Fall 2025 – Summer 2026 – Abstracts
Friday, Dec 5, 2025. Mathematics Research Seminar.
Rakvi, University of Maine
On Galois Representations of Elliptic Curves
Neville 421, 3:00 – 4:30pm.
Abstract: Given an elliptic curve over a number field K, there is a Galois representation associated to it that encodes information about Galois action on its torsion points. The talk will begin with an introduction of these representations followed by a survey of some recent progress on Mazur’s Program B and will conclude with a discussion of one of my own recent works on adelic indices joint with Kate Finnerty, Tyler Genao and Jacob Mayle.
Friday, Nov 21, 2025. Mathematics Research Seminar.
Shalin Parekh, University of Maine
The Dickman subordinator
Neville 421, 3:00 – 4:30pm.
Abstract: Caravenna-Sun-Zygouras recently constructed a random process which they dubbed the “critical stochastic heat flow” in a 2023 Inventiones paper. This is an example of a critical scaling limit, similar to the problem of constructing Yang-Mills in 4d which is a millennium problem. This talk will explain these notions of criticality, and then explain my attempt to understand the Caravenna-Sun-Zygouras construction, which in turn uses a stochastic process called the Dickman subordinator.
Friday, Nov 14, 2025. Mathematics Research Seminar.
Sheela Devadas, University of Maine
Covers of curves, Ceresa cycles, and unlikely intersections
Neville 421, 3:00 – 4:30pm.
Abstract: The Ceresa cycle is a canonical homologically trivial algebraic cycle in the Jacobian of a curve. In his 1983 thesis, Ceresa showed that this cycle is algebraically nontrivial for a very general complex curve of genus at least 3. In the last few years, there have been many new results shedding light on the locus in the moduli space of genus g curves where the Ceresa cycle becomes torsion. I will survey recent results regarding when the Ceresa cycle of a curve becomes torsion, and provide new examples of positive dimensional families of curves where only finitely many members of the family have torsion Ceresa cycle. This is joint work with Tejasi Bhatnagar, Toren D’Nelly Warady, and Padma Srinivasan.
Wednesday, Nov 12, 2025. Mathematics Colloquium.
Bob Franzosa, Professor Emeritus, University of Maine
Topological Spatial Relations in Geographic Information Systems: A Story about a Collaboration between Mathematics and Survey Engineering at UM a Few Decades Ago
Hill Auditorium, Barrows Hall
Refreshments at 3pm; Talk: 3:15 – 4:05 pm
Abstract: In this talk I will tell the story about the collaboration, introduce the idea behind topological spatial relations in geographic information systems, and present some basic foundational definitions and results. Following that, I will present some related and further results in topology.
Friday, Nov 7, 2025. Mathematics Research Seminar.
Krishnendu Khan, University of Maine
Calculation of some invariants for embedding of property (T) factors
Neville 421, 3:00 – 4:30pm.
Abstract: In this talk I’ll report on calculations of one-sided fundamental group for embeddings of group factors associated with a large class of relatively hyperbolic groups. This is based on a recent joint work with N. Amaraweera, J. F. Ariza Mejia and I Chifan. If time permits, I’ll share some structural results from an ongoing work of mine
Wednesday, Nov 5, 2025. Mathematics Colloquium.
Prof Shalin Parekh, University of Maine
The directed landscape is a black noise
Hill Auditorium, Barrows Hall
Refreshments at 3pm; Talk: 3:15 – 4:05 pm
Abstract: In probability and statistical physics, a stochastic process is called linearizable if it can be driven by some combination of Brownian motions and Poisson processes. It was once believed that all processes with independent and stationary increments are linearizable. In this talk I will use a framework of Boris Tsirelson to give examples of non-linearizable systems that arise quite naturally by considering scaling limits of last passage percolation and interacting particle systems. Based on joint work with Zoe Himwich.
Friday, Oct. 31, 2025. Mathematics Research Seminar.
Michael Cerchia, University of Maine
Low degree points on modular curves
Neville 421, 3:00 – 4:30pm.
Abstract: I will describe recent work under the broad category of “low degree points on modular curves”. This will include classifying the torsion subgroups of elliptic curves over quartic fields and determining quadratic points on 
-adic modular curves. I will also explain the moduli interpretation of the latter problem, which gives us a way to understand the possible images of Galois representations attached to elliptic curves over quadratic extensions. I will assume a little knowledge of basic group and Galois theory, but nothing about elliptic or modular curves.
