Math Colloquium - Spring 2025

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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

Zoi-Heleni Michalopoulou, NJIT

Source localization and geoacoustic inversion with virtual sound

The ability to localize a sound source in the ocean and infer seabed properties is essential for both antisubmarine warfare and environmental monitoring. A widely used mathematical technique for these applications is matched-field inversion (MFI). MFI maximizes the correlation between signals received at an array of hydrophones and numerically computed acoustic fields, termed replicas, generated with a sound propagation model. These computations account for various environmental conditions as well as the source’s location. The parameter values that maximize the correlation provide the estimates of the geoacoustic properties of the environment and the position of the source.  We will explore the mechanics and potential of MFI and demonstrate how its performance can be enhanced using Gaussian Process interpolation. This approach introduces the concept of virtual receiving arrays and virtual sound, exploiting the spatial coherence of the acoustic field and enabling denser sampling of the water column.

While MFI can be highly effective, it can also become computationally demanding under certain conditions, necessitating alternative strategies for solving the inverse problem. We will present scenarios where straightforward linearization techniques offer advantages, localizing a sound emitting source and providing geoacoustic property estimates of the propagation medium with minimal computations.

March 7

Hau-Tieng Wu, New York University (NYU)

Host: Amir Sagiv

Ridge analysis of time-frequency representations in the presence of Noise and its clinical application

The advancement of wearable devices has provided physicians with extensive diagnostic insights for healthcare. While time-frequency analysis is widely used to examine nonstationary time series in clinical applications, it remains challenging, particularly in the presence of noise. In this talk, I will discuss recent progress in addressing a key yet often overlooked step, ridge analysis, under a noise model, extending the Borell-TIS inequality and Dudley’s theorem to time-frequency representations. Ridge analysis serves as a bridge between time-frequency analysis, signal processing, and statistical inference, playing a crucial role in extracting reliable and actionable clinical information on a strong scientific foundation. The assessment of photoplethysmogram signal quality in digital health will be presented as an example.

March 14

Harry Dankowicz, University of Maryland [Mechanical Engineering]

Host: Roy Goodman

Using Self-Excited Template Dynamics and Root-Finding Algorithms for Sensor Design

I describe a new approach to dynamic sensor design that characterizes the steady-state sensor behavior in terms of a mapping onto a subset of degrees of freedom of a template nonlinear dynamical system with self-excited dynamics. The mapping is computed using a root-finding algorithm that can be made insensitive to the use of unmodeled actuators to drive the sensor dynamics. The sensor gain, as captured by the sensitivity of the components of the mapping to a parameter of interest, may be tuned by modifying parameters of the template system, without any changes to the sensor itself. An example application to mass sensing using a single, forced, linear, mass-spring-damper oscillator illustrates the general approach. The results show that the root-finding algorithm may be initialized without knowledge of the system damping or the properties of the unmodeled actuator and converges rapidly over a range of parameter values.

March 28

Doron Levy, University of Maryland

Host: Yuan-Nan Young

Fighting Drug Resistance with Math

The emergence of drug-resistance is a major challenge in chemotherapy. In this talk we will overview some of our recent mathematical models for describing the dynamics of drug-resistance in solid tumors. These models follow the dynamics of the tumor, assuming that the cancer cell population depends on a phenotype variable that corresponds to the resistance level to a cytotoxic drug.  Under certain conditions, our models predict that multiple resistant traits emerge at different locations within the tumor, corresponding to heterogeneous tumors. We show that a higher drug dosage may delay a relapse, yet, when this happens, a more resistant trait emerges. We will show how mathematics can be used to propose an efficient drug schedule aiming at minimizing the growth rate of the most resistant trait, provide extensions to the multi-drug setting, and discuss the competition between cancer cells and healthy cells.

April 4

Weilin Li, City University of New York

Host: Amir Sagiv

How to optimally invert a non-harmonic Fourier sum from data?

Spectral estimation is a fundamental problem of inverting a Fourier sum from noisy samples. Major difficulties arise if the frequencies are unknown and non-harmonic, which is usually the case in many imaging and signal processing problems. The stability of this nonlinear inverse problem behaves differently depending on the relationship between the number of samples 2m-1 versus the separation between the frequencies $\Delta$. This talk will mainly focus on the favorable well-separated case where $m \Delta \gg 2 \pi$. We develop a novel algorithm called gradient-MUSIC, which is a non-convex implementation of the classical-MUSIC algorithm. We provide a geometric analysis of its associated objective function and prove that gradient-MUSIC estimates the frequencies and amplitudes at the minimax optimal rates, for a variety of deterministic and stochastic noise. Time permitting, we discuss some work on the super-resolution regime where $m \Delta \ll 2 \pi$. This is joint work with Albert Fannjiang and Wenjing Liao.

April 11

Tomoki Ohsawa, University of Texas-Dallas

Host: Roy Goodman

Relative Dynamics and Stability of Point Vortices

I will talk about a Hamiltonian formulation of the relative dynamics of the planar $N$-vortex problem as well as a stability condition for its relative equilibria. The relative dynamics is the dynamics of the ``shape'' formed by the point vortices: For example, if $N = 3$, it is the dynamics of the shape of the triangle formed by three vortices, regardless of their position and orientation of the triangle. A relative equilibrium is a solution in which the vortices move in such a manner that the shape formed by the vortices does not change in time. The main result is a sufficient condition for the stability of relative equilibria using the Hamiltonian formulation of the relative dynamics.

April 25

Per-Olof Persson, University of California, Berkeley

Host: David Shirokoff

High-Order Methods on Unstructured Meshes for Fluid and Solid Mechanics

High-order accurate methods, such as the discontinuous Galerkin (DG) method, offer significant advantages over traditional low-order methods in applications like turbulent flows, multiphysics simulations, and wave propagation. However, their high computational cost and sensitivity to under-resolved features, such as shocks, remain major challenges. This talk presents recent advances in numerical schemes and solvers aimed at addressing these issues, with applications to Wall-Resolved Large Eddy Simulation of turbulence.

I will first discuss the importance of unstructured curved meshes, highlighting both the DistMesh algorithm and our recent work using Deep Reinforcement Learning for multiblock mesh generation. I will then introduce new discretization schemes, including the naturally sparse Line-DG and half-closed DG methods, which achieve better computational scaling at high polynomial degrees. These methods are coupled with efficient parallel solvers based on a new static condensation technique and optimally ordered incomplete factorizations. I will also discuss the use of adjoint methods for gradient-based optimization, demonstrated in our work on optimal designs for flapping flight, and how these techniques extend to high-order implicit shock tracking (HOIST).

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