Algebra Seminar

Algebra Seminar

Title: Primality tests, elliptic curves and field theory

Abstract: This talk would be of interest to anyone interested in algebraic varieties, field theory or number theory. We'll start by reviewing Pepin's primality test for Fermat numbers, which are numbers of the form F_n = 2^n+1. This test features a 'universal point' on an algebraic group (a multiplicative group) whose powers determine the primality of the given number. We will show how this is related to a field theory problem involving the splitting of rational primes for quadratic extensions. I'll also describe some directly analogous primality tests for other special types of numbers, which use elliptic curves. The field theory part of the problem is slightly elevated to the more general splitting of primes in abelian extensions. I will be working this summer on a project to develop a primality test that uses results about elliptic curves and non-abelian extensions, which would provide many more opportunities to develop tests for specific types of numbers.

Date:
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Event Series:

Algebra Seminar

Title: Subspace Polynomials and Cyclic Subspace Codes
 
Abstract: Subspace codes are collections of subspaces of the finite vector space $\mathbb{F}^n_q$ under the subspace metric. Of particular interest due to their efficient encoding/decoding algorithms are constant dimension cyclic codes where each codeword is a subspace of the same dimension and which are invariant under an action of $\mathbb{F}^{*}_{q^n}$.  We will see how to represent subspace codes using particular polynomials and deduce properties of the codes from the structure of these polynomials based on the work of Ben-Sasson et. al. in 2016. Using these results, we will give a construction of a constant dimension cyclic code containing multiple orbits under the $\mathbb{F}^{*}_{q^n}$ action with highest possible subspace distance.
Date:
-
Location:
745 POT

Algebra Seminar

Title: Subspace Polynomials and Cyclic Subspace Codes
 
Abstract: Subspace codes are collections of subspaces of the finite vector space $\mathbb{F}^n_q$ under the subspace metric. Of particular interest due to their efficient encoding/decoding algorithms are constant dimension cyclic codes where each codeword is a subspace of the same dimension and which are invariant under an action of $\mathbb{F}^{*}_{q^n}$.  We will see how to represent subspace codes using particular polynomials and deduce properties of the codes from the structure of these polynomials based on the work of Ben-Sasson et. al. in 2016. Using these results, we will give a construction of a constant dimension cyclic code containing multiple orbits under the $\mathbb{F}^{*}_{q^n}$ action with highest possible subspace distance.
Date:
-
Location:
745 POT

Algebra Seminar

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.
Date:
-
Location:
745 POT
Event Series:

Algebra Seminar

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.
Date:
-
Location:
745 POT
Event Series:

Algebra Seminar

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.
Date:
-
Location:
745 POT
Event Series:

Algebra Seminar

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.

 

Date:
-
Location:
745 POT
Event Series:

Algebra Seminar

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.

Date:
-
Location:
POT 745
Event Series:

Algebra Seminar

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.

Date:
-
Location:
CP 287
Event Series:

Algebra Seminar

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.
 
Date:
-
Location:
POT 745
Event Series: