Speaker: Daniel Miller, Cornell University
Title: Taniyama-Shimura, the $R=T$ theorem and Fermat-Wiles
Abstract: I will explain the statement of the Taniyama-Shimura conjecture, discuss Galois representations, deformation theory and the $R=T$ theorem, and explain how Taniyama-Shimura implies Fermat's Last Theorem. I will not assume the audience knows anything about these topics.
Speaker: Christine McMeekin, Cornell University
Title: The average rank of elliptic curves
Abstract: I will discuss the results of Manjul Bhargava and Arul Shankar regarding the average rank of elliptic curves. I will not be proving these results, but will introduce necessary terminology to understand their statements and will discuss some history and the importance of these results.
Speaker: Theodore Hui, Cornell University
Title: Distribution of the partition function modulo $m$.
Abstract: Let $p(n)$ be the partition function. Ramanujan discovered the identities $p(5 n+4)\equiv 0\pmod 5$, $p(7 n+5)\equiv 0\pmod 7$ and $p(11 n+6)\equiv 0\pmod{11}$. However, little was known for larger primes. Erdös conjectured that there are infinitely many primes $m$ such that $p(n_m)\equiv 0\pmod m$ for some $n_m$. In 2000, Ken Ono completely settled the problem by considering a special sequence of generating functions. I will present the terminology he uses and outline his proof. I will then focus on how to use those generating functions to both prove the periodicity of $p(n)$ modulo $m$ and generate some interesting congruences.
Speaker: Ling Long, Iowa State University and Cornell University
Title: Weakly holomorphic modular forms and congruences
Abstract: We will discuss a recent preprint by Kazalicki and Scholl on Hecke-like congruences satisfied by weakly holomorphic modular forms (allowing poles at cusps). Unlike the case of original congruences for cusp form, these congruences are nontrivial even for congruence subgroups.
Speaker: Ravi Ramakrishna, Cornell University
Title: A gentle introduction to $p$-adic Local Langlands, part I
Abstract: You may have run into expressions like "the Langlands program" or "the local Langlands program" or "the $p$-adic Local Langlands program." In this talk I will ignore the first expression but try to give some sense of what the latter two expressions mean by tying them to something more familiar, local class field theory.
Speaker: Joel Dodge, Binghamton University
Title: An introduction to the Carlitz module and explicit class field theory for $\mathbb{F}_q(T)$
Abstract: The study of the Carlitz module can be used to provide an explicit class field theory for the rational function field $k=\mathbb{F}_q(T)$ in the sense that: (1) it provides an explicit collection of polynomials whose roots generate the maximal abelian extension of $k$ and (2) it gives an explicit description of the action of the galois group of $k^{ab}/k$ on the roots of these polynomials. This is in perfect analogy with the construction of the maximal abelian extension of $\mathbb{Q}$ via the cyclotomic fields. I will introduce the basic notions of the Carlitz module and state many of the basic theorems of the subject. This will already go a long way towards developing the analogy between extensions of $k$ built from the Carlitz moduel and the cyclotomic extensions of $\mathbb{Q}$. In the end, I will state the analog of the Kronecker-Weber theorem for $k$.
Speaker: Ravi Ramakrishna, Cornell University
Title: A gentle introduction to $p$-adic Local Langlands, part II
Spring break
Speaker: Ravi Ramakrishna, Cornell University
Title: A gentle introduction to $p$-adic Local Langlands, part III
No seminar
Speaker: Daniel Vallieres, Binghamton University
Title: The abelian Stark conjecture
Abstract: Harold Stark wrote a series of four papers in the 70s where he pioneered the study of the special value $s=0$ of Artin $L$-functions. In this talk, we will try to give some of Stark's insights. It will be a gentle introduction to the abelian rank one Stark conjecture.
Speaker: Ravi Ramakrishna, Cornell University
Title: A gentle introduction to $p$-adic Local Langlands, part IV
Speaker: Daniel Miller, Cornell University
Title: Towards perfectoid spaces
Abstract: Recently Peter Scholze was able to prove a wide range of special cases of Deligne's weight monodromy conjecture by introducing a class of spaces he called perfectoid spaces. The theory of perfectoid spaces is quite complicated, so I will not say much about it. Instead, I will explain some of the motivation behind Scholze's work. In particular, I will discuss a beautiful result of Fontaine and Wintenberger relating the Galois groups of certain fields in characteristic $p$ and characteristic zero.
Seminar cancelled
Speaker: Everyone
Title: Summaries of the Upstate New York Number Theory Conference.
Abstract: Everyone will be giving summaries / additional thoughts on the topics spoken on at the Upstate Number Theory Conference.