i will talk about a result that is motivated by the work of de simoi-kaloshin-wei. it concerns inverse spectral problems for strictly convex domains with one reflectional symmetry. the two key ingredients are wave trace formulas of guillemin-melrose and asymptotic of periodic billiard orbits of rotation numbers 1/q.

in this talk we will cover the topic of hamiltonian variational
integrators and a framework for adaptive symplectic integrators. first, i
will introduce variational integrators from the perspective of duality in
optimization. then a variety of results will be presented on hamiltonian
variational integrators and the role they can play in developing adaptive
symplectic integrators.

formal schemes are, for admissible rings, the analogues of schemes for
ordinary rings. formal schemes were originally introduced alongside
ordinary schemes in ega, largely as tools for understanding the latter.
while they served well in such applications, more recent developments,
particularly in rigid analytic geometry, have motivated a study of
formal schemes in their own right. in this talk i will discuss some
vanishing theorems for formal schemes as work towards establishing a
minimal model program for (some suitable class of) them.

tensors are multiway arrays, and tensor decompositions are powerful tools for data analysis and compression. in this talk, we demonstrate the wide-ranging utility of both the canonical polyadic (cp) and tucker tensor decompositions with examples in neuroscience, chemical detection, and combustion science. the cp model is extremely useful for interpretation, as we show with an example in neuroscience. however, it can be difficult to fit to real data for a variety of reasons. we present a novel randomized method for fitting the cp decomposition to dense data that is more scalable and robust than the standard techniques. the tucker model is useful for compression and can guarantee the accuracy of the approximation. we show that it can be used to compress massive data sets by orders of magnitude; this is done by determining the latent low-dimensional multilinear manifolds.

this talk features joint work with woody austin (university of texas), casey battaglino (georgia tech), grey ballard (wake forrest), alicia klinvex (sandia), hemanth kolla (sandia), and alex williams (stanford university).

a geometric flow is a way of evolving a geometry on a manifold.
the hope is that as time goes on, the geometry converges to something
recognizable. i will talk about what's known, and what's not known, for two
geometric flows in three dimensions. the first flow is the ricci flow, used by
perelman to prove thurston's geometrization conjecture. the second flow
is the einstein flow, which generates solutions of the vacuum einstein equations
on a four dimensional spacetime. no prior knowledge of ricci flow or the
einstein equations will be assumed.