Given a planar domain on the rectangular grid, how many ways are there of tiling it by dominos (that is, by 1x2 rectangles)? And how does a generic tiling of a given domain look like?

It turns out that these questions are related to the determinants-based formulas, and that likewise formulas appear in many similar situations. In this direction, one obtains the famous arctic circle theorem, describing the behaviour of a generic domino tiling of an aztec diamond, and a statement for the lozenges tilings on the hexagonal lattice, giving the shape of a corner of a cubic crystal.

Over the past century an effort to understand dimension and structure of the harmonic measure spanned many spectacular developments in Analysis and in Geometric Measure Theory. Uniform rectifiability emerged as a natural geometric condition, necessary and sufficient for classical estimates in harmonic analysis, boundedness of the harmonic Riesz transform in L^2, and, in the presence of some background topological assumptions, for suitable scale invariant estimates on harmonic functions. While many of geometric and analytic problems remain relevant in sets of higher co-dimension (e.g., a curve in $\RR^3$), the concept of the harmonic measure is notoriously missing. In this talk, we introduce a new notion of a "harmonic" measure, associated to a linear PDE, which serves the higher co-dimensional sets. We discuss its basic properties and give large strokes of the argument to prove that our measure is absolutely continuous with respect to the Hausdorff measure on Lipschitz graphs with small Lipschitz constant.

Many gauge field theories can be described using a multisymplectic Lagrangian formulation, where the configuration manifold is the space of Lorentzian metrics. Group-equivariant interpolation spaces are critical to the construction of geometric structure-preserving discretizations of such problems, since they can be used to construct a variational discretization that exhibits a discrete Noether's theorem. We approach this problem more generally, by considering interpolation spaces for functions taking values in a symmetric space --- a smooth manifold with an inversion symmetry about every point.

Key to our construction is the observation that every symmetric space can be realized as a homogeneous space whose cosets have canonical representatives by virtue of the generalized polar decomposition -- a generalization of the well-known factorization of a real nonsingular matrix into the product of a symmetric positive-definite matrix times an orthogonal matrix. By interpolating these canonical coset representatives, we derive a family of structure-preserving interpolation operators for symmetric space-valued functions. As applications, we construct interpolation operators for the space of Lorentzian metrics, the space of symmetric positive-definite matrices, and the Grassmannian. In the case of Lorentzian metrics, our interpolation operators provide a family of finite elements for numerical relativity that are frame-invariant and have signature which is guaranteed to be Lorentzian pointwise. We illustrate their potential utility by interpolating the Schwarzschild metric numerically.

Kontsevich-Witten tau-function and the Hodge tau-function
are generating functions for two types of intersection numbers on
moduli spaces of stable curves. Both of them are tau functions for the
KP hierarchy. In this talk, I will describe how to connect these two
tau-functions by differential operators belonging to the
$\widehat{GL(\infty)}$ group. Indeed, these two tau-functions can be
connected using Virasoro operators. This proves a conjecture posted by
Alexandrov. This is a joint work with Gehao Wang.

I will describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth- order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. I first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. I will then apply the general the theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. I will demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection and the square phase field crystal model – are carried out to verify the efficiency of the scheme.

This is joint work with W. Feng (UTK), A. Salgado (UTK), and C. Wang (UMassD).