Georgia Institute of Technology College of Sciences

I will briefly survey the theory of matroids with coefficients, which was introduced by Andreas Dress and Walter Wenzel in the 1980s and refined by the speaker and Nathan Bowler in 2016. This theory provides a unification of vector subspaces, matroids, valuated matroids, and oriented matroids. Then I will outline an intriguing connection between Lorentzian polynomials, as defined by Petter Brändén and June Huh, and matroids with coefficients.  The second part of the talk represents ongoing joint work with June Huh, Mario Kummer, and Oliver Lorscheid.

Thompson links are links arising from elements of the Thompson group. They were introduced by Vaughan Jones as part of his effort to construct a conformal field theory for every finite index subfactor. In this talk I will first talk about Jones' construction of Thompson links. Then I will talk about grid diagrams and introduce a notion of half grid diagrams to give an equivalent construction of Thompson links and further associate with each Thompson link a canonical Legendrian type. Lastly, I will talk about some applications about the maximal Thurston-Bennequin number and presentation of link group. This is joint work with Yangxiao Luo.

In this talk, I will give an introduction to the implicit boundary integral method based on the co-area formula and it provides a simple quadrature rule for boundary integral on general surfaces.  Then, I will focus on the application of solving the linearized Poisson Boltzmann equation, which is used to model the electric potential of protein molecules in a solvent. Near the singularity, I will briefly discuss the choices of regularization/correction and illustrate the effect of both cases. In the end, I will show the numerical analysis for the error estimate. 

We give an overview of the interplay between structural graph theory, first-order logic, and parameterized complexity. We focus on introducing the subject. Time permitting, one particular topic will be the neighborhood complexity of monadically stable graph classes. 

Paper link: https://arxiv.org/abs/2405.03549

Abstract: Generative modeling via stochastic processes has led to remarkable empirical results as well as to recent advances in their theoretical understanding. In principle, both space and time of the processes can be discrete or continuous. In this work, we study time-continuous Markov jump processes on discrete state spaces and investigate their correspondence to state-continuous diffusion processes given by SDEs. In particular, we revisit the Ehrenfest process, which converges to an Ornstein-Uhlenbeck process in the infinite state space limit. Likewise, we can show that the time-reversal of the Ehrenfest process converges to the time-reversed Ornstein-Uhlenbeck process. This observation bridges discrete and continuous state spaces and allows to carry over methods from one to the respective other setting. Additionally, we suggest an algorithm for training the time-reversal of Markov jump processes which relies on conditional expectations and can thus be directly related to denoising score matching. We demonstrate our methods in multiple convincing numerical experiments.

We will discuss a problem of estimation of functionals of the form $\tau_f(\Sigma):= {\rm tr} (f(\Sigma))$ of unknown covariance operator $\Sigma$ of a centered Gaussian random variable $X$ in a separable Hilbert space ${\mathbb H}$ based on i.i.d. observation $X_1,\dots, X_n$ of $X,$ where $f:{\mathbb R}\mapsto {\mathbb R}$ is a given function. A naive plug-in estimator $\tau_f(\hat \Sigma_n)$ based on the sample covariance operator $\hat \Sigma_n$ has a large bias and bias reduction methods are needed to construct estimators with better error rates. We develop estimators with reduced bias based on linear aggregation of several plug-in estimators with different sample sizes and obtain the error bounds for such estimators with explicit dependence on the sample size $n,$ the effective rank ${\bf r}(\Sigma)= \frac{tr(\Sigma)}{\|\Sigma\|}$ of covariance operator $\Sigma$ and the degree of smoothness of function $f.$

Three fifteen-minute talks by local speakers.

Equidistribution problems, originating from the classical works of Kronecker, Hardy and Weyl about equidistribution of sequences mod 1, are of major interest in modern number theory. 

We will discuss how some of those problems relate to unipotent flows and present a conjecture by Margulis, Sarnak and Shah regarding an analogue of these results for the case of the horocyclic flow over a Riemann surface. Moreover, we provide evidence towards this conjecture by bounding from above the Hausdorff dimension of the set of points which do not equidistribute.

The talk will be accessible, no prior knowledge is assumed.