We discuss results generalizing a result of Cordero-Erausquin and Klartag involving transport of log-concave measures to the free probabilistic setting. We also discuss open problems in extending it further.

We develop and analyze a numerical method proposed for solving high-dimensional Fokker-Planck equations by leveraging the generative models from deep learning. Our starting point is a formulation of the Fokker-Planck equation as a system of ordinary differential equations (ODEs) on finite-dimensional parameter space with the parameters inherited from generative models such as normalizing flows. We call such ODEs "neural parametric Fokker-Planck equations". The fact that the Fokker-Planck equation can be viewed as the 2-Wasserstein gradient flow of the relative entropy (also known as KL divergence) allows us to derive the ODE as the 2-Wasserstein gradient flow of the relative entropy constrained on the manifold of probability densities generated by neural networks. For numerical computation, we design a bi-level minimization scheme for the time discretization of the proposed ODE. Such an algorithm is sampling-based, which can readily handle computations in higher-dimensional space. Moreover, we establish bounds for the asymptotic convergence analysis as well as the error analysis for both the continuous and discrete schemes of the neural parametric Fokker-Planck equation. Several numerical examples are provided to illustrate the performance of the proposed algorithms and analysis.

Chromatic homotopy theory is a powerful tool to study periodic phenomena in the stable homotopy groups of spheres. Under this framework, the homotopy groups of spheres can be built from the fixed points of Lubin--Tate theories $E_h$.  These fixed points are computed via homotopy fixed points spectral sequences.  In this talk, we prove that at the prime 2, for all heights $h$ and all finite subgroups $G$ of the Morava stabilizer group, the $G$-homotopy fixed point spectral sequence of $E_h$ collapses after the $N(h,G)$-page and admits a horizontal vanishing line of filtration $N(h,G)$.

This vanishing result has proven to be computationally powerful, as demonstrated by Hill--Shi--Wang--Xu’s recent computation of $E_4^{hC_4}$.  Our proof uses new equivariant techniques developed by Hill--Hopkins--Ravenel in their solution of the Kervaire invariant one problem.  As an application, we extend Kitchloo--Wilson’s $E_n^{hC_2}$-orientation results to all $E_n^{hG}$-orientations at the prime 2. This is joint work with Zhipeng Duan and XiaoLin Danny Shi.
 

Key polynomials were introduced by Demazure for Weyl groups. They are non-symmetric generalizations of Schur polynomials, which are important in representation theory and geometry. Lascoux polynomials are K-theoretic analogues of key polynomials. In this talk, we describe several rules to compute key polynomials and Lascoux polynomials using tableaux. 

We discuss Kurt Heegner's work on the "class number 1 problem", and other fun stories like why ${e^{\pi\sqrt{163}}}$ is pretty much an integer

The concept of period growth was defined by Grigorchuk in the 80s, but still there are only a few examples of groups where we can estimate this invariant. We will sketch a connection to the Burnside problems and introduce a family of groups with very small period growth, answering a question by Bradford.

The defining feature of biological systems is their ability to replicate, which has at its foundation the process of cell division.  We are focused on understanding the inherent "two-ness" of cells and how accuracy and optimality are ensured during the trilliions of cell divisions that are needed to build and maintain multicellular organisms.  Our recent work highlights temporal optimization during cell division that is frequently disrupted in human cancers, highlighting a new type of tumor suppressor mechanism.

https://mathweb.ucsd.edu/~bli/research/mathbiosci/MBBseminar/

Branching Brownian motion (BBM) is a random particle system which incorporates both the tree-like structure and the diffusion process. BBM has a natural interpretation as a population model. In this talk, we will see how one variant model of BBM, BBM with an inhomogeneous branching rate can be used to study the evolution of populations undergoing selection. We will provide a mathematically rigorous justification for the biological observation that the distribution of the fitness levels of individuals in a population evolves over time like a traveling wave with a profile defined by the Airy function. This talk is based on joint work with Jason Schweinsberg.