moving boundary (or often called “free boundary”) problems are ubiquitous in nature and technology. a computational perspective of moving boundary problems can provide insight into the “invisible” properties of complex dynamics systems, advance the design of novel technologies, and improve the understanding of biological and chemical phenomena. however, challenges lie in the numerical study of moving boundary problems. examples include difficulties in solving pdes in irregular domains, handling moving boundaries efficiently and accurately, as well as computing efficiency difficulties. in this talk, i will discuss three specific topics of moving boundary problems, with applications to ecology (population dynamics), plasma physics (iter tokamak machine design), and cell biology (cell movement). in addition, some techniques of scientific computing will be discussed.

positroids are certain representable matroids originally studied by postnikov in connection with the totally nonnegative grassmannian and now used widely in algebraic combinatorics.  the positroids give rise to determinantal equations defining positroid varieties as subvarieties of the grassmannian variety. rietsch, knutson-lam-speyer, and pawlowski studied geometric and cohomological properties of these varieties.  in this talk, we continue the study of the geometric properties of positroid varieties by establishing several equivalent conditions characterizing smooth positroid varieties using a variation of pattern avoidance defined on decorated permutations, which are in bijection with positroids.  this allows us to give several formulas for counting the number of smooth positroids according to natural statistics on decorated permutations.  furthermore, we give a combinatorial method for determining the dimension of the tangent space of a positroid variety at the torus fixed points using an induced subgraph of the johnson graph.  we will conclude with some open problems in this area.

this talk is based on joint work with jordan weaver and christian krattenthaler.  

we investigate the analogs of optimal transport theory in the setting of multivariable asymptotic random matrix theory.  asymptotic random matrix theory concerns the behavior of randomly chosen $n \times n$ matrices in the limit as $n \to \infty$.  for several random matrices $x_1^{(n)}$, $\dots$, $x_d^{(n)}$, one can study the asymptotic behavior of expressions like $(1/n) \tr(x_{i_1} \dots x_{i_k})$, and the appropriate limiting object is a non-commutative probability space, that is, a von neumann algebra $a$ of "random variables" together with an expectation map $e: a \to \mathbb{c}$, analogous to the expected trace of a random matrix.  meanwhile, optimal transport theory asks for the most efficient way to rearrange one distribution of mass $\mu$ on $\mathbb{r}^d$ into another such distribution $\nu$.  such a scheme is often given by transporting the mass at point $x$ to point $f(y)$, for a smooth function $f: \mathbb{r}^d \to \mathbb{r}^d$.

optimal transport is more challenging to make sense of in the non-commutative setting because, unlike classical probability theory, there are many non-isomorphic atomless non-commutative probability spaces, and in fact, space of non-commutative probability distributions fails basic separability and finite-dimensional approximation properties that one is used to in classical probability.  so there is often no possibility of transporting given non-commutative probability distribution $\mu$ to $\nu$ by some map $f$; nonetheless, for the relaxed problem of optimal couplings, we can recover a non-commutative analog of the monge-kantorovich duality characterizing optimal couplings. furthermore, in the regime of convex free gibbs laws (an analog of smooth log-concave probability measures on $\mathbb{r}^d$), non-commutative transport can be achieved by non-commutative smooth functions obtained as solutions to differential equations much like the classical case.  moreover, the non-commutative analog of triangular transformations of measures led to new insight into the structure of the underlying von neumann algebras.

we address a system of equations modeling a compressible fluid interacting with an elastic body in dimension three. we prove the local existence and uniqueness of a strong solution when the initial velocity belongs to the space $h^{2+\epsilon}$ and the initial structure velocity is in $h^{1.5+\epsilon}$, where $\epsilon\in(0,1/2)$. 

the learn to optimize paradigm leverages machine learning to accelerate the discovery of new optimization methods. this method's core idea is to use a neural network to simulate the optimization process or provide critical decisions during the process to solve the optimization problem. this talk will introduce two recent research works in learning to optimization.

the first is a theoretical work discussing the application of graph neural networks (gnns) to linear and mixed integer programming. we prove that well-trained gnns can solve linear programs but not mixed integer programs without proper fixes. the other is constructing a fixed-point iterative neural network to solve inverse problems and game problems.

