Talks

This lecture discusses the stratification of regions in the space of real symmetric matrices. The points of these regions are Mandelstam matrices for momentum vectors in particle physics. The kinematic strata are indexed by signs and rank two matroids. Matroid strata of Lorentzian polynomials arise when all signs are nonnegative. We describe the posets of strata, for massless and massive particles, with and without momentum conservation. This is joint work with Veronica Calvo and Hadleigh Frost.

High school students learn how to express the solution of a quadratic equation in one unknown in terms of its three coefficients.  Why does this formula matter? We offer an answer in terms of discriminants and data. This lecture invites the audience to a journey towards non-linear algebra.

The Harris–Venkatesh plus Stark conjecture says that the action of the derived Hecke algebra on weight-1 cusp forms describes Stark units modulo p for all but finitely many primes p. These derived Hecke operators H^0 → H^1 on the cohomology of modular curves are defined by Shimura classes arising from the cover of X_1(p) over X_0(p). I will report on in-progress work to describe p-adic Shimura classes, define derived Hecke operators on completed cohomology, and formulate a similar conjecture for p-adic regulators of Stark units.

Certain issues arise while considering Local Tate duality over characteristics $p>0$, when the Galois module has $p$-torsion. A solution was given by Shatz, where he considered finite flat group schemes instead of Galois modules and the dual group is Cartier dual. The duality theorem is then a topological duality of the cohomology groups. We give a more natural construction and proof of the topological aspects of the duality theorem. This is joint work with Manodeep Raha.

We describe how the relative Satake isomorphism due to Sakellaridis gives a conceptual way of choosing test vectors when constructing the tame part of an Euler system. This is joint work with Li Cai and Yangyu Fan.

Let p be an odd prime. In this talk, I will explain some recent progress towards Mazur's conjecture on the growth of the Mordell-Weil ranks of an elliptic curve E/Q over Zp-extensions of an imaginary quadratic field, where p is a prime of good reduction for E. If time permits, I will also discuss results towards generalization of this conjecture for abelian varieties.

Let p and N be two odd primes >=5 such that N is congruent to 1 mod p. While studying Eisenstein congruences at prime level N extends from Mazur's work on the Eisenstein ideal (1977) continuing on to more recent work of Wake--Wang-Erickson (2020), we study Eisenstein congruences at prime square level N^2. We prove precise R = T theorems identifying suitable universal pseudo-deformation rings with Hecke algebras, both at level Gamma0(N^2) and Gamma1(N^2). Our study requires working with (cyclic p-group) group ring valued Eisenstein series, which in turn necessitates us to establish a new module-theoretic criterion to prove an R = T theorem. This is joint work with Jaclyn Lang and Katharina Mueller.

We describe how Euler added up all positive integers into a mysterious fraction when he was 28 years old, and I try to legitimize his method “p-adically”. This is a story of Number Theory from the 17th century on. We only need some knowledge of polynomials and fractions of polynomials and very basics of differentiation. If time allows, I enter into some results related to Ramanujan I found when I was 28 years old. For the results exposed here, a detailed proof can be found in my book: “Elementary Theory of L-functions and Eisenstein Series,” LMSST vol. 26, 1993, Cambridge U. Press.