Fall 2024 – Summer 2025 – Abstracts
Friday, Dec. 13, 2024. Mathematics Research Seminar.
Prof. Francisco Gancedo, University of Seville; IAS
Evolution of vortex filament
Neville 421, 3:05 – 4:05.
Abstract: In this talk we show two new results of vortex filament evolution for incompressible Navier-Stokes and Euler equations. For Navier-Stokes, we prove global-in-time regularity for initial helical vortex filament. For Euler, we give the existence of weak dissipative solutions with initial vorticity concentrated in a circle.
Wednesday, Dec. 11, 2024. Mathematics Colloquium.
Prof. David Hiebeler, University of Maine
Some structured population/epidemiological models
Hill Auditorium, Barrows (ESRB)
Refreshments at 3pm, talk 3:15 – 4:05 pm.
Abstract: I will discuss some basic differential equation models in population ecology and epidemiology, first for well-mixed populations, then for populations with spatial structure or patch/group structure. I will show that a straightforward approach to such models misses some important details of interest, and one way to address the issue, and some problems that arise along the way.
Wednesday, Dec. 4, 2024. Mathematics Colloquium.
Dr. Rakvi, University of Maine
Elliptic Curves, Isogenies, and Adelic Indices
Hill Auditorium, Barrows (ESRB)
Refreshments at 3pm, talk 3:15 – 4:05 pm.
Abstract: Let E be an elliptic curve without complex multiplication, defined over the rationals. A well-known theorem of Serre bounds the largest prime ℓ for which the mod ℓ Galois representation of E is nonsurjective. After proving the theorem, Serre asked whether a universal bound on the largest non-surjective prime might exist. Although this question remains open, significant partial progress has been made. In particular, Lemos proved that the question has an affirmative answer for all E admitting a rational cyclic isogeny. In light of the considerable recent progress in understanding Galois representations of elliptic curves, Zywina proposed a more ambitious conjecture about the possible adelic indices that can occur as E varies. In this talk, we will give an overview of some of the work in this area and discuss an ongoing project (joint with Tyler Genao and Jacob Mayle) that aims to extend Lemos’s result to prove Zywina’s conjecture for certain families of elliptic curves.
Wednesday, Nov. 20, 2024. Mathematics Colloquium.
Prof. Naomi Tanabe, Bowdoin College
Large Sums of Multiplicative Functions
Hill Auditorium, Barrows (ESRB)
Refreshments at 3pm, talk 3:15 – 4:05 pm.
Abstract: Studying the asymptotic behavior of the multiplicative functions is a classical problem in analytic number theory, as they often arise in various areas of mathematics, including number theory, combinatorics, and even mathematical physics. One of the common approaches is to analyze the asymptotic of their summatory functions. In this talk, we explore large sums of divisor-bounded multiplicative functions, with a special focus on sums of Fourier coefficients of modular forms. This work originated from the Women In Numbers 6 Workshop and is a collaboration with Claire Frechette, Mathilde Gerbelli-Gauthier, and Alia Hamieh.
Wednesday, Nov. 13, 2024. Mathematics Colloquium.
Dr. Max Weinreich, Harvard University
Chaos in mathematical billiards: dynamics, algebra, and arithmetic
Hill Auditorium, Barrows (ESRB)
Refreshments at 3pm, talk 3:15 – 4:05 pm.
Abstract: Billiards is a mathematical model for anything that bounces: light, molecules, or the cue ball in the game of pool. In this talk, we will learn how to play billiards over algebraically closed fields. An old conjecture of Birkhoff predicts that the only non-chaotic billiard is the ellipse, but what is “chaos” for an algebraic dynamical system, and how do we measure it? We will survey three ways to quantify chaos (a.k.a. entropy) coming from classical dynamics, algebraic geometry, and number theory, and explain how to compute the algebraic entropy of the billiard in a Fermat curve.
Wednesday, Oct. 30, 2024. Mathematics Colloquium.
Prof. Joseph Hoisington, Colby College
Isoperimetric Inequalities and Spaces of Non-positive Curvature
101 Neville Hall
Talk 3:15 – 4:05 pm, refreshments at 3pm
Abstract: The classical isoperimetric inequality states that among all domains in Euclidean space with the same volume, the ball is the unique domain whose boundary is as small as possible. We will discuss several results and questions that involve extending the isoperimetric inequality to spaces of non-positive curvature. We will then present a new, strengthened version of one of these results, the linear isoperimetric inequality, and discuss its applications in metric geometry and group theory.
Wednesday, Oct. 23, 2024. Mathematics Colloquium.
