Seminars and Colloquia Schedule

Bertini theorems, connectivity of tropical varieties, and multivariate Puiseux series

Series
Algebra Seminar
Time
Monday, March 11, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Josephine YuGeorgia Tech

A theorem of Bertini says that an irreducible algebraic variety remains irreducible after intersecting with a generic hyperplane.  We will discuss toric Bertini theorems for intersections with generic algebraic subtori (defined by generic binomial equations) instead of hyperplanes. As an application, we obtain a tropical Bertini theorem and a strengthening of the Structure Theorem of tropical algebraic geometry, by showing that irreducible tropical varieties remain connected through codimension one even after removing some facets.  As part of the proof of the Toric Bertini over prime characteristics, we constructed a new algebraically closed field containing the multivariate rational functions, which is smaller than previously known constructions.  This is based on joint works with Diane Maclagan, Francesca Gandini, Milena Hering, Fatemeh Mohammadi, Jenna Rajchgot, and Ashley Wheeler.

Intersections of balls and a no-dimensional Tverberg theorem

Series
Other Talks
Time
Monday, March 11, 2024 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alexander PolyanskiiEmory University

The aim of my talk is to discuss the following result, its variations and its connections with a no-dimensional Tverberg theorem. For any n red and n blue points in the Euclidean d-space, there exists a perfect red-blue matching M such that the balls whose diameters are edges of M share a common point.

(Joint works with O. Pirahmad, A. Vasilevskii, and P. Barabanshchikova.)

Spectral minimal partitions, nodal deficiency and the Dirichlet-to-Neumann map

Series
PDE Seminar
Time
Tuesday, March 12, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jeremy L. MarzuolaUniversity of North Carolina at Chapel Hill

The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to compute, or even estimate. Here we compare two recently obtained formulas for the nodal deficiency, one in terms of an energy function on the space of equipartitions of the manifold, and the other in terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. We relate these two approaches by giving an explicit formula for the Hessian of the equipartition energy in terms of the Dirichlet-to-Neumann map. This allows us to compute Hessian eigenfunctions, and hence directions of steepest descent, for the equipartition energy in terms of the corresponding Dirichlet-to-Neumann eigenfunctions. Our results do not assume bipartiteness, and hence are relevant to the study of spectral minimal partitions.  This is joint work with Greg Berkolaiko, Yaiza Canzani and Graham Cox.

Convergence times for random walks on the unitary group

Series
Math Physics Seminar
Time
Wednesday, March 13, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Shivan MittalDepartment of Physics, The University of Texas at Austin

Available online at: https://gatech.zoom.us/j/98258240051

Consider the following question of interest to cryptographers: A message is encoded in a binary string of length n. Consider a set of scrambling operations S (a proper subset of permutations on n bits). If a scrambling operation is applied uniformly at random from S at each step, then after how many steps will the composition of scrambling operations look like a random permutation on all the bits? This question asks for the convergence time for a random walk on the permutation group. Replace the binary string with a quantum state and scrambling operations in S with Haar random quantum channels on two bits (qudits) at a time. Broadly speaking, we study the following question: If a scrambling operation is applied uniformly at random from S at each step, then after how many steps will the composition of scrambling operations (quantum channels) look like a Haar random channel on all qudits? This question asks about the convergence time for a random walk on the unitary group. Various protocols in quantum computing require Haar random channels, which motivates us to understand the number of operations one would require to approximately implement that channel.

More specifically, in our study, we add a connected-graph structure to scrambling operations (a step on the random walk), where qudits are identified by vertices and the allowed 2-qudit scrambling operations are represented by edges. We develop new methods for lower bounds on spectral gaps of a class of Hamiltonians and use those to derive bounds on the convergence times of the aforementioned random walk on the unitary group with the imposed graph structure. We identify a large family of graphs for which O(poly(n)) steps suffice and show that for an arbitrary connected graph O(n^(O(log(n))) steps suffice. Further we refute the conjectured O(n log(n)) steps for a family of graphs.

Virtual Knot Theory and the Jones Polynomial

Series
Geometry Topology Student Seminar
Time
Wednesday, March 13, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jacob GuyneeGeorgia Tech

Virtual knot theory is a variant of classical knot theory in which one allows a new type of crossing called a "virtual" crossing. It was originally developed by Louis Kauffman in order to study the Jones polynomial but has since developed into its own field and has genuine significance in low dimensional topology. One notable interpretation is that virtual knots are equivalent to knots in thickened surfaces. In this talk we'll introduce virtual knots and show why they are a natural extension of classical knots. We will then discuss what virtual knot theory can tell us about the both the classical Jones polynomial and its potential extensions to knots in arbitrary 3-manifolds. An important tool we will use throughout the talk is the knot quandle, a classical knot invariant which is complete up to taking mirror images.

Optimal transport map estimation in general function spaces

Series
Stochastics Seminar
Time
Thursday, March 14, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan Niles-WeedNew York University

We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure P satisfy a Poincare inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for Holder transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when P is the normal distribution, between log-smooth and strongly log-concave distributions, and when the transport map is given by an infinite-width shallow neural network. (joint with Vincent Divol and Aram-Alexandre Pooladian.)

 

Riemannian geometry and contact topology II

Series
Geometry Topology Working Seminar
Time
Friday, March 15, 2024 - 14:00 for 2 hours
Location
Skiles 006
Speaker
John EtnyreGeorgia Tech

This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them. There will be many open problems discussed (probably later in the series). 

Colorful Borsuk--Ulam Theorems (Zoe Wellner, CMU)

Series
Combinatorics Seminar
Time
Friday, March 15, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Speaker
Zoe WellnerCarnegie Mellon University

 The classical Borsuk--Ulam theorem states that for any continuous map  from the sphere to Euclidean space, $f\colon S^d\to R^d$, there is a pair of antipodal points that are identified, so $f(x)=f(-x)$. We prove a colorful generalization of the Borsuk–Ulam theorem. The classical result has many applications and consequences for combinatorics and discrete geometry and we in turn prove colorful generalizations of these consequences such as the colorful ham sandwich theorem, which allows us to prove a recent result of B\'{a}r\'{a}ny, Hubard, and Jer\'{o}nimo on well-separated measures as a special case, and Brouwer's fixed point theorem, which allows us to prove an alternative between KKM-covering results and Radon partition results.  This is joint work with Florian Frick.

Enhanced diffusion for time-periodic alternating shear flows

Series
CDSNS Colloquium
Time
Friday, March 15, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Kyle LissDuke

The dynamics of a passive scalar, such as temperature or concentration, transported by an incompressible flow can be modeled by the advection-diffusion equation. Advection often results in the formation of complicated, small-scale structures and can result in solutions relaxing to equilibrium at a rate much faster than the corresponding heat equation in regimes of weak diffusion. This phenomenon is typically referred to as enhanced diffusion. In this talk, I will discuss a joint work with Tarek Elgindi and Jonathan Mattingly in which we construct an example of a divergence-free velocity field on the two-dimensional torus that results in optimal enhanced diffusion.  The flow consists of time-periodic, alternating piece-wise linear shear flows. The proof is based on the probabilistic representation formula for the advection-diffusion equation, a discrete time approximation, and ideas from hyperbolic dynamics.