I will present a collection of my favorite equations in mathematical physics and discuss why they are interesting from my own perspective.

 In recent years there has been a considerable interest in questions regarding approximate homomorphisms between groups, going under the name of group stability. For our setting, a group G is said to be HS-stable if any approximate finite dimensional unitary representation of G is close to a true unitary representation of G, where proximity is measured by the (normalized) Hilbert-Schmidt norm. In the situation of amenable groups, this question can be translated into a finite dimensional approximation property of the character space of G, an object originating in harmonic analysis. We will present an analysis of the character space of B. H. Neumann groups, an uncountable family of Z-by-locally finite groups, and as a result deduce they are HS-stable. The analysis involves both character bounds of finite symmetric groups, as well as character theory of infinite symmetric groups.

  Mean-field game (MFG) systems offer a framework for modeling multi-agent dynamics, but unknown parameters pose challenges. In this work, we tackle an inverse problem, recovering MFG parameters from limited, noisy boundary observations. Despite the problem's ill-posed nature, we aim to efficiently retrieve these parameters to understand population dynamics. Our focus is on recovering running cost and interaction energy in MFG equations from boundary measurements. We formalize the problem as an constrained optimization problem with L1 regularization. We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method.  Numerical experiments illustrate the effectiveness and robustness of the algorithm. This is a joint work with Yat Tin Chow (UCR), Samy Wu Fung (Colorado School of Mines), Levon Nurbekyan (Emory), and Stanley J.
  Osher (
 UCLA).

Totally positive matrices are matrices in which each minor is positive. Lusztig extended the notion to reductive Lie groups. He also proved that specialization of elements of the dual canonical basis in representation theory of quantum groups at $q=1$ are totally non-negative polynomials. Thus, it is important to investigate classes of functions on matrices that are positive on totally positive matrices. I will discuss several sources of such functions. One has to do with multiplicative determinantal inequalities (joint work with M. Gekhtman). Another deals with certain partial sums of Plucker relations (joint work with P. K. Vishwakarma). The third source deals with majorizing monotonicity of symmetrized Fischer's products which are a natural generalization of Hadamard-Fischer inequalities. Majorizing monotonicity of symmetrized Fischer's products was already known for hermitian positive semidefinite case which brings additional motivation to verify if they hold for totally positive matrices as well (joint work with M. Skandera). The main tools we employed are network parametrization, Temperley-Lieb and monomial trace immanants.

Let G be an algebraic group over a local field. We will show that the image of G under an arbitrary continuous homomorphism into any (Hausdorff) topological group is closed if and only if the center of G is compact. We will show how mixing properties for unitary representations follow from this topological property.

The reflection coupling method on Riemannian manifolds is a powerful tool in the study of harmonic functions and elliptic operators. In this talk, we will provide an overview of some fundamental ideas in stochastic analysis and the coupling method. We will then focus on applying these ideas to the study of the fundamental gap problem on the sphere. Based on joint work with Gang Yang and Guofang Wei.

Mirror symmetry and the Breuil-Mezard Conjecture Abstract: The Breuil-Mezard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" that should govern congruences between mod p automorphic forms on a reductive group G. Most of the progress thus far has been concentrated on the case G = GL_2, which has several special features. I will talk about joint work with Bao Le Hung on a new approach to the Breuil-Mezard Conjecture, which applies for arbitrary groups (and in particular, in arbitrary rank). It is based on the intuition that the Breuil-Mezard conjecture is analogous to homological mirror symmetry.

Given a smooth surface in R^3, a classical question in differential geometry asks whether the surface can be continuously deformed through a smooth, nontrivial family of isometric surfaces. If such a family exists and does not arise from rigid motions of R^3, then the surface is said to be flexible. An old conjecture asserts that flexible, smooth closed surfaces do not exist. In this talk, we survey this question and the general uniqueness problem for isometric immersions. We then present new examples of flexible, smooth immersed cylinders in R^3 which are neither minimal nor developable. We conclude with a discussion of speculative approaches to the construction of flexible, smooth closed surfaces. These results are part of upcoming work with Andrew Sageman-Furnas.

Ricci flow is an important tool in geometric analysis. There have been remarkable topological applications of Ricci flow on closed manifolds, such as the Poincaré Conjecture resolved by Perelman, and the recent Generalized Smale Conjecture resolved by Bamler-Kleiner. In contrast, much less is known about the Ricci flow on open manifolds. Solitons produce self-similar Ricci flows, which often arise as singularity models. Collapsed singularities and solitons create additional difficulties for open manifolds. In this talk, I will survey some recent developments in Ricci flow on open manifolds. In particular, I will talk about the resolution of Hamilton's flying wing Conjecture, and the resulting collapsed steady solitons.

The explicit descriptions of low degree K3 surfaces lead to natural compactifications coming from geometric invariant theory (GIT) and Hodge theory. The relationship between these compactifications for degree two K3 surfaces was studied by Shah and Looijenga, and revisited by Laza and O’Grady, who also provided a conjectural description for the case of degree four K3 surfaces. I will discuss these results, as well as a verification of this conjectural picture using tools from K-moduli. This is joint work with Kristin DeVleming and Yuchen Liu.