Seminars and Colloquia by Series

Symmetrization for functions of bounded mean oscillation

Series
School of Mathematics Colloquium
Time
Thursday, September 24, 2020 - 11:00 for
Location
https://us02web.zoom.us/j/89107379948
Speaker
Almut BurchardUniversity of Toronto

Spaces of bounded mean oscillation (BMO) are relatively
large function spaces that are often used in place
of L^\infinity to do basic Fourier analysis.
It is not well-understood how geometric properties
of the underlying point space enters into the functional
analysis of BMO.  I will describe recent work with
Galia Dafni and Ryan Gibara, where we take some
steps towards geometric inequalities.
Specifically, we show that the symmetric decreasing
rearrangement in n-dimensions is bounded, but not
continuous in BMO. The question of sharp bounds
remains open. 

 

 

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

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 LogunovPrinceton 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 ArdilaSan 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 WarnkeGeorgia 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.

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