Wednesday, Oct. 29, 2025. Mathematics Colloquium.
Maricela Best McKay, University of British Columbia
Sketchy natural gradient descent for Physics informed Neural Networks and beyond
Hill Auditorium, Barrows Hall
Refreshments at 3pm; Talk: 3:15 – 4:05 pm
Abstract: Mathematical models and computer simulations are key tools in modern science. They help us understand how systems work, test ideas, and make predictions when experiments are too expensive or impossible. Many physical systems are described by partial differential equations (PDEs), which usually cannot be solved exactly, so simulations are used to approximate their solutions.
A new approach called Physics-Informed Neural Networks (PINNs) uses neural networks and optimization to solve PDEs. PINNs can easily combine data with physical laws, making them appealing in applications like data assimilation, uncertainty quantification, control, and inverse problems. However, they can be difficult to train, especially for complex systems that vary across different scales or frequencies.
To address this, we use a method called natural gradient descent, which leverages the geometry of the problem to help the network learn more efficiently and accurately than standard optimization algorithms like Adam or BFGS. Because natural gradients are expensive to compute, we designed a faster version using randomized sketching—a mathematical tool for approximating large matrix computations.
With this method, we trained PINNs up to several orders of magnitude more accurately and hundreds of times faster. For example, a model that once took several hours to train can now be done in under two minutes. Our approach scales to very large networks. While we focused on PINNs, sketching with natural gradients is a scalable framework with implications for regression tasks beyond PINNs.
Friday, Oct. 24, 2025. Mathematics Research Seminar.
Justin Trias, University of East Anglia, UK
The universal Harish-Chandra j-function
Neville 421, 3:00 – 4:30pm.
Abstract: The Harish–Chandra μ-function plays a central role in the explicit Plancherel formula for a p-adic group G. It arises as the normalising factor for the Plancherel measure on the unitary dual of G, and is defined through the theory of intertwining operators.
In this talk, we show how to extend the construction of the μ-function—or more precisely its inverse, the j-function—to all finitely generated representations, and over general coefficient rings such as Z[1/p]. This leads to a universal j-function with values in the Bernstein centre, which specialises to the classical j-function. Beyond its role in harmonic analysis, the universal j-function also encodes arithmetic information: it reflects aspects of the local Langlands correspondence for classical groups, via criterions of Muić and Moeglin-Tadić connecting it to the reducibility points of parabolically induced representations. Time permitting, we will illustrate how this perspective applies to the study of the local Langlands correspondence in families. This is joint work with Gil Moss.
Wednesday, Oct. 22, 2025. Mathematics Colloquium.
Sunrose Shreshtha, Carleton College
Periodic straight-line trajectories on flat surfaces
Hill Auditorium, Barrows Hall
Refreshments at 3pm; Talk: 3:15 – 4:05 pm
Abstract: Consider an orientable surface formed by gluing Euclidean polygons edge-to-edge via translations. We call such surfaces translation surfaces. Now, consider picking a point and a direction on the surface, and moving forward in a straight-line. The long-term dynamics of such straight-line flows on finite translation surfaces (built using finitely many polygons) has been well-studied due to its connections to the dynamics of polygonal billiards. However, less is known in general regarding straight-line flows on non-compact, infinite area translation surfaces (built using infinitely many polygons).
In this talk, we will consider periodic straight line trajectories on a few different surfaces built using infinitely many polygons with a special focus on the Mucube – an infinite Z^3 periodic surface – first discovered by Coxeter and Petrie and more recently studied by Athreya-Lee and Gutierrez-Romo-Lee-Sanchez. We will give a complete characterization of the periodic straight-line trajectories on the Mucube in terms of an infinitely generated infinite index subgroup of SL(2,Z).
Friday, Oct. 17, 2025. Mathematics Research Seminar.
Prof Neel Patel, University of Maine
Self-Similarity and Fluid Boundaries in Porous Media (Part 2)
Neville 421, 3:00 – 4:30pm.
Abstract: From the self-similar ODE we derived in Part 1, we will discuss how to prove an actual solution exists with the properties that we want. Namely, we will show that the ODE gives a solution to the PDE that has a corner at the origin at time t = 0 and is a smooth function for all t > 0.
Friday, Oct. 10, 2025. Mathematics Research Seminar.
Prof Neel Patel, University of Maine
Self-Similarity and Fluid Boundaries in Porous Media (Part 1)
Neville 421, 3:00 – 4:30pm.