 a {\em permutation statistic} is a complex-valued function on the symmetric group $\mathfrak{s}_n$.  we describe a notion of `locality' which measures the complexity of permutation statistics. applications are given to the asymptotic behavior of families of statistics as the parameter $n$ grows. the key technical tool is an irreducible character evaluation on the symmetric group algebra which involves a novel combinatorics of `monotonic ribbon tableaux'. joint with zach hamaker.

bypassing the formalities of sigma-algebra and measures, i will show how one can see the lebesgue integration as an initial object of some category. the talk is based on a paper by tom leinster.

teichmüller curves are algebraic curves in the moduli space of genus $g$ riemann surfaces that are isometrically immersed for the teichmüller metric. these curves arise from $\mathrm{sl}(2,\mathbb{r})$-orbits of highly symmetric translation surfaces, and the underlying surfaces have remarkable dynamical and algebro-geometric properties. a teichmüller curve is algebraically primitive if the trace field of its affine symmetry group has degree $g$. in genus $2$, calta and mcmullen independently discovered an infinite family of algebraically primitive teichmüller curves. however, in higher genus, such curves seem to be much rarer. we will discuss a result that shows that the regular $14$-gon yields the unique algebraically primitive teichmüller curve in genus $3$ of a particular combinatorial type. all relevant notions will be explained during the talk.

asymptotically cylindrical flows are ancient solutions to the mean curvature flow whose tangent flow at $-\infty$ are shrinking cylinders. in this talk, we study quantized behavior of asymptotically cylindrical flows. we show that the cylindrical profile function u of these flows have the asymptotics $u(y,\omega, \tau) = \frac{y^{t}qy - 2tr q}{|\tau|} + o(|\tau|^{-1})$ as $\tau\rightarrow -\infty$, where $q$ is a constant symmetric $k\times k$ matrix whose eigenvalues are quantized to be either 0 or $-\frac{\sqrt{2(n-k)}}{4}$. assuming non-collapsing, we can further draw two applications. in the zero rank case, we obtain the full classification. in the full rank case, we obtain the $so(n-k+1)$ symmetry of the solution. this is joint work with wenkui du.

expected shortfall, measuring the average outcome (e.g., portfolio loss) above a given quantile
of its probability distribution, is a common financial risk measure. the same measure can be
used to characterize treatment effects in the tail of an outcome distribution, with applications
ranging from policy evaluation in economics and public health to biomedical investigations.
expected shortfall regression is a natural approach of modeling covariate-adjusted expected
shortfalls. because the expected shortfall cannot be written as a solution of an expected loss
function at the population level, computational as well as statistical challenges around
expected shortfall regression have led to stimulating research. we discuss some recent
developments in this area, with a focus on a new optimization-based semiparametric approach
to estimation of conditional expected shortfall that adapts well to data heterogeneity with
minimal model assumptions.

prismatic cohomology is a new p-adic cohomology theory introduced by bhatt and scholze that specializes to various well-known cohomology theories such as étale, de rham and crystalline. i will roughly recall the properties of this cohomology and explain how to prove its poincaré duality.

[pre-talk at 1:20pm]

the matrix completion problems are about completing a partially filled matrix to achieve the lowest possible rank. as they can be interpreted as an understanding of a certain algebraic variety, we consider a corresponding algebraic matroid and desire to characterize its bases. polyhedralizing via tropical algebra may help us to figure out this characterization. we begin the talk with brief introductions on matrix completion problems, algebraic matroids, and tropical algebra. no pre-knowledge is assumed. this is based on ongoing work with may cai, cvetelina hill, and josephine yu. 

for the past 90 years, a fundamental question in classical homotopy theory is to understand the stable homotopy groups of spheres.  the most modern method to study these groups is to compare them with the ``motivic stable homotopy groups of spheres".  motivic homotopy theory has its roots in algebraic geometry.  as a result of the recent advances, there is a reintegration of algebraic topology and algebraic geometry, with close connections to equivariant homotopy theory and number theory.

in this talk, i will introduce the classical and motivic stable homotopy categories and the connections between the two.  i will then talk about the rich properties and extra structures that are present in the motivic stable homotopy category.  the presence of these extra structures gives new computational tools that dramatically improve our understanding of the classical stable homotopy groups.  moreover, the flow of information can be reversed as well, producing new results in motivic stable homotopy theory for general fields.