Introduction to Research Seminar

AMS Intro to Research Seminar has returned to Vincent 570!  Find us Wednesdays at 12:20.  🍕

Upcoming speakers and events:

4/26/2023 - Speaker: Carme Calderer

Past speakers and events:

4/19/2023 - Speaker: Casey Garner

Title: An Optimized Overview of Optimization with an Optimizer 

Abstract: Optimization is a vast discipline which impacts multiple mathematical domains and intersects constantly with our lives. I shall present a brief sampling of classical questions in optimization, detail typical problems optimizers pursue, and provide an overview of my past, current, and future research plans, from algorithm development to humanitarian assistance. This talk will not assume any background in optimization theory or analysis of algorithms. 

4/12/2023 - Speaker: Robbie Angarone 

Title: An invitation to the box-positive cone

Abstract: In this talk, I will discuss an open problem posed my advisor, Pasha Pylyavskyy. While the question is motivated by total positivity, cluster algebras, and symmetric functions, we will see that this domain is largely governed by simple combinatorial pictures called Young diagrams. After reviewing the basic setup and playing with some examples, we’ll discuss the algebraic motivation in more detail. This talk will assume no fancy background in algebraic combinatorics.

4/5/2023 - Movie: Flatland (2007)

Based on Edwin Abott's book "Flatland", this is an animated film about geometric characters living in a two-dimensional world. When a young girl named "Hex" decides to "think outside the box" (in a world where such thought is forbidden), her life becomes in danger and it is up to her grandfather to save her life.

3/29/2023 - Applied Game Theory

3/22/2023 - Panel: How to Choose an Advisor (co-hosted with GeMM)

3/15/2023 - Speaker: Ben Brubaker

Title: Automorphic Forms meet Statistical Mechanics

Abstract: I'll give one example of how the study of automorphic forms leads to interesting combinatorial problems. In this case, their resolution arises from techniques from mathematical physics,  particularly statistical mechanics. No prior acquaintance with these objects will be assumed, but it serves as a reminder that number theorists have to (and get to(!)) learn all kinds of interesting mathematics.

3/8/2023 - SPRING BREAK

3/1/2023 - Speaker: Ratul Biswas

Title: The ‘what’s, ‘why’s and ‘wow’s of Spin Glasses

Abstract: In this intro talk, I’ll explain what a spin glass is and how it naturally comes up in various problems in computer science and statistics. I’ll introduce several models of spin glasses, talk about some known results and finally mention some problems open for exploration.

2/22/2023 - SNOW DAY

2/14/2023 - Speaker: Dan Spirn  **NOTE THE UNUSUAL DATE -- THIS IS A TUESDAY**

Title: Harmonic maps: from superconductors to computer graphics

Abstract: In many physical problems, one looks for mappings from a domain into a particular manifold.  Harmonic maps generate least energy mappings into these manifolds; however, topology can induce the formation of singularities. On the other hand, these singularities typically provide important qualitative information about the original problem.  I'll provide motivation for studying harmonic maps, including condensed matter physics and computer graphics. Our toolbox for these problems included methods from the calculus of variations, partial differential equations and geometric measure theory.  I'll end with some ideas about how one can prove some results that give intuition to some of these problems. 

2/1/2023 - Applied Game Theory

1/25/2023 - Speaker: Olivia Cannon

Title: Center Manifolds and Mob Mentality

Abstract: Differential equations are a wide and varied field of study which apply to almost everything in our world. Mathematicians seek to take the complicated and make it simple. I will present on an application of DEs this summer to bias in social dynamics, and attempt to answer this: Can math be used to both describe people AND prove things? In doing so, I will also describe a technique where infinite-dimensional systems are reduced to a small finite number of variables. I will also try to posit that convolutions can be used for good. 

This talk assumes no knowledge of ODEs, dynamical systems, or the like. 

1/18/2023 - Applied Game Theory

12/7/2022 - Applied Crystallography and Symmetry Theory

11/30/2022 - Speaker: Lilly Webster

Title: A Gentle Introduction to Cluster Algebras

Abstract: Since their introduction in 2000, cluster algebras have been a very active area of research in combinatorics.  Despite their intimidating definition, cluster algebras turn out to illuminate a lot of common structure behind some familiar combinatorial objects.  In this talk, we’ll look at some examples of cluster algebras and start to explore their remarkable properties.  No background in combinatorics is necessary for this talk! 

11/16/2022 - Speaker: Arnd Scheel

Title: The Emergence of Order

Abstract: I'll talk about the emergence of self-organized collective behavior in large complex dynamical systems, focusing on the example of spiral waves. Extremely robust, fascinating, beautiful, omnipresent, they defy simple mathematical approaches. I'll present a variety of intriguing phenomena related to spiral wave dynamics and explain mathematical models that are amenable to analysis. On the technical side, I'll focus on questions of stability: spectral, linear, and nonlinear stability, why they're interesting, why they're hard, and how one could make progress.

