I am interested in developing numerical methods for solving
PDEs (e.g. Einstein's equation) on manifolds with topology $\mathbb{R} \times M$, where $M$ is a three-dimensional manifold with arbitrary topology.  This talk will describe the basic methods we have developed for constructing convenient representations of these manifolds suitable for this numerical work, and some simple examples will be shown.  There won't be time in this talk to describe everything we have done, so I will focus on just one issue: how to construct $C^0$ reference metrics on these manifolds.  We now have methods that can construct such metrics automatically for a fairly large collection of manifolds.  Unfortunately, these methods fail in general, so improved methods are needed.

I will discuss my ongoing joint work with Ionut Chifan and Daniel Drimbe on various rigidity aspects of von Neumann algebras arising from graph product groups whose underlying graph is a certain cycle of cliques and whose vertex groups are wreath-like product property (T) groups. In particular, I will describe all symmetries of these von Neumann algebras by establishing formulas in the spirit of Genevois and Martin’s results on automorphisms of graph product groups. In doing so, I will highlight the methods used from Popa’s deformation/rigidity theory as well as new techniques pertaining to graph product algebras.

The Kostka semigroup consists of pairs of partitions with at most r parts that have a positive Kostka coefficient. For this semigroup, Hilbert basis membership is an NP-complete problem. We introduce KGR graphs and conservative subtrees, through the Gale-Ryser theorem on contingency tables, as a criterion for membership. In our main application, we show that if a partition pair is in the Hilbert basis then the partitions are at most $r$ wide. We also classify the extremal rays of the associated polyhedral cone; these rays correspond to a (strict) subset of the Hilbert basis. In an appendix, the second and third authors show that a natural extension of our main result on the Kostka semigroup cannot be extended to the Littlewood-Richardson semigroup. This furthermore gives a counterexample to recent speculation of P. Belkale concerning the semigroup controlling nonvanishing conformal blocks.

A pervading question in the study of stochastic PDE is how small-scale random forcing in an equation combines to create nontrivial statistical behavior on large spatial and temporal scales. I will discuss recent progress on this topic for several related stochastic PDEs - stochastic heat, KPZ, and Burgers equations - and some of their generalizations. These equations are (conjecturally) universal models of physical processes such as a polymer in a random environment, the growth of a random interface, branching Brownian motion, and the voter model. The large-scale behavior of solutions on large scales is complex, and in particular depends qualitatively on the dimension of the space. I will describe the phenomenology, and then describe several results and challenging problems on invariant measures, growth exponents, and limiting distributions.

 Inspired by work of Rogers in the classical geometry of numbers, we'll describe how to obtain variance bounds for classical geometric counting problems in the settings of translation surfaces and hyperbolic surfaces, and give some applications to understanding correlations between special trajectories on these types of surfaces. Parts of this will be joint work with Y. Cheung and H. Masur; S. Fairchild and H. Masur; and F. Arana-Herrera, and all of this has been inspired by joint work with G. Margulis.

In this talk we define the fundamental geometric constructions behind the generalized Kahler geometry introduced by Hitchin and Gualtier and set up an appropriate generalization of the Calabi problem. Similarly to Cao's approach to the solution of the classical Calabi problem, we study the existence and convergence of the generalized Kahler-Ricci flow (GKRF) on relevant backgrounds. In particular, we prove that on a Kahler Calabi-Yau background, the GKRF converges to the unique classical Ricci-Flat structure. This result has non-trivial applications to understanding the space of generalized Kahler structures, and as a special case yields the topological structure of natural classes of Hamiltonian symplectomorphisms on hyperKahler manifolds. Based on a joint work with Apostolov, Fu and Streets.

We study the capacity of the range of a simple random walk in three and higher dimensions. It is known that the order of the capacity of the random walk range in n dimensions is similar to that of the volume of the random walk range in n-2 dimensions. We show that this correspondence breaks down for the law of the iterated logarithm for the capacity of the random walk range in three dimensions. We also prove the law of the iterated logarithm in higher dimensions.

This is joint work with Amir Dembo. 

