Georgia Institute of Technology College of Sciences

This pre-talk will be an introduction to infinite-type surfaces and big mapping class groups. I will have a prepared talk, but it will be extremely informal, and I am more than happy to take scenic diversions if the audience so desires!

In the world of finite-type surfaces, one of the key tools to studying the mapping class group is to study its action on the curve graph. The curve graph is a combinatorial object intrinsic to the surface, and its appeal lies in the fact that it is infinite-diameter and $\delta$-hyperbolic. For infinite-type surfaces, the curve graph disappointingly has diameter 2. However, all hope is not lost! In this talk I will introduce the omnipresent arc graph and we will see that for a large collection of infinite-type surfaces, the graph is infinite-diameter and $\delta$-hyperbolic. The talk will feature a new characterization of infinite-type surfaces, which provided the impetus for this project.

This is joint work with Federica Fanoni and Alan McLeay

A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property H, the speed of H is the function which sends an integer n to the number of distinct elements in H with underlying set {1,...,n}. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete "jumps" in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollobás, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss. This is joint work with Chris Laskowski.

Dynamical sampling is a framework for studying the sampling and reconstruction problems for vectors that evolve under the action of a linear operator. In the first part of the talk I will review a few specific problems that have been part of the framework or motivated by it. In the second part of the talk I will concentrate on the problem of recovering a burst-like forcing term in an initial value problem for an abstract first order differential equation on a Hilbert space. We will see how the ideas of dynamical sampling lead to algorithms that allow one to stably and accurately approximate the burst-like portion of a forcing term as long as the background portion is sufficiently smooth. 

After reviewing classical results about existence of completions of varieties, I will talk about a class of degenerations of toric varieties which have a combinatorial classification - normal toric varieties over rank one valuation rings. I will then discuss recent results about the existence of equivariant completions of such degenerations. In particular, I will show a result from my thesis about the existence of normal equivariant completions of these degenerations.

BlueJeans link: https://bluejeans.com/909590858?src=join_info

In 1980, Lennart Carleson introduced the following problem for the free Schrödinger equation: when does the solution converge to the initial datum pointwise almost everywhere? Of course, the answer is immediate for regular functions like Schwartz functions. However, the question of what Sobolev regularity is necessary and sufficient for convergence turned out to be highly non-trivial. After the work of many people, it has been solved in 2019, following important advances in harmonic analysis. But interesting variations of the problem are still open. For instance, what happens with periodic solutions in the torus? And what if we refine the almost everywhere convergence to convergence with respect to fractal Hausdorff measures? Together with Renato Lucà (BCAM, Spain), we tackled these two questions. In the talk, I will present our results after explaining the basics of the problem.

Evolutionary dynamics permeates life and life-like systems. Mathematical methods can be used to study evolutionary processes, such as selection, mutation, and drift, and to make sense of many phenomena in life sciences. I will present two very general types of evolutionary patterns, loss-of-function and gain-of-function mutations, and discuss scenarios of population dynamics  -- including stochastic tunneling and calculating the rate of evolution. I will also talk about evolution in random environments.  The presence of temporal or spatial randomness significantly affects the competition dynamics in populations and gives rise to some counterintuitive observations. Applications include origins of cancer, passenger and driver mutations, and how aspirin might help prevent cancer.

Bluejeans Link: https://gatech.bluejeans.com/348270750

Aldous-Broder algorithm is a famous algorithm used to sample a uniform spanning tree of any finite connected graph G, but it is more general: it states that given a reversible M Markov chain on G started at r, the tree rooted at r formed by the steps of successive first entrance in each node (different from the root) has a probability proportional to $\prod_{e=(e1,e2)∈Edges(t,r)}M_{e1,e2}$ , where the edges are directed toward r. As stated, it allows to sample many distributions on the set of spanning trees. In this paper we extend Aldous-Broder theorem by dropping the reversibility condition on M. We highlight that the general statement we prove is not the same as the original one (but it coincides in the reversible case with that of Aldous and Broder). We prove this extension in two ways: an adaptation of the classical argument, which is purely probabilistic, and a new proof based on combinatorial arguments. On the way we introduce a new combinatorial object that we call the golf sequences.

Based on joint work with Luis Fredes, see https://arxiv.org/abs/2102.08639

We study braided embeddings, which is a natural generalization of closed braids in three dimensions. Braided embeddings give us an explicit way to construct lots of higher dimensional embeddings; and may turn out to be as instrumental in understanding higher dimensional embeddings as closed braids have been in understanding three and four dimensional topology. We will discuss two natural questions related to braided embeddings, the isotopy and lifting problem.