We consider the problem of finding an elevated submatrix inside a large, noisy matrix. We are interested in both the detection problem (detecting the existence of the elevated submatrix) and the localization problem (find out the row and column index sets of the submatrix). Treating the elevated mean as the signal strength, we illustrate the fundamental signal boundaries of detection and localization, and propose estimators that reach the boundaries. The relationship of detection and localization problems will also be addressed.

I will describe some interesting selection problems for fully nonlinear, degenerate elliptic PDEs. In particular, I will focus on the vanishing discount procedure and show the convergence result via a new variational technique.

Abelian covers of finite volume hyperbolic manifolds are ubiquitous. We will discuss ergodic properties of the geodesic flow/ frame flow on such spaces. In particular, we will discuss the local mixing property of the geodesic flow/ frame flow, which we introduce to substitute the well-known strong mixing property in infinite volume setting. We will also discuss applications to measure classification problems and to counting and equidistribution problems. Part of the talk is based on the joint work with Hee Oh.

How complicated is a rational number? Its size is not a very good indicator for this. For instance, 1987985792837/1987985792836 is approximately 1, but so much more complicated than 1. We'll explain how to measure the complexity of a rational number using various notions of height. We'll then see how heights are used to prove some basic finiteness theorems in number theory.

We construct ``large'' Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct ``large'' Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function.

In this talk, we introduce the LRHB algorithm, which is an extension of the reduced-Hessian method of Gill and Leonard for unconstrained problems to problems with simple bound constraints. Numerical results for LRHB will be presented. We will also consider computational and practical issues with methods for nonlinear optimization and present results on a large test collection of problems indicating the reliability and efficiency of sequential quadratic programming methods and interior-point methods on certain classes of problems. This is joint work with Michael Ferry and Philip E. Gill.

In the pre-talk for graduate students, we will define discrete series representations and
give examples for $SL(2,\mathbb{R})$ and $GL(2,F)$, where $F$ is a local non-archimedean
field of characteristic $0$ with residue field of order not divisible by $2$. In the main talk,
first, we will review how the multiplicities of discrete series representations of
$SL(2,\mathbb{R})$ in $L^2(\Gamma \backslash SL(2,\mathbb{R}))$ are given by
dimensions of spaces of holomorphic cusp forms for $\Gamma$; we will take a look at
what happens if we try to use the formation of Poincar\'e series as an intertwiner; and we
will summarize some of the methods available for guaranteeing occurrence of discrete
series representations in $L^2(\Gamma \backslash G)$ when $G$ is a semisimple Lie
group other than $SL(2,\mathbb{R})$. Second, we will compute the product of the formal
dimension of two particular discrete series representations of $PGL(2,F)$ and the covolume
of a torsion-free lattice $\Gamma$ in $PGL(2,F)$ by dealing carefully with Haar measure
and applying standard facts from $\mathfrak{p}$-adic representation theory, thereby giving
the first explicit computation of multiplicities of those two discrete series representations in
$L^2(\Gamma \backslash PGL(2,F) )$; and we will say how the local Jacquet-Langlands
correspondence and the work of Corwin, Moy, and Sally could be used to carry out similar
calculations. (This material is part of our dissertation on representations of von Neumann
algebras coming from lattices in $SL(2,\mathbb{R})$ and $PGL(2,F)$.)

Continuous-time Markov chain models are often used to describe the stochastic dynamics of networks of reacting chemical species, especially in the growing field of systems biology. Discrete-event stochastic simulation of these models rapidly becomes computationally intensive. Consequently, more tractable diffusion approximations are commonly used in numerical computation, even for modest-sized networks. However, existing approximations (e.g., linear noise and Langevin), do not respect the constraint that chemical concentrations are never negative.

