A Model theorist looks at familiar mathematical systems

Carol Wood
April 15

Abstract: Model theory is a relatively young branch of logic, not much older than I am (but I am old). I would like to give a taste of its flavor by describing a consequence of the compactness theorem, a basic result in logic. Compactness allows nonstandard analogs e.g. of integers and real numbers. In these nonstandard structures we find infinitely large integers and both infinitely large and infinitesimally small real numbers.
Prerequisites: No prerequisites are required. It could be helpful if you ever wondered why the concept of limit in calculus is hard to understand, or maybe it was easy for you!

Congruent numbers and elliptic curves

Ken Ribet
April 8

Abstract: Congruent numbers are positive square-free integers that occur as areas of right triangles whose sides all have rational length. The numbers 1–4 are not congruent numbers, but 5, 6 and 7 all are congruent numbers. These numbers appear in Fibonacci's book "Liber Quadratorum (book of squares)" and were studied by Lagrange, Fermat and others. Determining the set of congruent numbers amounts to identifying elliptic curves of a certain type that have positive rank. The conjecture of Birch and Swinnerton-Dyer translates the identification problem into an analytic problem involving L-series of elliptic curves. I will explain how the main themes in this story fit together and point to some ongoing research about elliptic curves and their L-series. Although Fibonacci numbers appear prominently in Math 55, I had little idea about Fibonacci himself until I encountered his Book of Squares. Spoiler: he was born around 1170 and published his book in 1225. He acquired the name "Fibonacci" only in 1838!

Dispersion and global solutions in nonlinear dispersive flows

Daniel Tataru
April 1

Abstract: Dispersive flows are partial differential equations where linear waves travel in different directions, depending on their frequency. Nonlinear effects may potentially disrupt this pattern over long time scale. The aim of this talk will be to provide some insights into these competing phenomena, leading up to some very recent conjectures.
Prerequisites: Real analysis as well as some differential equations for the second part of the talk (e.g. math 123 or 126)

Representations of groups and tensor categories

Aleksandra Utiralova
March 18

Abstract: To any group \(G\) one can associate the category \(\operatorname{Rep}(G)\) of its finite-dimensional representations. It turns out that it is possible to reconstruct \(G\) from just this categorical data but to do this one should take into account the tensor product structure on representations of \(G\), which turns \(\operatorname{Rep}(G)\) into the so-called tensor category. I will give a brief description of this construction and then proceed to discuss tensor categories without the underlying group. The examples I'm interested in include the category of super vector spaces and Deligne categories \(\operatorname{Rep}(S_t)\), which give interpolations of the category of representations of the symmetric group \(S_n\) to complex values of the parameter \(n\).

Physics of flows for most efficient heat transfer between two walls

Anuj Kumar
March 11

Abstract: We are interested in the design of forcing in the Navier–Stokes equation such that the resultant flow maximizes the heat transfer between two differentially heated walls for a given power supply budget. Previous work established that heat transport cannot scale faster than 1/3-power of the power supply. Recently, Tobasco & Doering (PRL'17) constructed self-similar two-dimensional steady branching flows, saturating this upper bound up to a logarithmic correction to scaling. We present a construction of three-dimensional "branching pipe flows" that eliminates the possibility of this logarithmic correction and for which the corresponding heat transport scales as a clean 1/3-power law in power supply. Our flows resemble previous numerical studies of the three-dimensional wall-to-wall problem by Motoki, Kawahara & Shimizu (JFM'18). However, using an unsteady branching flow construction, it appears that the 1/3 scaling is also optimal in two dimensions. After carefully examining these designs, we extract the underlying physical mechanism that makes the branching flows "efficient," based on which we present a design of mechanical apparatus that, in principle, can achieve the best possible case scenario of heat transfer. We will further discuss some interesting implications of branching flows, for example, anomalous dissipation in turbulent flows and Rayleigh--B'enard convection.
Prerequisites: The talk will use a few concepts from PDEs and calculus of variations but I will try to keep it very simple.

Nash's \(C^1\) isometric embedding theorem

Sung-Jin Oh
March 4

Abstract: The goal of this talk is to present a proof of the remarkable Nash \(C^1\) embedding theorem, which states, for instance, that the unit sphere \(S^2\) can be "crumpled" in a \(C^1\) fashion into an arbitrarily small ball in \(\mathbb R^3\). Note that such a statement is obviously false if one replaces "\(C^1\)" by "smooth", by consideration of curvature. This theorem turned out to be more than a mere curiosity; its proof foreshadowed an important technique called "convex integration", which found remarkable applications in a wide array of fields, such as symplectic topology, calculus of variations, and fluid dynamics.

An overview of microlocal geometry

Kendric Schefers
February 26

Abstract: Notions of "smoothness" and "singularity" exist for a wide variety of objects in geometry. The first example one encounters in life are the notions for real-valued functions, where a smooth function is one which is differentiable, and a singular one is one which is not. There are also such notions for distributions, D-modules, coherent sheaves, etc. Working microlocally in each of these setting means trying to get a handle on the singularities in question by looking not only at where singularities occur, but also in what direction they arise. In this talk, I will try to give an impression of how the microlocal philosophy is implemented in modern geometry.
Prerequisites: should probably be linear algebra and multivariable calculus, but truthfully, the talk is going to be very impressionistic. Most of what I talk about would require a lot of advanced material to fully understand, but I'll be sort of hand-waving the whole time. No one who shows up will fully understand the talk, but conversely, anyone who shows up will understand at least the gist of it.

