we'll give multiple approaches to the problem of representing an integer as the sum of two squares. with this concrete motivation, we'll see examples of some important objects and theorems from the past 200+ years of number theory.

bi-free probability is a generalization of free probability to study pairs of left and right faces in a non-commutative probability space. in this talk, i will demonstrate a characterization of bi-free independence inspired by the ``vanishing of alternating centred moments'' condition from free probability. i will also show how these ideas can be used to introduce a bi-free unitary brownian motion and a liberation process which asymptotically creates bi-free independence.

the number of standard tableaux of a given shape and major index $r$ mod $n$ give the irreducible multiplicities of certain induced or restricted representations. we give simple necessary and sufficient conditions classifying when this number is zero. this result generalizes the $r=1$ case due essentially to klyachko (1974) and proves a recent conjecture due to sundaram (2016) for the $r=0$ case. indeed, we prove a stronger asymptotic uniform distribution result for ``almost all'' shapes.

we'll discuss aspects of the proof, including a representation-theoretic formula due to desarmenien, normalized symmetric group character estimates due to fomin-lulov, and new techniques involving ``opposite hook lengths'' for classifying $\lambda \vdash n$ where $f^\lambda \leq n^d$ for fixed $d$.

the solution set of a linear matrix inequality (lmi) is known as a spectrahedron. free spectrahedra, obtained by substituting matrix tuples instead of scalar tuples into an lmi, arise canonically in the theory of operator algebras, systems and spaces and the theory of matrix convex sets. indeed, free spectrahedra are the prototypical examples of matrix convex sets, set with are closed with respect to taking matrix convex combinations. they also appear in systems engineering, particularly in problems governed by a signal flow diagram.

extreme points are an important topic in convexity; they lie on the boundary of a convex set and capture
many of its properties. for matrix convex sets, it is natural to consider matrix analogs of the notion of an extreme point. these notions include, in increasing order of strength, euclidean extreme points, matrix extreme points, and arveson boundary points. this talk will, in the context of matrix convex sets over $\mathbb{r}^g$, provide geometric unified interpretations of euclidean extreme points, matrix extreme points, and arveson boundary points. additionally, methods for computing arveson boundary points of free spectrahedra will be discussed.

in this talk, a representation of local and regular dirichlet forms on real line, which are associated with symmetric linear diffusions, will be given and based on this, several related problems will be discussed.

in this paper, we propose a new weighted square-root lasso procedure to estimate the regression coefficient matrix in sparse multivariate regression model with high-dimensional responses. the key advantage of the methodology is that it does not require the knowledge of the error term and has the tuning-insensitive property. to account for the within-subject correlation between responses, we use a working precision matrix which can be easily obtained in practice. oracle inequalities of the estimators are derived. the performance of our proposed methodology is illustrated via extensive simulation studies. an application to dna methylation data is also provided.

i will survey recent progress aimed at understanding the tautological rings of the moduli spaces of curves and k3 surfaces.

a del pezzo surface of degree $d$ over a finite field of size $q$ has at most $q^2+(10-d)q+1$ $\mathbb{f}_q$-rational points. a surface attaining this maximum is called ‘split’, and if all of these rational points lie on the exceptional curves of the surface, then it is called ‘full’. can we count and classify these extremal surfaces? we focus on del pezzo surfaces of degree 3, cubic surfaces, and of degree 2, double covers of the projective plane branched over a quartic curve. we will see connections to the geometry of bitangents of plane quartics, counting formulas for points in general position, and error-correcting codes.

eikonal equations arise in the fields of computer vision, image processing, geoscience, seismic tomography, to name a few. in some applications, the equation needs to be solved on a billion-point grid, and for tens of thousand times. in this talk, i will first introduce the most popular fast marching method (fmm) by sethian in 1996 and fast sweeping method (fsm) by zhao in 2005. then i will briefly survey some modern variants and many parallelization techniques. in the last, i will describe a significant improvement when the algorithm is applied locally.

wadspurger has shown that the genuine automorphic cuspidal representations of the metaplectic cover s of $sl_2$ are divided naturally into packets, and that thse packets are indexed by the cuspidal automorphic representations of $pgl_2$. we construct packets of lambda-adic modular forms of half integral weight, indexed by lambda-adic forms on $pgl_2$. the elements of the lambda-adic packets are nonzero, but they have specializations that vanish, owing to a trivial zero phenomenon and the sign of a complex root number. this is in contrast to the usual trivial zero phenomenon which arises from the vanishing of a p-adic factor.