During this talk I will introduce the notion of a k-AOU space, which we may think of as a matricial Archimedean order unit space. I will then describe the relationship between the category of k-AOU spaces and k-positive maps, and the category of operator systems and completely positive maps. After demonstrating the existence of injective envelopes and C*-envelopes in the category of k-AOU spaces, I will describe a connection with quantum correlations. Combined with previous work, this yields a reformulation of Tsirelson's conjecture. 

I will describe some recent progress in periodic homogenization of Hamilton-Jacobi equations. First, we show that the optimal rate of convergence is $O(\varepsilon)$ in the convex setting. We then give a minimalistic explanation that the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts in two dimensions. Joint works with Wenjia Jing and Yifeng Yu.

Most biochemical reactions in living cells are open system interacting with environment through chemostats. At a mesoscopic scale, the number of  each species in those biochemical reactions can be modeled by the random time-changed Poisson processes. To characterize the macroscopic behaviors in the large volume limit, the law of large number in path space determines a mean-field limit nonlinear Kurtz ODE, while the WKB expansion yields a Hamilton-Jacobi equation and the corresponding Lagrangian gives the good rate function in the large deviation principle. A parametric variation principle can be formulated to compute the reaction paths. We propose a gauge-symmetry criteria for a class of non-equilibrium chemical reactions including  enzyme reactions, which identifies a new concept of balance within the same reaction vector due to flux grouping degeneracy. With this criteria,  we (i) formulate an Onsager-type gradient flow structure in terms of the energy landscape given by a steady solution to the Hamilton-Jacobi equation;  (ii) find transition paths between multiple non-equilibrium steady states (rare events in biochemical reactions).  We illustrate this idea through a bistable catalysis  reaction. In contrast to the standard diffusion approximations via Kramers-Moyal expansion, a new drift-diffusion approximation sharing the same gauge-symmetry is constructed based on the Onsager-type gradient flow formulation to compute the correct energy barrier.

We construct ramified families of curves to explicitly model the Lubin-Tate action, the action of a formal group law on its deformation space, for a maximal finite subgroup $G$. We will see that as a $G$-representation, this deformation space is a quotient of a regular representation of a finite cyclic group! This allows us to partially compute the $E_2$ page of the homotopy fixed point spectral sequence of the $K(h, p)$-local homotopy groups of spheres for height $h=p^{k-1}(p-1)$, for all such $h$ and $p$ simultaneously. Thus, we resolve a 40 year old computational stalemate.

In this talk, I will introduce two modeling approaches for biomedical diseases, one is pathophysiology-driven modeling, the other one is data-driven modeling. The former one is used when the pathophysiology of such a disease is well known. As an example, a mathematical model of atherosclerosis, based on this modeling approach, provides a personalized cardiovascular risk by solving a free boundary problem. Some interesting mathematical problems are also introduced by this new model to help us understand cardiovascular risk. The second modeling approach is used to learn the mathematical model based on clinical data when the pathophysiology of a particular disease is not well understood. I will use Alzheimer's disease as an example to illustrate the idea of this modeling approach and apply it to personalized treatment studies of aducanumab, a recently FDA-approved Alzheimer's medication.
 

The point of topology is to study shapes--and these shapes tend to have points. However, points aren't actually that cool. We will develop a theory of shapes called locales, which is 100% point-free. That is, completely pointless. Then we'll say a few words about the Banach-Tarski Paradox.

We study the decomposition methods for solving a class of nonconvex and nonsmooth two-stage stochastic programs, where both the objective and constraints of the second-stage problem are nonlinearly parameterized by the first-stage variable.  Due to the failure of the Clarke-regularity of the resulting nonconvex recourse function, classical decomposition approaches such as Benders decomposition and (augmented) Lagrangian-based algorithms cannot be directly generalized to solve such models. By exploring an implicitly convex-concave structure of the recourse function, we introduce a novel surrogate decomposition framework based on the so-called partial Moreau envelope. Convergence for both fixed scenarios and interior sampling strategy is established. Numerical experiments are conducted to demonstrate the effectiveness of the proposed algorithm.

Detecting differences and building classifiers between distributions, given only finite samples, are important tasks in a number of scientific fields. Optimal transport (OT) has evolved as the most natural concept to measure the distance between distributions, and has gained significant importance in machine learning in recent years.  There are some drawbacks to OT: Computing OT can be slow, and it often fails to exploit reduced complexity in case the family of distributions is generated by simple group actions.  In this talk, we discuss how optimal transport embeddings can be used to deal with these issues, both on a theoretical and a computational level.  In particular, we’ll show how to embed the space of distributions into an $L^2$-space via OT, and how linear techniques can be used to classify families of distributions generated by simple group actions in any dimension. The proposed framework significantly reduces both the computational effort and the required training data in supervised settings. We demonstrate the benefits in pattern recognition tasks in imaging and provide some medical applications.

This talk is based on joint work with Alex Cloninger, Harish Kannan, Varun Khurana, and Jinjie Zhang.

I will report on joint work with Spencer Leslie where we define an analogue of the Cohen-Zagier Eisenstein series to the exceptional group $G_2$. Recall that the Cohen-Zagier Eisenstein series is a weight $3/2$ modular form whose Fourier coefficients see the class numbers of imaginary quadratic fields. We define a particular modular form of weight $1/2$ on $G_2$, and prove that its Fourier coefficients see (certain torsors for) the 2-torsion in the narrow class groups of totally real cubic fields. In particular:

1) we define a notion of modular forms of half-integral weight on certain exceptional groups,
2) we prove that these modular forms have a nice theory of Fourier coefficients, and
3) we partially compute the Fourier coefficients of a particular nice example on $G_2$.

The RAS protein network presents a unique situation in biology where all of the critical reactions are very well characterized both for the wild-type versions of the RAS proteins and for the cancer causing mutant forms of RAS.  We have previously shown that mathematical models that build up from reaction mechanisms can be used to make non-obvious and novel predictions about the behaviors of RAS mutants.  Recently, we have used these mathematical models to understand why some RAS mutations respond to drugs, when others do not.

The fermionic diagonal coinvariant ring was introduced by Rhoades and Jongwon Kim and is a quotient of a polynomial ring in two sets of $n$ anticommuting variables modulo $\mathfrak{S}_n$ invariant polynomials with no constant term, where the action of $\mathfrak{S}_n$ permutes both sets of variables simultaneously. In this talk, we will introduce a basis of this ring for which the action of $\mathfrak{S}_{n-1} \subset \mathfrak{S}_n$ can be interpreted combinatorially and use this basis to determine the isomorphism type of the ring. We will also relate our basis to a cyclic sieving result by Thiel.

Stoichiometric theory includes multiple biological scales from elements to ecosystems, and allows the construction of robust mechanistic, predictive, and empirically testable models via rigorous chemical and physical laws. Experimental and fundamental evidence motivates the application of this microscopic approach to understand macroscopic phenomena. I will introduce a series of stoichiometric models and their novel dynamics that resolve some biological paradoxes and lead to new insights. Selected new mathematical development will be briefly described. “True” model validation will be presented in contrast to conventional methods with many freedoms. I will briefly mention my recent expansion on a new graduate program and research of data science and machine learning.