Abstract: Finding explicit solutions or solutions with an explicit type of behavior is something that is very useful when understanding a PDE. Searching for self-similar solutions is one way to achieve this. Self-similarity is a general concept, e.g the Koch snowflake and Romanesco broccoli. In this talk, I will introduce the idea of self-similarity for PDE, show how it gives the solution to the heat equation, and then introduce the notion of a self-similar solution to a fluid free boundary problem.
Wednesday, Oct. 8, 2025. Mathematics Colloquium.
Dr. Ashwin Iyengar, 2024-2025 American Mathematical Society Congressional Fellow
Numbers are political: a discussion of mathematics, advocacy, and American politics
Hill Auditorium, Barrows Hall
Refreshments at 3pm; Talk: 3:15 – 4:05 pm
Abstract: Politics, policy and advocacy are seldom seen as traditional career paths for a mathematician. This is in part due to the abstract and theoretical nature of mathematics, but it also has to do with the culture of the mathematical community. In this talk I will explore avenues from mathematics to policy by giving a few examples. I will primarily speak about my experience as the 2024-2025 American Mathematical Society Congressional Fellow, and my path from number theory to policy. I will talk about the role that mathematicians and numerically-literate people can play in policy and advocacy more broadly, for example through economics, data science, and data communication. I will also talk about the places where theoretical math actually does intersect policy, including the work of the Metric Geometry and Gerrymandering Group and the history of voting theory. I will conclude by reflecting on the political activism of mathematicians in the past and present (and future?). This talk will be non-technical, and will be aimed at anyone interested in using mathematics in the broader context of their political world.
Friday, Sept 26, 2025. Mathematics Research Seminar.
Prof Evan Miller, University of Maine
Permutation symmetric solutions of the incompressible Euler equation and related models
Neville 421, 3:00 – 4:30pm.
Abstract: In this talk, I will discuss some results for the incompressible Euler equation and related models under permutation symmetry. The Fourier-restricted Euler equation is a model equation for the Euler equation where the Helmholtz projection is replaced by a projection onto a subspace of divergence free vector fields, but the nonlinearity is otherwise unchanged. There are permutation symmetric solutions of the Fourier-restricted Euler equation that exhibit finite-time blowup. This is also true in the viscous case when the fractional dissipation is small enough. I will also discuss some conditional results for the full Euler equation.
Wednesday, Sept. 24, 2025. Mathematics Colloquium.
Prof. Evan Miller, University of Maine
Weak and strong solutions of the Navier-Stokes equation
Hill Auditorium
Refreshments at 3pm; Talk: 3:15 – 4:05 pm.
Abstract: In this talk, I will discuss solutions of the Navier-Stokes equation. I will introduce the concept of a weak solution of a PDE, with a particular focus on the Navier-Stokes equation. I will also discuss scaling properties of the Navier-Stokes equation. The global regularity problem for Navier-Stokes is particularly challenging in three dimensions because of the relationship between scaling and energy.
Friday, Sept 19, 2025. Mathematics Research Seminar.
Prof Jack Buttcane, University of Maine
An update on Bessel functions and minimax problems
Neville 421, 3:00 – 4:30pm.
Abstract: I’ll give an update on two problems I’ve already discussed in the seminar on the GL(n) Bessel functions and a minimax problem related to vector-valued automorphic forms.
Wednesday, Sept. 17, 2025. Mathematics Colloquium.
Prof. Sheela Devadas, University of Maine
Higher-weight Jacobians: Generalizing the idea of adding points on a curve
101 Neville Hall
Talk 3:15 – 4:05 pm, refreshments at 3pm
Abstract: An abelian variety is a geometric object which also has a way to “add” points. One-dimensional abelian varieties, elliptic curves, are important objects in number theory, with applications to cryptography, integer factorization, and primality proving. While we cannot “add” points on a general curve, we can embed any curve over the complex numbers inside a higher-dimensional abelian variety known as the Jacobian.
In this talk I will discuss my work with Max Lieblich, where we generalize the notion of the Jacobian of a complex curve to higher-dimensional varieties and higher “weights”. Jacobians of weight 2 are connected to ideas from number theory such as the Brauer group and Tate conjecture. We use the theory of CM elliptic curves and lattices in the complex numbers to compute certain higher-weight Jacobians as complex tori. Surprisingly, we find that they are always algebraic. However, we can see there is something non-algebraic about the construction by considering the minimal fields of definitions of 2-Jacobians of abelian surfaces.