Introduction to Research Seminar

4/29/19 - Elizabeth Kelley

Title: A crash course in cluster algebras

Abstract: I'll begin with a quick overview of the definition and major properties of cluster algebras, give some motivating examples, and then describe a related combinatorial object called a "snake graph" which can be used to verify the positivity phenomenon in some cases. I'll then define a type of generalized cluster algebra due to Chekhov and Shapiro, building on previous definitions and examples, and briefly summarize my work (joint with Esther Banaian) on extending the snake graph construction to this generalization. I won't assume any prior knowledge about cluster algebras.

4/17/19 - Daniel Spirn

Title: Some applied analysis problems from condensed matter and fluid dynamics Abstract: I will discuss a few research areas that I have been thinking about for the past few years. These include problems arising in condensed matter and the interactions of fluids with membranes. My primary goal is to develop intuition for the qualitative behavior of these systems. No background is necessary.

4/10/19 - Vic Reiner

Title: Algebraic combinatorics: using algebra for counting

Abstract: I'll talk about a general counting problem, with answer given by numbers generalizing binomial coefficients. Some of their many properties only become easy to see after you apply a bit of (linear and multilinear) algebra.

4/3/19 - Arnab Sen

Title: Mathematics on Random Structures Abstract: I will give brief overview of two areas of my research. In the first part, I will discuss a special type of interacting particle system called the majority dynamics. In this process, the voters keep updating their opinions according to the majority of their neighbors. In the second part, I will talk about spectral properties of random matrices where the entries enjoy some inherent structures. I will mainly focus on two models: adjacency matrices of random graphs and random Toeplitz matrices.

3/27/19 - Paul Garrett

Title: Physics applied to number theory

Abstract: Early in the 20th century, Polya and Hilbert separately observed that if we could somehow find a self-adjoint operator whose eigenvalues were s(s-1) for all zeros s of the zeta function, then we would have proven the Riemann Hypothesis, because eigenvalues of self-adjoint operators are real. However, any such operator could not be continuous, since the zeros of zeta are not bounded. By coincidence, in the 1920's, quantum physicists realized that a key role is played by unbounded self-adjoint operators on Hilbert spaces, motivating Stone, von Neumann, Friedrichs, and others to give this a rigorous foundation. In 1977, Stark and Hejhal observed that a numerical calculation of eigenvalues s(s-1) of the Laplacian on the modular curve by Haas appeared to include zeros of zeta and of an L-function. Hejhal debunked this, but in 1982,3,4 Y. Colin de Verdiere gave heuristics that hold out some hope. After a hiatus of a few decades, in recent years E. Bombieri and I have succeeded in clarifying what's going on.

2/27/19 - Adrienne Sands

Title: An Automorphic Hamiltonian

Abstract: Discrete eigenfunction expansions are used in acoustics, signal processing, and quantum mechanics to transform differential equations into (easier!) algebraic equations. For example, any L²-function on an interval [a,b] has a discrete Delta-eigenfunction expansion, its Fourier series. Since there are no non-trivial eigenfunctions on L²(ℝ), there is no basis of Delta-eigenfunctions to use in a discrete expansion. However, eigenfunctions of the Hamiltonian T=-Delta + x² form a basis for L²(ℝ) such that any L²-function has an exotic (but discrete!) T-eigenfunction expansion. In this talk, we highlight the analytic features of T which motivate the automorphic analogue.

2/6/19 - Dmitriy Bilyk

Title: Points on Spheres

Abstract: A very natural question of the optimal distribution of finitely many points on the sphere can have a variety of different interpretations and ways to quantify it. These include minimizing discrete energy (e.g. electrons on the sphere, maximizing the sum of distances etc), sphere pack- ings and coverings, partitions and tessellations of the sphere, discrepancy, random points, lattices, discrepancy, structured configurations (e.g. regular polytopes), approximation and numerical inte- gration of functions (e.g. spherical designs), reconstruction of vectors (e.g. tight frames) etc. I will survey some recent results and open problems related to these topics.

10/31/18 - Sam Stewart

Title: Lost in the Crowd: How Mathematicians Model Really Dense Crowds

Abstract: If you?ve seen Lord of the Rings, then you know that simulating massive crowds is now possible. But most of the close-up shots in the movie have real people in the really dense spots near gates, etc. This is because there are still few models that produce good pictures for dense crowds. The difficulty lies in capturing the subtle emergent behavior: individuals? local actions produce global behavior so that the crowd becomes more than a sum of its parts. From starling flocks to stock markets, emergent phenomena appear again and again. We?ll describe the cutting edge computational models (and see some cool simulation runs) that try to approximate emergence, and explore the questions that computers currently can't answer.

10/24/18 - Jay Stotsky

Title: Simulation and Modeling of the Biomechanics of Biofilms

Abstract: Bacterial biofilms are communities of bacteria that adhere and grow on a surface, usually in an aqueous environment. Biofilms are ubiquitous in nature, and are important in many settings such as the treatment of infections, the development of strategies to mitigate corrosion, and process control in bioreactors. I will focus in this talk on how to obtain useful information about bacte- rial biofilms through computational simulations. Key findings include the validation of a biofilm model with experimental data and an exploration of the effect that biofilm microstructure has on macroscopic properties.

10/3/18 - Esther Banaian

Title: Three Frieze Patterns

Abstract: We define frieze patterns and discuss their classical connections to polygons. Then we soup up this connection by considering both cluster algebras and quiver representations associated to a polygon, and show how each also gives a frieze pattern.