Title: Crepant resolutions via stacks

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.

Title: Weight Filtrations and Derived Motivic Measures

Asbtract: Weight Filtrations are mysterious: they record some shadow of how a variety might be recovered from smooth and projective ones. Some of the information recorded by weight filtrations can be understood via the motivic measures they define, i.e. group homomorphisms from the Grothendieck ring of varieties. With Zakharevich’s discovery of the higher K groups of the category of varieties, there is an ongoing project to understand these groups by lifting motivic measures (on the level of K_0) to so-called "derived" ones, i.e. on the level of K_i for all i. I will describe some of this work, which shows that if one closely studies how the Gillet-Soulé weight complex is constructed, then one can also obtain derived motivic measures to non-additive categories as well, such as the compact objects in the category of motivic spaces, along with that of compact objects in the stable homotopy category. These new derived motivic measures allow us to answer questions in the literature, providing new ways to understand the higher K groups of varieties, and relating them to other interesting algebro-geometric objects in the literature.

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