In this talk, we propose an approximation for such Markov chains, via reflected diffusion processes, that respects the fact that concentrations of chemical species are non-negative. This fixes a difficulty with Langevin approximations that they are frequently only valid until the boundary of the positive orthant is reached. Our approximation has the added advantage that it can be written down immediately from the chemical reactions. This contrasts with the linear noise approximation, which involves a two-stage procedure --- first solving a deterministic ordinary differential equation, followed by a stochastic differential equation for fluctuations around those solutions.

Under mild assumptions, we first prove that our proposed approximation is well defined for all time. Then we prove that it can be obtained as the weak limit of a sequence of jump-diffusion processes that behave like the Langevin approximation in the interior of the positive orthant and like a rescaled version of the Markov chain on the boundary of the orthant. For this limit theorem, we adapt an invariance principle for reflected diffusions, due to Kang and Williams, and modify a result on pathwise uniqueness for reflected diffusions, due to Dupuis and Ishii. Some numerical examples illustrate the advantages of our approximation over direct simulation of the Markov chain or use of the linear noise approximation.

Joint work with Saul Leite (Federal University of Juiz de Fora, Brazil), David Anderson (U. Wisconsin-Madison) and Des Higham (U. Strathclyde).

\def\R{ {\mathbb{R}} }
\def\bbS{ {\mathbb S}}

Let $M_n (\bbS)^g$ denote $g$-tuples of $n \times n$ complex self-adjoint matrices. Given tuples $X=(X_1, \dots, X_g) \in M_{n_1} (\bbS^g)$ and $Y=(Y_1, \dots, Y_g) \in M_{n_2}(\bbS)^g$, a matrix convex combination of $X$ and $Y$ is a sum of the form
\[
V_1^* XV_1+V_2^* Y V_2 \quad \quad \quad V_1^* V_1+V_2^* V_2=I_n
\]
where $V_1:M_{n} (\R) \to M_{n_1}$ and $V_2:M_n (\R) \to M_{n_2}$ are contractions. Matrix convex sets are sets which are closed under matrix convex combinations.

While in the classical setting there is only one good notion of an extreme point, there are three natural notions of extreme points for matrix convex sets: Euclidean, matrix, and absolute extreme points. A central goal in the theory of matrix convex sets is to determine if one of these notions of extreme points for matrix convex sets is minimal with respect to spanning.

Matrix extreme points are the most restricted type of extreme point known to span matrix convex sets; however, they are not necessarily the smallest set which does so. Absolute extreme points, a more restricted type of extreme points that are closely related to Arveson's boundary, enjoy a strong notion of minimality should they span. However, until recently it has been unknown if general matrix convex sets are spanned by their absolute extreme points.

This talk will give a class of closed bounded matrix convex sets which do not have absolute extreme points. The sets considered are noncommutative sets, $K_X$, formed by taking matrix convex combinations of a single tuple $X$. In the case that $X$ is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces, $K_X$ is a closed bounded matrix convex set with no absolute extreme points.

Markoff triples are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these, we will discuss joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular, that for almost all primes the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite. \\
Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.

The Ran space Ran(X) is the space of finite
subsets of X, topologized so that points can collide. Ran spaces have
been studied in diverse works from Borsuk-Ulam and Bott, to
Beilinson-Drinfeld, Gaitsgory-Lurie and others. The alpha form of
factorization homology takes as input a manifold or variety X together
with a suitable algebraic coefficient system A, and it outputs the
sheaf homology of Ran(X) with coefficients defined by A.
Factorization homology simultaneously generalizes singular homology,
Hochschild homology, and conformal blocks or observables in conformal
field theory. I'll discuss applications of this alpha form of
factorization homology in the study of mapping spaces in algebraic
topology, bundles on algebraic curves, and perturbative quantum field
theory. I'll also describe a beta form of factorization homology,
where one replaces Ran(X) with a moduli space of stratifications of X,
designed to overcome certain strict limitations of the alpha form. One
such application is to proving the Cobordism Hypothesis, after
Baez-Dolan, Costello, Hopkins-Lurie, and Lurie. This is joint work
with David Ayala.