From theoretical to applied, we present curiosity driven research which goes beyond classical dynamical systems theory and (i) extend the notion of chaos to actions of topological semigroups, (ii) model how the human bone renews, (iii) study transient dynamics as it occurs e.g. in oceanography, (iv) understand how to protect houses from hurricane damage. The talk introduces concepts from topological dynamics, mathematical biology, entropy theory and mechanics.

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