Events of the Week

In a polynomial ring, the d-th Veronese subring is generated as a k-algebra by all monomials whose degree is a multiple of d. In a polynomial ring with standard grading (degree of each variable is 1), the Veronese subrings have many nice properties: they are normal, Cohen-Macaulay, and Koszul. Furthermore, their defining ideals are quadratic, binomial, and determinantal, generated by 2x2 minors of suitable matrices that also form a Groebner basis for the ideal. In this talk, we will discuss Veronese subrings of a non-standard graded polynomial ring. We will see that many of the nice properties are satisfied in two-variable case, but no longer hold in general in more variables. This is based on joint work with Bek Chase, Luca Fiorindo, Thiago de Holleben, Emanuela Marangone, Alexandra Seceleanu, and Srishti Singh.

The square-root cancellation hypothesis, in its original form, concerns cancellation in certain GL(1) sums with applications to the distribution of zeros of L-functions associated with GL(2) cusp forms. Building on Ye’s work on a varying GL(2) cusp form and my work (jointly with Ye and Gillespie) on the Rankin Selberg convolution of two GL(2) cusp forms, both allowed to move, I will discuss a variant in which only one form is permitted to vary. This leads naturally to shifted convolution sums and new analytic challenges. I will outline my methods and preliminary results in this setup and discuss how these fit into the broader concept of the square root cancellation hypothesis.

The Andrews-Curtis conjecture is a long-standing open problem about presentations of the trivial group, related to some of the central problems in low-dimensional topology. I will discuss a generalization of the Andrews-Curtis conjecture and recent approaches using 4-manifolds and quantum topology.

We will share a story that brings together geometry, dynamics, and number theory in interesting and novel ways, and also creates some very compelling imagery- the talk will have lots of pictures, and all relevant background notions will be introduced and explained. Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat 3-manifolds which we call translation prisms. Using ideas of Furstenberg, and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot-Vijayraghavan expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular n-gon for n = 7, 9, 14, 16, 18, 20, 24, 30. This is joint work with Nicolas Bedaride, Pat Hooper, and Pascal Hubert.

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