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Department of Mathematics

Algebra and Number Theory Seminars

Tiling

Group theory is the formal mathematical study of symmetry. Groups are among the foundational objects composing abstract algebra, yet they also pervade nearly every discipline in pure mathematics as well as many areas of science and engineering. One striking result of group theory shows that there are exactly 17 different types of planar symmetry. This image illustrates one of these types of symmetry in a section of tilework at the Alhambra Palace in Granada, Spain. This particular symmetry is characterized by 3-fold rotational symmetry with no reflections (Photo credit The_Alhambra_and_The_Alcazar).

The Algebra and Number Theory Seminar is structured to include talks on a broad range of mathematical areas that are of interest to algebraists and number theorists, including analytic and algebraic number theory, algebra, combinatorics, algebraic and arithmetic geometry, cryptography, representation theory, and more. Talks are given by a variety of local, national, and international speakers in number theory and related areas.

See below for upcoming seminars or access the seminar archive.

Organizers

Mary Flahive, Clayton Petsche, Thomas Schmidt and Holly Swisher

Timing

We traditionally meet every Tuesday at 11:00 a.m.

A generalization of Franklin's partition identity and a Beck-type companion identity

Weniger 201

Speaker: Holly Swisher

Euler's classic partition identity states that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. We develop a new generalization of this identity, which yields a previous generalization of Franklin as a special case, and provide both a q-series and bijective proof. We further establish an accompanying Beck-type companion identity which gives the excess in the total number of parts of partitions of one kind over the other.This is joint work with Gabriel Gray, David Hovey, Brandt Kronholm, Emily Payne, and Ren Watson. Read more.

TBA

Weniger 201

Speaker: Justin Bloom

Read more.

Descartes’s Revenge: or, support varieties and what they’re good for

WNGR 201

Speaker: Jon Kujawa

Since Descartes, we’ve known that algebra and geometry are intertwined. Support varieties are a modern incarnation of this observation. Information about algebraic objects (e.g. representations of finite groups) can be revealed by introducing algebraic geometry via cohomology. This will be a very high level introduction focused on the big themes of the subject along with concrete examples. Read more.

Oscillating asymptotics and conjectures of Andrews

Zoom

Speaker: Amanda Folsom

In the 1980s, Andrews studied certain q-hypergeometric series from Ramanujan’s ``Lost" Notebook, and made several conjectures on their Fourier coefficients, which encode partition theoretic information. The corresponding conjectures on the coefficients of Ramanujan’s $\sigma(q)$ were resolved by Andrews, Dyson and Hickerson, who related them to the arithmetic of $\mathbb Q(\sqrt{6})$. Cohen also made connections to Maass waveforms. In this work, by new methods, we prove additional conjectures of Andrews, e.g., we blend novel techniques inspired by Garoufalidis' and Zagier's recent work on asymptotics of Nahm sums with classical techniques such as the Circle Method in Analytic Number Theory.This is joint work with Joshua Males, Larry, Rolen, and Matthias Storzer. Read more.

Methods in studying Hecke traces

Weniger 201

Speaker: Liubomir Chiriac

Hecke operators play a central role in the theory of modular forms, and appear in a number of important conjectures. In this talk I will discuss some methods used to study the traces of these operators acting on spaces of cusp forms. We will explore techniques from combinatorics, p-adic analysis and diophantine approximation, highlighting their applicability in broader contexts. Read more.

Modularity and Resurgence

Zoom

Speaker: Eleanor McSpirit

The study of asymptotics as q approaches roots of unity is central to the theories of mock and quantum modular forms. In a collection of works, Gukov, Pei, Putrov, and Vafa proposed a candidate for a q-series invariant of closed 3-manifolds coming from physics. Many of these invariants are known to be mock and quantum modular forms, and this modularity has been integral to their study. Resurgent analysis is a natural tool to study this invariant from the perspective of physics, and is a theory centrally concerned with the relationship of functions to their asymptotic series. This has led to several questions on the interrelationship of resurgence and modularity. While this has been discussed across the subject, many questions remain. This talk will discuss ongoing work to make this connection explicit and natural from the perspective of number theory. Read more.

Modular functions and the monstrous exponents

WNGR 201

Speaker: Holly Swisher

In 1973 Ogg initiated the study of monstrous moonshine with the observation that the prime divisors of the monster group are exactly those for which the Fricke quotient X_0(p)+p of the modular curve X_0(p) has genus zero. Here, motivated by Deligne's theorem on the p-adic rigidity of the elliptic modular j-invariant, we present a modular function-based approach to explaining some of the exponents that appear in the prime decomposition of the order of the monster.This is joint work with John Duncan. Read more.

TBA

TBA

Speaker: Derek Garton

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