Georgia Institute of Technology College of Sciences

In 2000, Mike Stillman conjectured that the projective dimension of a homogeneous ideal in a standard graded polynomial ring can be bounded just in terms of the number and degrees of its generators. I’ll describe the Ananyan-Hochster principle important to its proof, how to package this up using ultraproducts, and use this to give a characterization of the polynomial rings graded by any abelian group that possess a Stillman bound.

Geometric mechanics is a tool for mathematically modeling the locomotion of animals or robots. In this talk I will focus on modeling the locomotion of a very simple robot. This modeling involves constructing a principal SE(2)-bundle with a connection. Within this bundle, the base space is parametrized by variables that are under the control of the robot (the so-called control variables). A loop in the base space gives rise to some holonomy in the fiber, which is an element of the group SE(2). We interpret this holonomy as the locomotion that is realized when the robot executes the path in the base space (control) variables.

Now, we can put a metric on the base space and ask the following natural question: What is the shortest path in the base space that gives rise to a fixed amount of locomotion? This is an extension of the isoperimetric problem to a principal bundle with a connection.

In this talk I will describe how to compute holonomy of the simple robot model, described above. Then I will solve the isoperimetric problem to find the shortest path with a fixed holonomy.

No prior knowledge of geometric mechanics will be assumed for this talk.

Let $S^{2d-1}$ be the unit sphere in $\mathbb{R}^{2d}$, and $\sigma_{2d-1}$ the normalized spherical measure in $S^{2d-1}$. The (scale t) bilinear spherical average is given by 
$$\mathcal{A}_{t}(f,g)(x):=\int_{S^{2d-1}}f(x-ty)g(x-tz)\,d\sigma_{2d-1}(y,z).$$
There are geometric motivations to study bounds for such bilinear spherical averages, in connection to the study of some Falconer distance problem variants. Sobolev smoothing bounds for the operator 
$$\mathcal{M}_{[1,2]}(f,g)(x)=\sup_{t\in [1,2]}|\mathcal{A}_{t}(f,g)(x)|$$
 are also relevant to get bounds for the bilinear spherical maximal function
$$\mathcal{M}(f,g)(x):=\sup_{t>0} |\mathcal{A}_{t}(f,g)(x)|.$$
In a joint work with B. Foster and Y. Ou, we put that in a general framework where $S^{2d-1}$ can be replaced by more general smooth surfaces in $\mathbb{R}^{2d}$, and one can allow more general dilation sets in the maximal functions: instead of supremum over $t>0$, the supremum can be taken over $t\in \tilde{E}$ where $\tilde{E}$ is the set of all scales obtained by dyadic dilation of fixed set of scales $E\subseteq [1,2]$.

A set in the Euclidean plane is called an integer distance set if the distance between any pair of its points is an integer.  All so-far-known integer distance sets have all but up to four of their points on a single line or circle; and it had long been suspected, going back to Erdős, that any integer distance set must be of this special form. In a recent work, joint with Marina Iliopoulou and Sarah Peluse, we developed a new approach to the problem, which enabled us to make the first progress towards confirming this suspicion.  In the talk, I will discuss the study of integer distance sets, its connections with other problems, and our new developments.
 

In recent years, significant progress has been made in our understanding of the quantitative behavior of random matrices. Such results include delocalization properties of eigenvectors and tail estimates for the smallest singular value. A key ingredient in their proofs is a 'distance theorem', which is a small ball estimate for the distance between a random vector and subspace.  Building on work of Livshyts and Livshyts, Tikhomirov and Vershynin, we introduce a new distance theorem for inhomogeneous vectors and subspaces spanned by the columns of an inhomogeneous matrix. Such a result has a number of applications for generalizing results about the quantitative behavior of i.i.d. matrices to matrices without any identical distribution assumptions. To highlight this, we show that the smallest singular value estimate of Rudelson and Vershynin, proven for i.i.d. subgaussian rectangular matrices, holds true for inhomogeneous and heavy-tailed matrices.
This talk is partially based on joint work with Max Dabagia.

As highly tunable platforms with exotic rich phase diagrams, moiré materials have captured the hearts and minds of physicists. Moiré materials arise when 2D crystal layers are stacked at relative twists. Their almost periodicity and multiscale behavior make these materials particularly mathematically appealing. We will describe the challenges in establishing a framework to study (phonon/vibrational) wave propagation in these materials, and explain how to overcome them.

A basic result of probabilistic combinatorics, originally due to Erdős and Rényi, is the determination of the threshold at which the random graph $G_{n,p}$ contains a triangle with high probability. But one can also ask more refined versions of this question, where we ask not just for one triangle but for many triangles which interact in complicated ways. For example, what is the threshold at which we can no longer partition $G_{n,p}$ into two triangle-free subgraphs?


In this talk, I will discuss the proof of the Kohayakawa–Kreuter conjecture, which gives a general answer to all such questions. Rather surprisingly, a key step of the proof is a purely deterministic graph decomposition statement, closely related to classical results such as Nash-Williams' tree decomposition theorem, whose proof uses techniques from combinatorial optimization and structural graph theory.

Based on joint works with Micha Christoph, Eden Kuperwasser, Anders Martinsson, Wojciech Samotij, and Raphael Steiner.