Are you tired of graphs, paths, and flags on staffs? Have all those Dyck paths finally crossed the line? Are you ready to make a full binary tree and "leaf" the Catalan numbers behind? Then this is the talk for you! We are going to talk about concepts in combinatorics that have connections to areas of applied mathematics such as compressed sensing, quantization, and convex optimization; in particular, we're going to discuss some magnificent ways in which we can either solve (or approximately solve) certain problems in combinatorics by applying techniques from these areas. Conversely, we present cases where we can use ideas from combinatorics to prove results in applied math!

Let $q$ be a non-degenerate indefinite quadratic form over $ \mathbb{R}$
in $n \ge 3$ variables. Establishing a longstanding conjecture of Oppenheim, Margulis proved in 1986 that if $q$ is not a multiple of a rational form, then the set of values $q( \mathbb{Z}^n)$ is a dense subset of $ \mathbb{R}$.
Quantifying this result, Eskin, Margulis, and Mozes proved in 1986 that unless $q$ has signature $(2,1)$ or $(2,2)$, then the number $N(a,b;r)$ of integral vectors $v$ of norm at most $r$ satisfying $q(v) \in (a,b)$ has the asymptotic behavior $N(a,b;r) \sim \lambda(q) \cdot (b-a) r^{n-2}$.

Now, let $S$ is a finite set of places of $ \mathbb{Q}$ containing the Archimedean one, and $q=(q_v)_{v \in S}$
is an $S$-tuple of irrational isotropic quadratic forms over the completions $ \mathbb{Q}_v$. In this talk I will discuss the question of distribution of values of $q(v)$ as $v$ runes over $S$-balls in $ \mathbb{Z}[1/S]$.
This talk is based on a joint work with Seonhee Lim and Jiyoung Han.

The field of geometric numerical integration(GNI) seeks to exploit the
underlying (geometric)structure of a dynamical system in order to
construct numerical methods that exhibit desirable properties of
stability and/or preservation of invariants of the flow. Variational
Integrators are built for Hamiltonian systems by discretizing the
generating function of the symplectic flow, rather than discretizing
the differential equations directly. Traditionally, the generating
function considered is a type I generating function.
In this talk we will discuss the properties and
advantages/disadvantages of discretizing the type II/III generating
function of the flow. After establishing error analysis and adjoint
results, we consider the possible numerical resonance properties
corresponding to the different types of generating functions.

In Math 20, we learned how to differentiate and integrate functions defined on Euclidean spaces. There is a much wilder world of smooth spaces (manifolds) where a generalization of this calculus is possible, but it requires a steep learning curve and a lot of new language to understand. There is a class of manifolds, however, that is both large and interesting, and also retains enough Euclidean-like structure to do calculus almost the same way as in Math 20. These are called Lie groups.

I will discuss (with two or three guiding examples) how do to calculus on Lie groups, which can usually be realized as groups of square matrices. I will then discuss the most important differential equation in the world -- the heat equation -- in the context of matrix Lie groups, and the beautiful interplay between geometry and heat flow. Finally, I'll talk about my research into the heat flow of eigenvalues in matrix Lie groups -- and there'll be lots of cool pictures.

In this talk, we discuss results on gravitational perturbations of black holes by evaluating quasi-local mass on surfaces of fixed size at the null infinity in a gravitational radiation. In particular, a general theorem regarding the decay rate of the quasi-local energy-momentum at infinity is proved and is applied to study the gravitational perturbation of the Schwarzschild solution. The theorem associates a 4-vector to each loop near null infinity, which encodes the distinctive features of a gravitational wave.

Variational integrators provide a systematic way to derive geometric numerical methods for Lagrangian dynamical systems, which preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered dynamical system. Even though this is the case for a large class of systems, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type like they are often found in fluid dynamics or plasma physics.

We propose the application of the variational integrator method to so called formal Lagrangians, which allow us to embed any dynamical system into a Lagrangian system by doubling the number of variables. Thereby we are able to derive variational integrators for arbitrary systems, extending the applicability of the method significantly. A discrete version of the Noether theorem for formal Lagrangians yields the discrete momenta preserved by the resulting numerical schemes.

The theory is applied to dynamical systems from fluid dynamics and plasma physics like the vorticity equation, the Vlasov-Poisson system and magnetohydrodynamics, including numerical examples. Recent attempts of applying this method also to noncanonical Hamiltonian ODEs will be sketched.

In the theory of ``motives", algebraic cycles are central objects. For instance, the so-called ``motivic cohomology groups”, that give the universal bigraded ordinary cohomology on smooth varieties, are obtained from a complex of abelian groups consisting of certain algebraic cycles.

In this talk, we discuss how one can go beyond it, and we show that an infinitesimal version of the above complex of abelian groups of algebraic cycles can be identified with the big de Rham-Witt complexes after a suitable Zariski sheafification. This in a sense implies that the crystalline cohomology theory admits a description in terms of algebraic cycles, going back to a result of S. Bloch and L. Illusie in the 1970s. This is based on a joint work with Amalendu Krishna.

In symplectic manifolds, isotropic, coisotropic, and lagrangian submanifolds play a central role, and their study leads to deep problems in symplectic geometry and topology. It turns out that the linearized version of this study is already quite non-trivial.
The classification of pairs of isotropic subspaces in a symplectic vector space turns out to be rather simple, but for isotropic triples, it is much more complicated. In particular, there are families of inequivalent indecomposable isotropic triples depending on one parameter (but no more).

In these talks, I will report on progress on this problem in ongoing work with
Christian Herrmann (University of Dartmstadt) and Jonathan Lorand (University of Z\"urich).

In symplectic manifolds, isotropic, coisotropic, and lagrangian submanifolds play a central role, and their study leads to deep problems in symplectic geometry and topology. It turns out that the linearized version of this study is already quite non-trivial.
The classification of pairs of isotropic subspaces in a symplectic vector space turns out to be rather simple, but for isotropic triples, it is much more complicated. In particular, there are families of inequivalent indecomposable isotropic triples depending on one parameter (but no more).

In these talks, I will report on progress on this problem in ongoing work with
Christian Herrmann (University of Dartmstadt) and Jonathan Lorand (University of Z\"urich).

If R is the free associative algebra in two variables, say over the complex numbers, then the ring of two by two matrices over R is also a quotient of R by a differential ideal I. Playing off the two descriptions of R/I leads naturally to the Schwarzian Korteweg-DeVries equation, in which a function of x evolves in time driven by the Schwarzian derivative. I will present this abstract setup and then explain how ruled surfaces (appropriately chosen) over hyperelliptic curves provide solutions of the equation. I will also describe several commuting flows.