Math Colloquium - Spring 2025
Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall I, unless noted otherwise.
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January 24
Daniel Massatt, Louisiana State University
Host: Casey Diekman
Numerical Methods and Modeling for Moiré 2D Materials
Moiré 2D materials are nano-structures exhibiting several almost identical periodicities giving rise to long range moiré patterns. They have many tunable parameters including twist angle, species of layers, number of layers, pressure, strain, and external fields. Choices of parameters allow tuning of physical observables such as conductivity or density, but also can give rise to exotic physical phenomena arising from many-body effects such as correlated insulators, fractional quantum hall effect, and unconventional superconductivity.
In this work we construct and analyze algorithms for computing observables for several fundamental single particle ab initio models, and derive how to construct an efficient basis for applications in many-body physics. These moiré materials are incommensurate, or have no periodicity, making conventional algorithms inapplicable. We consider complex geometries including incommensurate bilayers and double-incommensurate trilayer materials. We exploit tools from ergodic theory, multi-scale analysis, spectral theory, and momentum space to construct effective algorithms.
January 31
Charles Parker, University of Oxford
Host: Casey Diekman
High-order structure-preserving finite element methods and applications
In the past ten years, structure-preserving methods have risen in popularity in the finite element community. With roots in geometric integrators and finite element exterior calculus, structure-preserving methods conserve a particular physical quantity of the underlying system, like Gauss's law in electromagnetism and the incompressibility condition in flow problems. Constructing such methods is a nontrivial task for many applications, and without assumptions on the mesh structure, high-order elements are often required.
In this talk, we will first focus on elements that preserve the divergence constraint arising in incompressible flow. In particular, we will address the uniform stability properties of the 2D Falk-Neilan and Scott-Vogelius elements in the context of the Stokes equations. Then, we will turn to fourth-order problems, such as plate bending, where H2-conforming discretizations are often not available in standard software. We will show that a careful, structure-preserving discretization of a particular reformulation of these problems can be used to compute H2-conforming approximations using standard software without ever implementing the elements.
February 7
Alla Borisyuk, University of Utah
Host: Victor Matveev
Effect of Astrocytes in Neuronal Networks
Astrocytes are glial cells playing multiple important roles in the brain, e.g. control of synaptic transmission. We are developing tools to include “effective” astrocytes in neuronal network models in an easy-to-implement computationally-efficient way. In our approach we first consider neuron-astrocyte interaction at fine spatial scale, and then extract essential ways in which the network is influenced by the presence of the astrocytes. For example, by using a DiRT (Diffusion with Recharging Traps) model we find that a synapse tightly ensheathed by an astrocyte makes neuronal connection faster, weaker, and less reliable, and subsequently astrocytes can push the network to synchrony and to exhibiting strong spatial patterns, possibly contributing to epileptic disorder. Further, the calcium signals in the astrocyte initiate a loop of calcium-sodium-potassium trans-membrane activity. This activity modulates the extracellular concentrations of these ions and, consequently, the excitability of the nearby postsynaptic cell, further modifying the neuronal network activity.
February 14
Gregory Berkolaiko, Texas A&M University
Host: Amir Saggiv
Morse Theory for Eigenvalues of Self-Adjoint Families
The question of optimizing an eigenvalue of a family of self-adjoint operators that depends on a set of parameters arises in diverse areas of mathematical physics. Among the particular motivations for this talk are the Floquet-Bloch decomposition of the Schroedinger operator on a periodic structure, nodal count statistics of eigenfunctions of quantum graphs, conical points in potential energy surfaces in quantum chemistry and the minimal spectral partitions of domains. In each of these problems one seeks to identify and/or count the critical points of the eigenvalue with a given label (say, the third lowest) over the parameter space which is often known and simple, such as a torus.
Classical Morse theory is a set of tools connecting the number of critical points of a smooth function on a manifold to the topological invariants of this manifold. However, the eigenvalues are not smooth due to presence of eigenvalue multiplicities or ``diabolical points''.We rectify this problem for eigenvalues of generic families of finite-dimensional operators. The ``diabolical contribution'' to the``Morse indices'' of the problematic points turns out to be universal:it depends only on the multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family. Using tools such as Clarke subdifferential, stratified Morse theory of Goresky--MacPherson, and homology of Grassmannians,we are able to derive explicit formulas for the said ``diabolical contribution''.
Based on a joint work with Igor Zelenko (Texas A&M University).
February 21
Gabriel Ocker, Boston University
Host: James MacLaurin
A constructive approach to neural field theories
The neuronal dynamics governing sensory, motor, and cognitive function are commonly understood through field theories for neural population activity. Classic neural field theories are integro-differential equations and have inspired a range of dynamical systems analyses. They are, however, derived strong and restrictive assumptions on the underlying biophysical dynamics. We will review classical models of theoretical and mathematical neuroscience. Then, we will discuss recent and ongoing work in a constructive approach to neural field theories that allows directly incorporating biophysically motivated nonlinearities and uncovering their impact on macroscopic, coordinated, and patterned activity in neuronal networks.
February 28
Nicholas Kotov, University of Michigan [Chemical Engineering]
Host: Lou Kondic
Title and Abstract Forthcoming
March 7
Hau-Tien Wu, New York University (NYU)
Host: Amir Sagiv
Title and Abstract Forthcoming
March 14
Harry Dankowicz, University of Maryland [Mechanical Engineering]
Host: Roy Goodman
Title and Abstract Forthcoming
March 28
Doron Levy, University of Maryland
Host: Yuan-Nan Young
Title and Abstract Forthcoming
April 4
Weilin Li, City University of New York
Host: Amir Sagiv
Title and Abstract Forthcoming
April 11
Tomoki Ohsawa, University of Texas-Dallas
Host: Roy Goodman
Title and Abstract Forthcoming
April 25
Per-Olof Persson, University of California, Berkeley
Host: David Shirokoff
Title and Abstract Forthcoming
May 2
Elizabeth Allman, University of Alaska, Fairbanks
Host: Kristina Wicke
Title and Abstract Forthcoming
May 9
Felix Parra Diaz, Princeton Plasma Physics Laboratory
Host: Michael Siegel
Theory and computation for magnetic confinement
Fusion energy is a promising technology: safe, based on abundant fuel, with no waste legacy and minimal land use. Due to the climate emergency, it has recently attracted much public attention and private funding. For efficient energy production, the right mixture of hydrogen isotopes must be heated to temperatures ten times larger than the temperature of the sun core. This is a grand technological and physics challenge in which we have made great progress. The most advanced concept for energy production is magnetic confinement fusion, an approach that utilizes large electromagnets to confine the hydrogen and powerful beams and radiofrequency emitters to heat it up. This talk will describe the principles behind magnetic confinement fusion and its two most illustrious representatives: the tokamak and the stellarator. The talk will emphasize the importance of theoretical and computational tools in the development of these concepts by describing how the community models energy losses in these systems then and uses these models to design better machines.
Last Updated: February 5, 2025