We will discuss Galois cohomology groups of “intermediate” quotients of an induced module, which sit between Iwasawa cohomology up a tower and cohomology over the ground field. Special elements in Iwasawa cohomology that arise from Euler systems become divisible by a certain Euler factor upon norming down to the ground field. In certain instances, there are reasons to wonder whether this divisibility can also hold for the image in intermediate cohomology. Using “intermediate” Coleman maps, we shall see that the situation locally at $p$ is as nice as one could imagine.

We will discuss Galois cohomology groups of “intermediate” quotients of an induced module, which sit between Iwasawa cohomology up a tower and cohomology over the ground field. Special elements in Iwasawa cohomology that arise from Euler systems become divisible by a certain Euler factor upon norming down to the ground field. In certain instances, there are reasons to wonder whether this divisibility can also hold for the image in intermediate cohomology. Using “intermediate” Coleman maps, we shall see that the situation locally at $p$ is as nice as one could imagine.

[pre-talk at 1:20PM]

Understanding the structure of conjugacy classes is essential in the study of a group. We will see how conjugacy classes of a group can be understood using group actions, and analyze the conjugacy classes for a variety of examples, including the group of symmetries of a tree and the group of almost symmetries of a tree, following a joint work with Waltraud Lederle.

For more than hundred years, various concepts were developed to understand the fields of geometric objects and invariant differential operators between them for conformal Riemannian, projective geometries, hypersurface type CR geometries, etc.. More recently, general tools were presented for the entire class of the so called parabolic geometries, i.e., the Cartan geometries modelled on homogeneous spaces G/P with P a parabolic subgroup in a semi-simple Lie group G. All these geometries determine a class of distinguished affine connections, which carry an affine structure modelled on differential 1-forms . They correspond to reductions of P to its reductive Levi factor, and we call them Weyl structures similarly to the conformal case. The standard definition of differential invariants in this setting is as affine invariants of these connections, which do not depend on the choice within the class. In the lecture, I shall describe a universal calculus which provides an important first step to determine such invariants. The lecture will follow the recent preprint https://arxiv.org/abs/2210.16652 with Andreas Cap, but I will try to stress the cases relevant for the CR structures.

This talk will feature an idiosyncratic take on an underlying theme in my research program, that topology and geometry in higher dimensions can be used in describing arithmetic phenomena in lower ones. I hope to explain why there might be such a phenomenon, while indicating how unexpectedly deep it appears to be. For instance, here’s an interesting question that doesn’t appear to have been much studied but ties in closely with joint work with Akshay Venkatesh: when do two integer polynomials in a single variable x that are products of powers of x and cyclotomic polynomials sum to a third? Curiously, the path towards an answer appears to intertwine with the homology of modular curves, as well as a chain complex computing the homology of a circle.

This talk will feature an idiosyncratic take on an underlying theme in my research program, that topology and geometry in higher dimensions can be used in describing arithmetic phenomena in lower ones. I hope to explain why there might be such a phenomenon, while indicating how unexpectedly deep it appears to be. For instance, here’s an interesting question that doesn’t appear to have been much studied but ties in closely with joint work with Akshay Venkatesh: when do two integer polynomials in a single variable x that are products of powers of x and cyclotomic polynomials sum to a third? Curiously, the path towards an answer appears to intertwine with the homology of modular curves, as well as a chain complex computing the homology of a circle.

Algebraic cycles on varieties are central objects in algebraic geometry and number theory. Problems around them are notoriously difficult. In the case of Shimura varieties, the study of modular forms whose coefficients are algebraic cycles and the closely related study of the automorphy of representations spanned by algebraic cycles are central to the advancement of knowledge in this area. I will discuss the background and history of these topics, as well as some recent progress and applications

Algebraic cycles on varieties are central objects in algebraic geometry and number theory. Problems around them are notoriously difficult. In the case of Shimura varieties, the study of modular forms whose coefficients are algebraic cycles and the closely related study of the automorphy of representations spanned by algebraic cycles are central to the advancement of knowledge in this area. I will discuss the background and history of these topics, as well as some recent progress and applications.