Seminars and Colloquia by Series

Geometry of nodal sets of Laplace eigenfunctions

Series
School of Mathematics Colloquium
Time
Thursday, September 17, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/89107379948
Speaker
Alexander Logunov – Princeton University –

We will discuss geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interestingrelation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. Zero sets of Laplace eigenfunctions on surfaces are unions of smooth curves with equiangular intersections. The topology of the zero set could be quite complicated, but Yau conjectured that the total length of the zero set is comparable to the square root of the eigenvalue for all eigenfunctions. We will start with open questions about spherical harmonics and explain some methods to study nodal sets, which are zero sets of solutions of elliptic PDE. 

Zoom: https://us02web.zoom.us/j/89107379948

A tale of two polytopes: The bipermutahedron and the harmonic polytope

Series
School of Mathematics Colloquium
Time
Thursday, September 3, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/89107379948
Speaker
Federico Ardila – San Francisco State University –

This talk's recording is available here.

The harmonic polytope and the bipermutahedron are two related polytopes which arose in our work with Graham Denham and June Huh on the Lagrangian geometry of matroids. This talk will explain their geometric origin and discuss their algebraic and geometric combinatorics.

The bipermutahedron is a (2n−2)-dimensional polytope with (2n!)/2^n vertices and 3^n−3 facets. Its f-polynomial, which counts the faces of each dimension, is given by a simple evaluation of the three variable Rogers-Ramanujan function. Its h-polynomial, which gives the dimensions of the intersection cohomology of the associated topic variety, is real-rooted, so its coefficients are log-concave.

The harmonic polytope is a (2n−2)-dimensional polytope with (n!)^2(1+1/2+...+1/n) vertices and 3^n−3 facets. Its volume is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges; or equivalently, a weighted sum of the lattice point counts of all the corresponding trimmed generalized permutahedra.

These two polytopes are related by a surprising fact: in any dimension, the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.

The talk will be as self-contained as possible, and will feature joint work with Graham Denham, Laura Escobar, and June Huh.

A dynamic view on the probabilistic method: random graph processes

Series
School of Mathematics Colloquium
Time
Thursday, May 7, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://gatech.bluejeans.com/344615810
Speaker
Lutz Warnke – Georgia Tech

 Random graphs are the basic mathematical models for large-scale disordered networks in many different fields (e.g., physics, biology, sociology).
Since many real world networks evolve over time, it is natural to study various random graph processes which arise by adding edges (or vertices) step-by-step in some random way.

The analysis of such random processes typically brings together tools and techniques from seemingly different areas (combinatorial enumeration, differential equations, discrete martingales, branching processes, etc), with connections to the analysis of randomized algorithms.
Furthermore, such processes provide a systematic way to construct graphs with "surprising" properties, leading to some of the best known bounds in extremal combinatorics (Ramsey and Turan Theory).

In this talk I shall survey several random graph processes of interest (in the context of the probabilistic method), and give a glimpse of their analysis.
If time permits, we shall also illustrate one of the main proof techniques (the "differential equation method") using a simple toy example.

Cancelled

Series
School of Mathematics Colloquium
Time
Thursday, March 12, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Oscar Bruno – Caltech, Computing and Mathematical Sciences

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 Leok – UCSD

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 Sly – Princeton 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 Huh – Princeton 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 Rolfsen – UBC

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.  

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