Georgia Institute of Technology College of Sciences

Three fifteen-minute talks by local speakers.

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.

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.

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.