Titles/Abstracts

Plenary talks

Daniel Goldston

Title: Gaps and differences between primes

Abstract: In 2013 Yitang Zhang proved there are infinitely often two primes that differ by some number less than 70,000,000. Using a different method (devised independently by Maynard and Tao), the joint effort of many mathematicians reduced this number to 246, which isn't too far from the twin prime conjecture that there are infinitely many primes differing by 2. The current methods, however, cannot prove the twin prime conjecture without a fundamentally new idea. In this talk I will describe some of the ideas behind this work, and also describe other problems on prime gaps and prime differences. Despite the recent breakthroughs, most of the important unproved problems on primes are still far out of reach.

Eugenia Rosu

Title: The sum of two cubes and other fun equations

Abstract: The aim of the talk is to give an introduction to some famous problems involving elliptic curves, such as the congruent number problem and the sum of two cubes. We will mention several of the current approaches for finding rational points of elliptic curves and in the process we will introduce some background for the famous Birch and Swinnerton-Dyer Conjecture.

Xinyi Yuan

Title: Introduction to the abc conjecture

Abstract: The goal of this talk is to survey some basic aspects of the abc conjecture including its history, relation to other famous conjectures, and current status. In particular, we will mention its relation with Fermat's last theorem, the effective Mordell conjecture, and the arithmetic Bogomolov--Miyaoka--Yau inequality.

Keith Conrad

Title: Ranks of elliptic curves

Abstract: The rank of an elliptic curve (with rational coefficients) is an important numerical invariant of the elliptic curve. In this talk we will discuss solved and unsolved problems in number theory that are related to ranks, how to think about the rank in several ways, and questions mathematicians are interested in directly about ranks.

Graduate student talks

Ravi Fernando

Title: Special values of L-functions

Abstract: L-functions are certain complex-analytic generating functions that generalize the Riemann zeta function and contain various kinds of arithmetic information. The values of L-functions at integer inputs often turn out to be interesting numbers, such as zeta(2) = pi^2/6. In this talk, we will discuss some concrete examples of L-functions and their values, and state some theorems and open conjectures about how they behave in general.

Minseon Shin

Title: Brauer groups of rings, varieties, and stacks

Abstract: The Brauer group Br(k) of a field k classifies the isomorphism classes of division algebras D over k, where D has finite dimension as a k-vector space and the center of D is k. In this talk we'll discuss Brauer groups of rings and algebraic varieties and state some recent progress on the Brauer group of the moduli stack of elliptic curves over an algebraically closed field.

Shelly Manber

Title: Rational Points on Varieties

Abstract: Hilbert's 10th problem, proposed in 1900, asks whether it can be determined in finite time if a Diophantine equation has integer solutions. We can generalize this problem to ask if such an equation has rational solutions, and if, so, what we can say about how many solutions there are. In this talk, we will explore some of the directions one can go to try to answer this question and some of the known results and open questions, including a discussion of the local-global principle and connections to the Birch and Swinnerton-Dyer Conjecture.

Alexander Youcis

Title: An invitation to Shimura varieties

Abstract: We discuss the role of representation theory in the study of modern number theory. In particular, we discuss the two most important types of groups attached to the rational numbers ℚ: adelic groups and Galois groups. We motivate the theory of Shimura varieties, a generalization of modular curves, which allows us to build and relate representations of adelic groups and Galois groups. Finally, we mention work of the speaker, building on the work of many others, which helps understand these representations.

Undergraduate talks

Yulia Alexandr (Wesleyan)

Title: Visibility Graphs of Staircase Polygons: Algorithm for Building Staircase Polygons from Slope-Ranking Balanced Tableaux

Abstract: There exists a relationship between staircase polygons, persistent graphs, and balanced tableaux. While many aspects of this relationship had been rigorously studied, the problem of recovering a simple staircase polygon whose visibility graph is isomorphic to the skeleton of a given slope-ranking balanced tableau remained open for 22 years and was previously known to be PSPACE. In this paper, we demonstrate a deterministic polynomial-time algorithm for constructing a staircase polygon with desired properties and prove that certain conditions required by the algorithm hold for any balanced tableau.

Eric Chen (Berkeley)

Title: Constructive Galois theory of algebraic groups (Coauthors: J.T. Ferrara, Liam Mazurowski)

Abstract: A fundamental aspect of the inverse Galois problem is describing all extensions of a base field K with a given Galois group G. A constructive approach to this problem known as the theory of generic extensions seeks to explicitly parametrize all Galois extensions of K with the given group G.

In our work, we show the existence of and explicitly construct these parameterizations for a broad class of connected algebraic groups over fields of positive characteristic. An attractive consequence of our work is the parameterization of all cyclic 2-groups over all fields of positive characteristic using an "optimal" number of parameters. This contrasts with a theorem of Lenstra, which states that this is not possible for cyclic 2-groups of order ≥ 8 over ℚ.