Abstract: Consider an invariant that behaves nicely for smooth varieties, such as Euler number, Betti numbers, or Hodge numbers.
Suppose we want a version of this invariant for singular varieties that sees interesting information about the singularities. I will discuss how this naturally leads to the notion of crepant resolutions of singularities. However, crepant resolutions by smooth varieties are rare in practice. I will discuss joint work with M. Satriano in which we show that crepant resolutions actually exist in broad generality, as long as one is willing to consider algebraic stacks. Specifically, any variety with log-terminal singularities admits a crepant resolution by a smooth algebraic stack. This talk will not assume familiarity with stacks.
Title: Rational torsion and reducibility for abelian varieties associated to newforms
Abstract: Let f be a newform and A its associated abelian variety.
We have shown before that for certain primes r, if A is an optimal semistable elliptic curve with reducible torsion subgroup A[r], then A has rational r-torsion. In this talk we define necessary terms and how this result generalizes to other abelian varieties. We will sketch the proof and discuss the differences that arise when A is not necessarily an elliptic curve.
Title: 3-dimensional hypersurfaces and intermediate Jacobians.
Abstract: Clemens-Griffiths discovered in 1971 a surprising connection
between two questions (all terms will be defined in the talk)
1. When is a 3-dimensional hypersurface rational (i.e. birational to
P^3)?
2. When is a flat torus isometric to the Jacobian of an algebraic curve?
I will explain this connection and how I used it recently to give a simple proof of the (already known) irrationality of the general quartic 3-fold. More importantly, we will see some fundamental classical ideas in
action: resolution of singularities, the theory of periods, automorphism groups of varieties, and a beautiful example of Felix Klein.
This talk will be aimed at beginners (like the speaker) in algebraic geometry.
Title: Relations between Frobenius eigenvalues of abelian varieties over finite fields
Abstract: Elliptic curves over a finite field F_q come in two flavours: ordinary and supersingular. As q varies over powers of a fixed prime p, the eigenvalues of Frobenius of an ordinary elliptic curve are uniformly distributed on a circle, while those of a supersingular elliptic curve are supported are finitely many places. For higher dimensional abelian varieties, this dichotomy branches out in a more interesting way. Further, the multiplicative relations between the Frobenius eigenvalues capture facets of the geometry of the underlying abelian variety. In joint work with Santiago Arango-Piñeros and Deewang Bhamidipati, we study this phenomenon and provide a classification of the possible scenarios in low dimension. In this talk, I will discuss some of our results and some open questions in this area.