Ricci solitons are generalizations of the Einstein manifolds and are self similar solutions to the Ricci flow. In particular, steady Ricci solitons are eternal solutions to the Ricci flow. In this talk, we will discuss the flying wing construction of some Kahler and Riemannian steady Ricci solitons of nonnegative curvature. This is based on joint work with Ronan Conlon and Yi Lai, as well as with Yi Lai and Man Chun Lee.

We review some recent results on the problem of reconstructing a variety from its topology.  This includes some recent work with Fanjun Meng and Lingyao Xie.

I will review recent constructions of oligomorphic tensor categories generalizing Deligne's Rep(S_t). Then, I will lean into the model theoretic part of the question. Specifically, I will explain where there are no continuous families like the original Rep(S_t) and where you should look for n-parameter families, i.e., depending on n free variables. Ultimately, these questions are closely related to classes of structures in model theory.

My talk will be about two open questions and a (perhaps surprising) link between them:

(1) Connes' rigidity conjecture, that in particular predicts that the von Neumann algebras of PSL_n(Z) are not isomorphic for different values of n. Ancient works with Vincent Lafforgue and Tim de Laat suggest a possible approach to it: does the non-commutative Lp space of the von Neumann of SL(n,Z) have the completely bounded approximation property for some non-trivial p?

(2) Kakeya conjecture : every subset of R^d containing a unit segment in every direction has dimension d.Both questions are open for large values of the parameters (n>2 and >3). I will explain why (1) is difficult: it implies some form of (2) for d<=(n+1)/2.

The magical ability of a spinning top to stay upright fascinated people for many thousands of years and in many civilizations. Clay spinning tops dated to about 6,000 years ago were excavated in Iraq.  And for all these millenia another similarly counterintuitive phenomenon went undiscovered until 1908, when Stephenson observed that an upside-down pendulum becomes stable if its pivot is subjected to vertical vibrations (there is no feedback involved). Later other counterintuitive effects were discovered – and used in physical experiments. Traditionally, this subject has been treated by formal computations, sometimes quite long. This is an effective practical tool, but it does not explain what is really going on.  As an alternative to this approach, I will give a geometrical explanation of the seemingly mysterious effect in which an inverted multiple pendulum stands upright when its pivot undergoes vertical vibrations.  I will also describe the recently discovered “ponderomotive Lorentz force”: a point mass in a rapidly oscillating potential force field behaves as if it were electrically charged and in the presence of magnetic field. This is a purely mathematical effect: there is no electricity or magnetism involved; superficially this looks like the Faraday effect in which a changing electric field generates a magnetic field.

In this work, we consider an inverse problem in mean-field games (MFG). We aim to recover the MFG model parameters that govern the underlying interactions among the population based on a limited set of noisy partial observations of the population dynamics under the limited aperture.  Due to its severe ill-posedness, obtaining a good quality reconstruction is very difficult.   Nonetheless, it is vital to recover the model parameters stably and efficiently in order to uncover the underlying causes for population dynamics for practical needs.

Our work focuses on the simultaneous recovery of running cost and interaction energy in the MFG equations from a finite number of boundary measurements of population profile and boundary movement.  To achieve this goal, we formalize the inverse problem as a constrained optimization problem of a least squares residual functional under suitable norms 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 Samy W. Fung (Colorado School of Mines), Siting Liu (UCR), Levon Nurbekyan (Emory University), and Stanley J. Osher (UCLA).

Free convolution is a fundamental operation in free probability. It expresses the distribution of the sum of two freely independent random variables in terms of the distributions of the summands. Compared to classical convolution of probability measures, free convolution is considerably more difficult to analyze and calculate. To untangle this complicated operation, we introduce a precise functional comparison between free and classical convolutions. This comparison states that the expectation of f w.r.t. classical convolution is larger than the expectation w.r.t. free convolution as long as f has non-negative fourth derivative. The comparison is based on the existence of convolution comparison measures, novel measures on the plane whose positivity depends on a peculiar identity involving Hermitian matrices.

In 1966, Mark Kac asked the famous question “Can one hear the shape of a drum?”
In his article with this question as the title, he translated it into eigenvalue problems for planar domains.
This question highlighted the relationship between eigenvalues and geometry.
One can then ask how eigenvalues are related to the geometry of the boundary.
In this talk, we consider a special type of eigenvalues, called Steklov eigenvalues, that are closely tied to boundary geometry.
We will introduce Steklov eigenvalues and explain their basic background and applications.
Then we will discuss our recent results on inequalities relating Steklov eigenvalues to the boundary area of compact manifolds.

This dissertation consists of four papers that develop computational and structural results in equivariant stable homotopy theory. The results include the computation of the reduced ring of the $RO(C_2)$-graded $C_2$-equivariant stable stems, the construction of the first family of $C_{p^n}$-equivariant ``$v_1$''-self maps, the computation of the $C_{p^n}$-equivariant Mahowald invariants of all elements in the Burnside ring, extending the classical computations of Bredon--Landweber and Iriye, and the computation of the spoke-graded $C_3$-equivariant stable stems.