Prof. Mayan Maiti, Purdue University
A historical context of the Langlands Program
Hill Auditorium, Barrows (ESRB)
Talk 3:15 – 4:05 pm, refreshments at 3pm
Abstract: Euler introduced analytical tools to investigate the nature of the prime distribution. One of those analytical tools is sum of reciprocal of prime integers (now known as L-series). Later, Dirichlet, Dedekind, Weber, Artin, Weil, Hecke etc. generalized this analytical tool (L-functions) in various contexts to better understand other mathematical objects (such as Galois groups, Elliptic curves etc.) containing various amount of Arithmatic information. Later Robert P. Langlands, by investigating the relations among these, put forward a theory that connects these tools via Automorphic representations in more general settings. In this talk I will talk about these various L-series, a connection among them, and introduce a basic information of the Langlands program.
Wednesday, Oct. 16, 2024. Mathematics Colloquium.
Prof. Brandon Hanson, University of Maine
The puzzles of Paul Erdős
Hill Auditorium, Barrows (ESRB)
Talk 3:15 – 4:05 pm, refreshments at 3pm
Abstract: Paul Erdős is among the most prolific mathematicians to have ever lived, collaborating so extensively that mathematicians now track their Erdős number. He made many insightful contributions to mathematics, perhaps most notably in his posing of problems that could steer a research area for years. Motivated by the recent launch of erdosproblems.com, I will discuss some of my favourite Erdős problems and the progress surrounding them.
Wednesday, Oct. 9, 2024. Mathematics Colloquium.
Prof. Chandrika Sadanand, Bowdoin College
You can hear the shape of a polygonal billiard table
Hill Auditorium, Barrows (ESRB)
Talk 3:15 – 4:05 pm, refreshments at 3pm
Abstract: Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).
Wednesday, Oct. 2, 2024. Mathematics Colloquium.
Prof. Michael Cerchia, University of Maine
Rational points on curves: classifying torsion subgroups of elliptic curves
Hill Auditorium, Barrows (ESRB)
Talk 3:15 – 4:05 pm, refreshments at 3pm
Abstract: The problem of finding rational and integral points on curves is ancient. It goes at least as far back as 1800BC Mesopotamia, where a clay tablet was constructed containing a list of Pythagorean triples — integers (a,b,c) such that a2+b2=c2. Yet it wasn’t until the late 20th century, with the help of modern mathematical developments in algebra and geometry, that this subfield really took off, leading to Andrew Wiles’s proof of Fermat’s Last Theorem. In this talk, I will discuss the general problem of finding rational points on curves before focusing on a special class of cubic curves called elliptic curves. What’s remarkable about these is that their rational points form finitely generated abelian groups. Towards the end of my talk, I will discuss progress on classifying all possible torsion subgroups of elliptic curves as we vary the base field.
Wednesday, Sept. 25, 2024. Mathematics Colloquium.
Prof. Krishnendu Khan, University of Maine
A Journey around the fundamental group of von Neumann Algebras
Hill Auditorium, Barrows (ESRB)
Talk 3:15 – 4:05 pm, refreshments at 3pm
A consequence of the strengthening of Connes Rigidity conjecture by Sorin Popa implies that the fundamental group F(L(Γ)) of type II1 factor L(Γ) associated to all countable, discrete, property (T) groups Γ with infinite conjugacy class is trivial. We’ll talk about the mentioned conjecture and how it has been, and still is, a driving force behind the striking development of beautiful mathematics in operator algebras let alone von Neumann Algebras during more than a couple of decades.
Wednesday, September 25, 2023. Math Club Speaker.
Dr. Colin Carroll, Google Research.
Automatic differentiation and machine learning
Neville Hall 208
Talk is 5:00-6:00pm and there will be pizza!
Abstract: Derivatives and gradients are ubiquitous in modern machine learning. We will talk about some of their applications and some of the reasons they are so useful. The main event will be walking through how to implement a computer program that will automatically and efficiently compute derivatives (hint: it is the chain rule all the way down). Some (Python) code will be shown, but no programming background is assumed. Familiarity with single variable calculus, and a lack of fear of multivariable calculus will be helpful.
Wednesday, Sept. 18, 2024. Mathematics Colloquium.
Andrew Knightly, University of Maine
Statistical properties of elliptic curves and modular forms
101 Neville Hall
Talk 3:15 – 4:05 pm, refreshments at 3pm
Abstract: If E is an elliptic curve 
for integers 
, then the number of solutions 
over the field 
has interesting statistical behavior as we vary 
and/or E. We will give a leisurely survey of this and related phenomena for families of modular forms. If time allows, we will also give a non-technical overview of the trace formula, which is the main tool for studying such families. No prior familiarity with modular forms or elliptic curves will be assumed.