i will introduce some recent progress towards understanding the scalability of markov chain monte carlo (mcmc) methods and their comparative advantage with respect to variational inference. i will discuss an optimization perspective on the infinite dimensional probability space, where mcmc leverages stochastic sample paths while variational inference projects the probabilities onto a finite dimensional parameter space. three ingredients will be the focus of this discussion: non-convexity, acceleration, and stochasticity. this line of work is motivated by epidemic prediction, where we need uncertainty quantification for credible predictions and informed decision making with complex models and evolving data.

in the 1980s, mahowald and kane used brown-gitler spectra to construct splittings of $bo \wedge bo$ and $l \wedge l$.these splittings helped make it feasible to do computations using the $bo$- and $l$-based adams spectral sequences.i will discuss progress towards an analogous splitting for $bp\langle 2 \rangle \wedge bp \langle 2 \rangle$ at odd primes.

capillary surfaces model interfaces between incompressible immiscible fluids. the euler-lagrange equations for the capillary energy functional reveals that such surfaces are solutions of the prescribed mean curvature equation, with prescribed contact angle where the interface meets the container of the fluids. min-max methods have been used with great success to construct unstable critical points of various energy functionals, particularly for the special case of closed minimal surfaces. we will discuss the development of min-max methods to construct general capillary surfaces.

do you ever hear your number theorist friends say the word "perfectoid"? does it make you feel confused? afraid? if you have some vague idea that perfectoid spaces are an important concept, but have no idea what purpose they serve, then this talk is for you. we will attempt to describe how the theory of perfectoid spaces can be used to prove a simple statement in ring theory, which a priori has nothing to do with perfectoid spaces. we will assume familiarity with some basic notions regarding rings and modules at the level of math 200, but we will do our best to strip away all unnecessary jargon and communicate plainly the role of perfectoid geometry in the proof.

a group is said to have the infinite conjugacy class (icc) property if every non-identity element has an infinite conjugacy class. in this talk i will survey some ideas in geometric group theory, harmonic functions on groups, and topological dynamics and show how the icc property sheds light on these three seemingly distinct areas. in particular i will discuss when a group has only constant bounded harmonic functions, when every proximal dynamical system has a fixed point, and what this all has to do with the growth of a group. no prior knowledge of harmonic functions on groups or topological dynamics will be assumed.

this talk will include joint work with anna erschler, yair hartman, omer tamuz, and pooya vahidi ferdowsi.

we say that a finitely generated group is self-simulable if every action of the group on a zero-dimensional space which is effectively closed (this means it can be described by a turing machine in a specific way) is the topological factor of a subshift of finite type on said group. even though this seems like a property which is very hard to satisfy, we will show that these groups do exist and that their class is stable under commensurability and quasi-isometries of finitely presented groups. we shall present several examples of well-known groups which are self-simulable, such as thompson's v and higher-dimensional general linear groups. we shall also show that thompson's group f satisfies the property if and only if it is non-amenable, therefore giving a computability characterization of this well-known open problem. joint work with mathieu sablik and ville salo.

we use ideas from the theory of absolute concentration robustness to control a species of interest in a given chemical reaction network. the results are based on the network topology and the deficiency of the system, independent of reaction parameter values. the control holds in the stochastic regime and the quasistationary distribution of the controlled species is shown to be approximately poisson under a specific scaling limit.

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

the theory of local theta correspondence is built up from two main ingredients: a reductive dual pair inside a symplectic group, and a weil representation of its metaplectic cover. exceptional correspondences arise similarly: dual pairs inside exceptional groups can be constructed using so-called freudenthal jordan algebras, while the minimal representation provides a suitable replacement for the weil representation. the talk will begin by recalling these constructions. focusing on a particular dual pair, we will explain how one obtains howe duality for the correspondence in question. finally, we will discuss applications of these results. the new work in this talk is joint with gordan savin.

in both normal tissue and disease states, cells interact with one another, and other tissue components. these interactions are fundamental in determining tissue fates, and the outcomes of normal development, and cancer metastasis. i am interested in collective cell behaviours, which i view as swarms with a twist: (1) cells are not simply point-like particles but have spatial extent, (2) interactions between cells go beyond simple attraction-repulsion, and (3) cells “live” in a regime where friction dominates over inertia. examples include: wound healing, embryogenesis, the immune response, and cancer metastasis. in this seminar, i will give an overview of my computational, modelling, and theoretical contributions to tissue modelling at the sub-cellular, cellular, and population level.

in the first part, i focus on the nonlocal “armstrong adhesion model” (armstrong et al. 2006) for adhering tissue (an example of an aggregation-diffusion equation). since its introduction, this approach has proven popular in applications to embyonic development and cancer modeling. however many mathematical questions remain. combining global bifurcation results pioneered by rabinowitz, equivariant bifurcation theory, and the mathematical properties of the non-local term, we prove a global bifurcation result for the non-trivial solution branches of the scalar armstrong adhesion model. i will demonstrate how we used the equation’s symmetries to classify the solution branches by the nodal properties of the solution’s derivative.

in the second part, i focus on agent-based modelling of cell migration. small gtpases, such as rac and rho, are well known central regulators of cell morphology and motility, whose dynamics play a role in coordinating collective cell migration. experiments have shown gtpase dynamics to be affected by both spatio-temporally heterogeneous chemical and mechanical cues. while progress on understanding gtpase dynamics in single cells has been made, a major remaining challenge is to understand the role of gtpase heterogeneity in collective cell migration. motivated by recent one-dimensional experiments (e.g. microchannels) we introduce a one-dimensional modelling framework allowing us to integrate cell bio-mechanics, changes in cell size, and detailed intra-cellular signalling circuits (reaction-diffusion equations). we use numerical simulations, and analysis tools, such as bifurcation analysis, to provide insights into the regulatory mechanisms coordinating collective cell migration.

hilbert schemes of points on a smooth projective curve are simply symmetric powers of the curve itself; they are smooth and we know essentially everything about them. we propose a variation by studying double nested hilbert schemes of points, which parametrize flags of 0-dimensional subschemes satisfying certain nesting conditions dictated by young diagrams. these moduli spaces are almost never smooth but admit a virtual structure à la behrend-fantechi. we explain how this virtual structure plays a key role in (re)proving the correspondence between gromov-witten invariants and stable pair invariants for local curves, and say something on their k-theoretic refinement.