11/9/2022 - Speaker: Peter Olver

Title: Fractalization and Quantization in Dispersive Systems

Abstract: The evolution, through spatially periodic linear dispersion, of rough initial data produces fractal, non-differentiable profiles at irrational times and, for asymptotically polynomial dispersion relations, quantized structures at rational times.  Such phenomena have been observed in dispersive wave models, optics, and quantum mechanics, and lead to intriguing connections with exponential sums arising in number theory.   Ramifications and recent progress on the analysis, numerics, and extensions to nonlinear wave models, both integrable and non-integrable, will be presented.  

11/2/2022 - SIAM Panel

10/26/2022 - Speaker: Tyler Lawson

Title: Using algebra to study pictures and using pictures to study algebra

Abstract: Homotopy theory is an unusual combination subject; it formed from geometry and algebra, but ultimately it has developed new and unusual connections with a wide variety of topics. I'll spend some time talking a little bit about where homotopy theory came from, what kinds of things it touches on now, and the culture of it.

10/19/2022 - Applied Game Theory

10/12/2022 - Speaker: Christine Berkesch

Title: Geometry of toric syzygies 

Abstract: Free resolutions, or syzygies, with a graded structure are algebraic objects that encode many geometric properties. This correspondence lies at the heart of classical projective algebraic geometry. By analogy, multigraded resolutions should also provide powerful geometric tools. I will discuss some foundational results from the classical story and give an overview of recent work to extend these tools to the multigraded setting of toric geometry. This is joint work with Daniel Erman and Gregory G. Smith.

10/5/2022 - Speaker: Max Engelstein

Title: How much does a soap bubble jiggle?

Abstract: We (who is we? mathematicians? engineers? bees?) have known for some time that the way to enclose a region of area one with least perimeter is the circle. However, proving this rigorously isn't something that was really done until the 20th century. There is also the question of HOW much less perimeter the circle has, and what exactly that seemingly nonsensical question means. I will survey some progress on (quantitative) isoperimetric inequalities, ending, if there is time, with some recent(-ish) work joint with O. Chodosh (Stanford), L. Spolaor (UCSD) and R. Neumayer (CMU).

9/28/2022 - Speaker: Dick McGeehee

Title: Mathematics and the Climate Emergency

Abstract: The Climate Emergency has begun and it will continue to worsen.  The last seven years have been the hottest since records started being kept in the mid-nineteenth century.  Floods, fires, droughts, and heat waves are becoming more common and more severe. Simple mathematical models can be instructive in understanding the phenomena we are experiencing and can be used to get a glimpse of the future in store for the planet.

9/21/2022 - Applied Game Theory

9/14/2022 - Speaker: Paul Garrett

Title: RH, distributions, and singular perturbations of Laplacians on non-Euclidean spaces

Abstract:  In 1977, in a flawed numerical solution of a eigenvalue problem (Delta-lambda)u=0 (on the modular curve), H. Haas apparently found many eigenvalues lambda=s(s-1) with zeta(s)=0. Soon after, D. Hejhal observed that also zeros of L(s,chi) appeared, for a certain Dirichlet L-function. This would have been numerical support for the Polya-Hilbert speculation/program to prove RH by relating the zeros of zeta to the eigenvalues of a self-adjoint operator.

By 1981, Hejhal had redone the calculations, and found that NONE of the zeros of zeta actually appeared. On the other hand, Hejhal DID see that Haas had solved (Delta-lambda)u=delta, with a Dirac delta at a cube root of unity in the upper half-plane. This is not a homogeneous equation, so the lambdas are not necessarily eigenvalues, and not necessarily real.

In 1983, Y. Colin de Verdiere recalled a semi-miraculous way to convert certain such inhomogeneous equations to homogeneous equations, with the same lambdas, thus, magically causing the lambdas to become eigenvalues of a self-adjoint operator. This can be described in terms of the singular perturbations of Laplacians, already used by P. Dirac, H. Bethe, and other physicists around 1930, even before Stone and vonNeumann's rigorization of unbounded self-adjoint operators. Namely, rewrite the inhomogenous equation as ((Delta-delta.delta)-lambda)u=0, where the lowercase delta term is a singular potential (as opposed to confining potential) added to the Laplacian. As an operator, it maps u to delta(u).delta. Awkwardly, delta is not in L^2. This (correctly) suggests that the domain of this singular perturbation of Delta is a subset of functions u such that delta(u)=0. 

In the 1930's, S. Sobolev recast similar issues in rigorous form, in fact, resembling Dirac's intuitive-but-correct heuristics. G. Levi had already executed a prototype in 1906 to prove a correct version of Dirichlet's minimum principle. In the 1940's, L. Schwartz gave a robust context for most of the known "generalized function" arguments, including Dirac's delta.

Returning to the original context, in fact, yes, in 1983 Colin de Verdiere sketched a proof, and in 2020 E. Bombieri and PG gave a complete argument, that lambda's for which (Delta-lambda)u=delta has an L^2 solution u, with delta(u)=0, are of the form s(s-1) for s a zero of either zeta or L(s,chi)! This is just the tip of an iceberg. :)