## Anna Skorobogatova : Area-minimizing currents: structure of singularities and uniqueness of tangent cones

- Geometry and Topology ( 28 Views )The problem of determining the size and structure of the interior singular set of area-minimizing surfaces has been studied thoroughly in a number of different frameworks, with many ground-breaking contributions. In the framework of integral currents, when the surface has higher codimension than 1, the presence of singular points with flat tangent cones creates an obstruction to easily understanding the interior singularities. Until recently, little was known in this direction, particularly for surfaces of dimension higher than two, beyond Almgren’s celebrated dimension estimate on the interior singular set. In this talk I will discuss joint works with Camillo De Lellis and Paul Minter, where we establish (m-2)-rectifiability of the interior singular set of an m-dimensional area-minimizing integral current and classify tangent cones at \mathcal{H}^{m-2}-a.e. interior point.

## Samit Dasgupta : Ribets Lemma and the Brumer-Stark Conjecture

- Number Theory ( 34 Views )In this talk I will describe my recent work with Mahesh Kakde on the Brumer-Stark Conjecture and certain refinements. I will give a broad overview that motivates the conjecture and gives connections to explicit class field theory. I will conclude with a description of recent work (joint w/ Kakde, Jesse Silliman, and Jiuya Wang) in which we complete the proof of the conjecture. Moreover, we deduce a certain special case of the Equivariant Tamagawa Number Conjecture, which has important corollaries. The key aspect of the most recent results, which allows us to handle the prime p=2, is the proof of a version of Ribet's Lemma in the case of characters that are congruent modulo p.

## Erik Bates : The Busemann process of (1+1)-dimensional directed polymers

- Probability ( 27 Views )Directed polymers are a statistical mechanics model for random growth. Their partition functions are solutions to a discrete stochastic heat equation. This talk will discuss the logarithmic derivatives of the partition functions, which are solutions to a discrete stochastic Burgers equation. Of interest is the success or failure of the “one force-one solution principle” for this equation. I will reframe this question in the language of polymers, and share some surprising results that follow. Based on joint work with Louis Fan and Timo Seppäläinen.

## Chun-Hsien Hsu : Weyl algebras on certain singular affine varieties

- Number Theory ( 45 Views )The module theory of the Weyl algebra, known as the theory of $D$-modules, has profound applications in various fields. One of the most famous results is the Riemann-Hilbert correspondence, establishing equivalence between holonomic $D$-modules and perverse sheaves on smooth complex varieties. However, when dealing with singular varieties, such correspondence breaks down due to the non-simplicity of Weyl algebras on singular varieties. In our ongoing work, we introduce a new ring of differential operators on certain singular affine varieties, whose definition is analytically derived from harmonic analysis. It should contain the Weyl algebra as a proper subring and shares many properties with the Weyl algebra on smooth varieties. In the talk, after a brief review of the Weyl algebra, I will explain how the new ring of differential operators arises as a consequence of an explicit form of the Poisson summation conjecture and discuss its properties.