Seminars and Colloquia by Series

The Connections Between Discrete Geometric Mechanics, Information Geometry and Machine Learning

Series
School of Mathematics Colloquium
Time
Thursday, March 5, 2020 - 11:00 for
Location
Speaker
Melvin LeokUCSD

Please Note: Melvin Leok is a professor in the Department of Mathematics at the University of California, San Diego. His research interests are in computational geometric mechanics, computational geometric control theory, discrete geometry, and structure-preserving numerical schemes, and particularly how these subjects relate to systems with symmetry. He received his Ph.D. in 2004 from the California Institute of Technology in Control and Dynamical Systems under the direction of Jerrold Marsden. He is a three-time NAS Kavli Frontiers of Science Fellow, and has received the NSF Faculty Early Career Development (CAREER) award, the SciCADE New Talent Prize, the SIAM Student Paper Prize, and the Leslie Fox Prize (second prize) in Numerical Analysis. He has given plenary talks at the Society for Natural Philosophy, Foundations of Computational Mathematics, NUMDIFF, and the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control. He serves on the editorial boards of the Journal of Nonlinear Science, the Journal of Geometric Mechanics, and the Journal of Computational Dynamics, and has served on the editorial boards of the SIAM Journal on Control and Optimization, and the LMS Journal of Computation and Mathematics.

Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-$h$ flow map of a Hamiltonian system on the space of probability distributions.

Replica Symmetry Breaking for Random Regular NAESAT

Series
School of Mathematics Colloquium
Time
Thursday, February 13, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Allan SlyPrinceton University

Ideas from physics have predicted a number of important properties of random constraint satisfaction problems such as the satisfiability threshold and the free energy (the exponential growth rate of the number of solutions).  Another prediction is the condensation regime where most of the solutions are contained in a small number of clusters and the overlap of two random solutions is concentrated on two points.  We establish this phenomena for the random regular NAESAT model.

Lorentzian polynomials

Series
School of Mathematics Colloquium
Time
Thursday, January 9, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
June HuhPrinceton University

Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions. Although no specific background beyond linear algebra and multivariable calculus will be needed to enjoy the presentation, I advertise the talk to people with interests in at least one of the following topics: graphs, convex bodies, stable polynomials, projective varieties, Potts model partition functions, tropicalizations, Schur polynomials, highest weight representations. Based on joint works with Petter Brändén, Christopher Eur, Jacob Matherne, Karola Mészáros, and Avery St. Dizier.

Ordered groups and n-dimensional dynamics

Series
School of Mathematics Colloquium
Time
Friday, December 6, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dale RolfsenUBC

A group is said to be torsion-free if it has no elements of finite order.  An example is the group, under composition, of self-homeomorphisms (continuous maps with continuous inverses) of the interval I = [0, 1] fixed on the boundary {0, 1}.  In fact this group has the stronger property of being left-orderable, meaning that the elements of the group can be ordered in a way that is nvariant under left-multiplication.  If one restricts to piecewise-linear (PL) homeomorphisms, there exists a two-sided (bi-)ordering, an even stronger property of groups.

I will discuss joint work with Danny Calegari concerning groups of homeomorphisms of the cube [0, 1]^n fixed on the boundary.  In the PL category, this group is left-orderable, but not bi-orderable, for all n>1.  Also I will report on recent work of James Hyde showing that left-orderability fails for n>1 in the topological category.  

A solution to the Burr-Erdos problems on Ramsey completeness

Series
School of Mathematics Colloquium
Time
Thursday, November 21, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jacob FoxStanford University

A sequence A of positive integers is r-Ramsey complete if for every r-coloring of A, every sufficiently large integer can be written as a sum of the elements of a monochromatic subsequence. Burr and Erdos proposed several open problems in 1985 on how sparse can an r-Ramsey complete sequence be and which polynomial sequences are r-Ramsey complete. Erdos later offered cash prizes for two of these problems. We prove a result which solves the problems of Burr and Erdos on Ramsey complete sequences. The proof uses tools from probability, combinatorics, and number theory. 

Joint work with David Conlon.

An inverse problems approach to some questions arising in harmonic analysis

Series
School of Mathematics Colloquium
Time
Tuesday, November 12, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Betsy StovallUniversity of Wisconsin

 One strategy for developing a proof of a claimed theorem is to start by understanding what a counter-example should look like.  In this talk, we will discuss a few recent results in harmonic analysis that utilize a quantitative version of this approach.  A key step is the solution of an inverse problem with the following flavor.  Let $T:X \to Y$ be a bounded linear operator and let $0 < a \leq \|T\|$.  What can we say about those functions $f \in X$ obeying the reverse inequality $\|Tf\|_Y \geq a\|f\|_X$?  

New invariants of homology cobordism

Series
School of Mathematics Colloquium
Time
Thursday, October 31, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kristen HendricksRutgers

This is a talk about 3-manifolds and knots. We will begin by reviewing some basic constructions and motivations in low-dimensional topology, and will then introduce the homology cobordism group, the group of 3-manifolds with the same homology as the 3-dimensional sphere up to a reasonable notion of equivalence. We will discuss what is known about the structure of this group and its connection to higher dimensional topology. We will then discuss some existing invariants of the homology cobordism group coming from gauge theory and symplectic geometry, particularly Floer theory. Finally, we will introduce a new invariant of homology cobordism coming from an equivariant version of the computationally-friendly Floer-theoretic 3-manifold invariant Heegaard Floer homology, and use it to construct a new filtration on the homology cobordism group and derive some structural applications. Parts of this talk are joint work with C. Manolescu and I. Zemke; more recent parts of this talk are joint work with J. Hom and T. Lidman.

Total Curvature and the isoperimetric inequality: Proving the Cartan-Hadamard conjecture

Series
School of Mathematics Colloquium
Time
Thursday, October 3, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mohammad GhomiGeorgia Institute of Technology

The classical isoperimetric inequality states that in Euclidean space spheres provide unique enclosures of least perimeter for any given volume. In this talk we discuss how this inequality may be extended to spaces of nonpositive curvature, known as Cartan-Hadamard manifolds, as conjectured by Aubin, Gromov, Burago, and Zalgaller in 1970s and 80s. The proposed proof is based on a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for the smooth approximation of the signed distance function, via inf-convolution and Reilly type formulas among other techniques. Immediate applications include sharp extensions of Sobolev and Faber-Krahn inequalities to spaces of nonpositive curvature. This is joint work with Joel Spruck.

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