Title: FI-algebras

Abstract: An FI-algebra encodes a family of algebras with symmetric group actions, and an ideal of an FI-algebra represents an infinite family of ideals with symmetry.  I will give an overview of some results about when such ideals are finitely generated, and how to compute with them.  Then I will explain how to compute Hilbert series of these ideals.  Along the way we will see some surprising connections to combinatorics, such as well-partial orders and regular languages.

Title: The Combinatorics, Algebra, and Geometry of Conformal Blocks

Abstract: For any choice of smooth, marked projective curve and some representation-theoretic data, the  Wess-Zumino-Novikov-Witten (WZNW) model of conformal field theory produces a finite dimensional vector space called a space of conformal blocks.  As the marked curve is varied, the conformal blocks form a vector bundle over the space of curves; this gives rise to the so-called "fusion rules" of the WZNW theory.  I will explain how these rules allow us to count the dimensions of the space of conformal blocks using Ehrhart theory in a case associated to the Lie algebra sl_2.  Hiding behind this surprising connection are deep algebraic properties of the total coordinate ring of a closely related moduli space associated to a curve: it's spaces of parabolic SL_2 principal bundles.  An honorary appearance will be made by the chain of loops.

Title: Rationality of the Poincare series for lattice simplices

Abstract: We often study rings by resolving their defining ideal as a module over a polynomial ring. We may also study a ring by resolving the ground field as a module over the ring itself. Such resolutions do not generally have finite length, and so new interesting questions arise about the growth of Betti sequences. In particular, it is interesting to study the Poincare series, a generating function for the Betti numbers.

In this talk we present a sketch of this phenomenon and a result about the Poincare series of rings associated to a particular family of lattice simplices.

This is joint work with Ben Braun.

Title: The Kodaira dimension of the moduli space of curves

Abstract: After introducing the moduli space of curves and some notions from birational geometry, we will describe recent progress on the Kodaira dimension of the moduli space of curves.

Title: Explicit rational point calculations for certain hyperelliptic curves

Abstract: Given a curve of genus at least 2, it was proven in 1983 by Faltings that it has only finitely many rational points. Unfortunately, this result is ineffective, in that it gives no bound on the number of rational points. 40 years earlier, Chabauty proved the same result under the condition that the rank of the Jacobian of the curve is strictly smaller than the genus. While this is obviously a weaker result, the methods behind that proof could be made effective, and this was done by Coleman in 1985 using p-adic analysis. Coleman's work led to a procedure known as the Chabauty-Coleman method, which has shown to be extremely effective at determining the set of rational points exactly, particularly in the case of hyperelliptic curves. In this talk I will discuss how we implement this method using Magma and Sage to provably determine the set of rational points on a large set of genus 3, rank 1 hyperelliptic curves, and how these calculations fit into the context of the state of the art conjectures in the field. The subject of this talk is joint work with Jennifer Balakrishnan, Francesca Bianchi, Victoria Cantoral-Farfan, and Mirela Ciperiani.

Note the different room!!

Title: Tropical geometry of the Hodge bundle

Abstract: The Hodge bundle is a vector bundle over the moduli space of smooth curves (of genus $g$) whose fiber over a smooth curve is the space of abelian differentials on this curve. We may define a tropical analogue of its projectivization as the moduli space of pairs $(\Gamma, D)$ consisting of a stable tropical curve $\Gamma$ and an effective divisor $D$ in the canonical linear system on $\Gamma$. This tropical Hodge bundle turns out to be of dimension $5g-5$, while the classical projective Hodge bundle has dimension $4g-4$. This means that not every pair $(\Gamma, D)$ in the tropical Hodge bundle arises as the tropicalization of a suitable element in the algebraic Hodge bundle.

In this talk I am going to outline a comprehensive (and completely combinatorial) solution to the realizability problem, which asks us to determine the locus of points in the tropical Hodge bundle that arise as tropicalizations. Our approach is based on recent work of Bainbridge-Chen-Gendron-Grushevsky-M\”oller on compactifcations of strata of abelian differentials. Along the way, I will also develop a moduli-theoretic framework to understand the specialization of divisors to tropical curves as a natural tropicalization map in the sense of Abramovich-Caporaso-Payne.



This talk is based on joint work with Bo Lin as well as with Martin Moeller and Annette Werner.

Title: Improvements to the Brill-Noether Theorem

Abstract: In 1980 Griffiths and Harris proved what is known as the "Brill-Noether Theorem," which essentially says that for a general curve C of genus g that the dimension of a variety of special linear series on C is precisely equal to the Brill-Noether number of that variety. However, it is also known that a general curve C of genus g must have a particular gonality k, so the next natural question to ask is, "Can we compute the dimension of a variety of special linear series on a general curve C of genus g with specified gonality k?" This week we will see a result from Nathan Pfleuger that says we can, at least, bound the dimension above via a modification to the Brill-Noether number using recent results from Tropical Geometry.

Title: The Maroni Invariant of Trigonal Chains of Loops

Abstract: What is a Maroni invariant? What are trigonal chains of loops? Why should you care? We'll answer all these questions and more in our excursion into divisor theory on graphs, complete with an explicit computation of the Maroni invariant of a chain of loops and an explanation of what I've been doing for the last 6 months of my life.

Title: Markov Bases of Hierarchical Models

Abstract: We will start by discussing the significance of Markov Bases for investigating Hierarchical Models occurring in Algebraic Statistics. Markov Bases are often very large and hard to compute. This talk is going to introduce an alternative way of thinking about them using tools from Algebraic Geometry and present some of the latest computational results.

Title: The strange consequences of Siegel zeros



Abstract: If you believe the Generalised Riemann Hypothesis, then there are no zeros of L-functions with real part bigger than 1/2, but unfortunately we don't know how to show this. A `Siegel zero' is a putative strong counterexample to GRH, and if such exceptional zeros do exist, then there are many strange consequences for the distribution of prime numbers. However, prime numbers would also become very regular, and this allows us to prove things which go beyond even the consequences of GRH, if these exceptional zeros exist! I will survey some of these results, including recent joint work showing we can prove results towards the horizontal Sato-Tate conjecture for Kloosterman sums in this alternative world where Siegel zeros exist.