Algebra, Geometry, and Number Theory Seminar

Abstracts

Frank Thorne - I will discuss the `geometric sieve’ of Bhargava. For example, suppose that $Y$ is a subscheme of $Z^n$ of codimension at least 2; then Bhargava’s sieve can be used to prove that for “most” lattice points x, none of the reductions of x (mod p) are in Y(F_p) for any p > 100. The paper is a beautiful mix of (easy) algebraic geometry and analytic number theory; I will discuss both the proofs and some applications.

Alexander Duncan - Essential dimension measures the minimal number of parameters required to describe an algebraic object. I will discuss the essential dimension of finite groups and its connections to constructive inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations. Techniques from birational geometry will play a prominent role in the talk.

Alexandra Seceleanu - The problem of describing the set of hypersurfaces passing through a finite set of points with given multiplicity leads to challenging mathematical questions. For example, one can ask what the minimum degree of such a hypersurface is or how many independent hypersurfaces there are of any given degree. The most general forms of these questions are still open and have given rise to longstanding conjectures in algebraic geometry. Searching for structural reasons to explain some of the these conjectures, Harbourne and Huneke proposed an approach based on comparisons between the set of all polynomials vanishing at the points to a prescribed order, which is called a symbolic power ideal, and algebraically better understood counterparts, namely the ordinary powers of the ideal of base points. Two questions will be shown to be related: How tight can this comparison be made? Which arrangements of lines in the plane have no points where only two lines meet? I will answer these and many more questions while considering some special arrangements of lines with unexpected combinatorial and algebraic properties.​

Nicolas Addington - The question of which cubic 4-folds are rational is one of the foremost open problems in algebraic geometry. I'll start by explaining what this means and why it's interesting; then I'll discuss three approaches to solving it (including one developed in the last year), my own work relating the three approaches to one another, and the troubles that have befallen each approach.

Cameron Atkins - Totally reflexive modules have very nice homological properties. Auslander used them to generalize his Auslander-Buchsbaum formula. These modules are difficult to find, and even stranger, some rings do not admit non-free totally reflexive modules. Yoshino gives very specific ring invariants that are necessary for a ring to admit non-free totally reflexive modules, and in the case when the ring is an embedded deformation, Yoshino showed these conditions are sufficient. This seminar explains Yoshino's argument, as well as, provides explicit examples (and non-examples) of rings that admit non-free totally reflexive modules.​​

Tyler Lewis - The maximal Cohen-Macaulay modules of a ring simultaneously provide: an easy class of modules to work with and a good reflection of the structure of the ring. Hence, we are interested in classifying local Cohen-Macaulay rings that have a finite number of indecomposable maximal Cohen-Macaulay modules. Currently, there is no complete classification of such rings; however, there is a complete classification for hypersurface singularities. In this seminar we will see how to construct maximal Cohen-Macaulay modules over hypersurfaces. Also, we will consider parts of the proof that for a complete hypersurface over an algebraically closed field of characteristic 0, having finite Cohen-Macaulay type, being a simple hypersurface singularity, and being an ADE singularity are equivalent.

Matthew Ballard - What is the least efficient generating set for the ring of $n \times n$ matrices over a field? How does this relate to the Orlov spectrum of the derived category of representations of the $A_n$ quiver? What do these terms mean? Come find out.

Harsh Mehta - The talk will be about properties of the product of binomial coefficients and essentially how the exponent of a prime dividing the quantity changes. We notice a connection of the product of binomial coefficients of the form $\prod_{j = 0}^n \binom{n}{j}$ and the product of Farey fractions with the greatest denominator being $n$. We then study properties of the products of all Farey fractions and notice that its size encompass information related to RH. We also look at properties of the exponent of $p$ dividing the product.

Robert Wilcox - A universal Hilbert set, H, is an infinite set of the integers with the following property: for any f(x,y) in Z[x,y] irreducible in Q[x,y], the polynomial f(x,h) in Z[x] is irreducible in Q[x] for all but finitely many h in H. Bilu in 1996 and Debes and Zannier in 1998 showed non-constructively the existence of such a set of asymptotic density 1. In this talk, we describe a correspondence between irreducible bivariate polynomials and integer points on a curve. Using Siegel's theorem on integer points, we present an explicit construction of a universal Hilbert set of asymptotic density 1.

Jesse Thorner - Let $\pi$, $\pi'$ be cuspidal automorphic representations of GL(d) and GL(d') over a field K. We prove a log-free zero density estimate for the Rankin-Selberg convolution L-function $L(s,\pi\otimes\pi')$ under the assumption of the Ramanujan-Petersson conjecture. Using this estimate, we prove automorphic analogues of Hoheisel's prime number theorem for short intervals and Linnik's theorem on the least prime in an arithmetic progression. We consider specific applications to the Sato-Tate Conjecture and the Chebotarev Density Theorem. This is joint work with Robert Lemke Oliver.

Xiaofei Yi - We will discuss unique factorization rings in the seminar. First I will talk about some techniques to determine whether a Noetherian domain is a UFD, and some properties stronger than unique factorization. At the end, I will introduce a unique factorization ring with zero divisors and discuss some of its properties.

Cynthia Vinzant - A reciprocal linear space is the image of a linear space under coordinate-wise inversion. This nice algebraic variety appears in many contexts and its structure is governed by the combinatorics of the underlying hyperplane arrangement. A reciprocal linear space is also an example of a hyperbolic variety, meaning that there is a family of linear spaces all of whose intersections with it are real. This special real structure is witnessed by a determinantal representation of its Chow form in the Grassmannian. In this talk, I will introduce reciprocal linear spaces and discuss the relation of their algebraic properties with their combinatorial and real structure.

Thomas Schnibben - Matrix Factorizations were introduced by David Eisenbud as a way to describe minimal free resolutions over a hypersurface. In this seminar we describe a matrix factorization of a complete intersection and see how it relates to minimal free resolutions of maximal Cohen-Macaulay modules.

Yongqiang Zhao - Although sieve methods in classical analytic number theory have a long and very fruitful history, its appearance in algebraic geometry is relatively new, and was introduced by Bjorn Poonen about ten years ago. In this talk, we will first discuss Poonen's sieve through a concrete example, then we will introduce a new interpolation technique to sieve methods for varieties over finite fields, which extends the applicable range of the existing sieve methods. As an application, we apply this new technique to count trigonal curves with fixed genus over finite fields or cubic function fields with bounded discriminant.

Carrie Finch - For a fixed dimension d, the Nexus numbers are differences of consecutive dth powers. In this talk, we explore the properties of these numbers modulo m. In addition, we consider the interactions between Nexus numbers and other sequences, including integer sequences and polynomial sequences.

Anastassia Etropolski - Let K be a number field and let E/K be an elliptic curve. It is a result of Katz from 1981 that if E has a torsion point of exact order p modulo almost every prime of K, then E does not necessarily have a p-torsion point over K, but it must be isogenous to an elliptic curve with a K-rational p-torsion point. More recently, Sutherland proved an analogous theorem for elliptic curves which admit an isogeny of order p. In his case, however, over K = Q, the local-global principle holds for p not equal to 7 and E in a single isomorphism class. This counterexample arises by studying the points on a particular modular curve. These two results sit in a natural framework of images of Galois representations. Specifically, if E has a torsion point of order p, then the image of the mod p Galois representation attached to E lands in a certain subgroup, up to choice of basis. Similarly, if E admits a p-isogeny, then its image lands in a Borel subgroup. This talk will answer the analogous local-global problem when the image lands in the split and nonsplit Cartan subgroups and their normalizers, which gives a complete answer to this family of questions.

Last semester's seminar.