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

Felix Parra Diaz, Princeton Plasma Physics Laboratory

Host: Michael Siegel

Title and Abstract Forthcoming

February 7

Alla Borisyuk, University of Utah

Host: Victor Matveev

Title and Abstract Forthcoming

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

Title and Abstract Forthcoming

February 28

Nicholas Kotov, University of Michigan [Checmical 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 Universirty 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

Last Updated: January 15, 2025