Generalized Parking Function Polytopes

Andrés R. Vindas Meléndez
February 12

Abstract: A classical parking function is a list of positive integers whose nondecreasing rearrangement (b1, ..., bn) satisfies b <= i. The convex hull of {parking functions of length n} is an n-dimensional polytope (think high dimensional polygon), called the classical parking function polytope. We can loosen our notion of "parking function" to create generalized parking function polytopes. These new polytopes are relatives of the Pitman-Stanley polytope and some partial permutohedra. We leverage these connections to compute the volume of special parking polytopes.

Can we use math to improve traffic flow?

Franziska Weber
February 5

Abstract: Braess’s paradox is a proposed explanation for when the modification of a road system by for example adding a new road, leads to a worse traffic situation instead of an improvement. It was proposed in 1968 by the mathematician Dietrich Braess and can mathematically be formulated as a Nash Equilibrium that is a worse situation than the best overall flow through the road network. In this talk, we review Braess’s paradox and its emergence in real life situations and then explore whether it can be observed in traffic flow models involving partial differential equations.

A mirror into the higher dimensional world

Catherine Cannizzo
January 29

Abstract: We live in a three dimensional world. If we consider time as a fourth “coordinate”, we have four dimensions. These four dimensions are known as “space-time.” In physics, string theory conjectures that, at scales much smaller than an electron (of a similar order of magnitude of the universe to the atom), there are small strings vibrating. For the theory to work, the strings must vibrate in 6 compact extra dimensions, for a total of 10 dimensions! It turns out that two geometric models for the strings give the same physics. These pairs opened up mathematicians to the notion of “mirror symmetry” which gives us a lot of interesting information about geometric spaces. In this talk, I will define a “geometric space”, give some examples, and illustrate how to visualize higher dimensions as well as touch on the connection to my research in homological mirror symmetry.

Moduli

Martin Olsson
January 22

Abstract: One of the key notions in modern mathematics is the notion of moduli. It arises in algebraic geometry from the fact that classification of various objects of interest often leads to the exploration of other spaces (called moduli spaces). I will discuss some elementary examples of moduli in this talk.

Some Mathematics in Condensed Matter Physics

Mengxuan Yang
November 27

Abstract: Discovery of superconductivity and its physics theory has led to interesting mathematical studies of condensed matter physics. In this talk, using a model of twisted bilayer graphene recently developed by Becker-Embree-Wittsten-Zworski, I will introduce some mathematical ideas behind condensed matter physics, including representation theory and Bloch-Floquet theory. I will tell you some mathematical story behind "magic angles", at which you may find superconductivity.
Prerequisites: My talk only needs some linear algebra background. Some functional analysis would be helpful but that should not be necessary.

Differential Geometry of Surfaces

Alexander B. Givental
November 20

Abstract: In one hour, I'll try to formulate and prove the main result of Math 140, the celebrated Gauss--Bonnet theorem.
Prerequisites: The lecture will be very elementary; in particular, knowledge of calculus will not be assumed.

Ruler and compass constructions and abstract algebra

Emiliano Gomez
November 13

Abstract: This will be a pretty elementary and informal talk. We will see why some of the famous classical Greek ruler and compass constructions are impossible (doubling the square, squaring the circle, trisecting the angle). If time allows, we will also see which regular polygons are constructible and which ones are not. These problems were open for 2000 years, and their solution exemplifies the beauty and power of the interaction between geometry and algebra.
Prerequisites: There are very minimal requirements for the talk: I assume the audience knows a few Math 113 and Math 110 concepts: groups, rings, fields, rings of polynomials, vector spaces, and basic familiarity with complex numbers.

Algebraic Curves and the Riemann-Roch theorem

David Eisenbud
November 6

Abstract: The Riemann-Roch theorem is probably the most important single result in algebraic geometry. I'll try to explain what it says, and why it is important, without too many technical details.
Prerequisites: Math 113

The math behind a periodic table for 2-dimensional quantum matter

Colleen Delaney
October 30

Abstract: Ordinarily we think of particles like electrons as zipping around in 3 dimensions of space. But when these particles lose the freedom to move in 1 of these 3 dimensions, their collective behavior can form what is called a topological quantum phase of matter, the discovery of which was awarded the 2016 Nobel Prize in Physics. These exotic 2-dimensional phases (or states) of matter can support virtual or quasiparticles called anyons whose physics have a very mathematical flavor: (1) they depend only on the topology (global shape) and not the length or angles of the path the anyons carve out in spacetime and (2) they can be described using a branch of abstract algebra called category theory. The harmony between these topological and algebraic aspects of anyons can be expressed through a picture calculus obeyed by diagrams of anyon spacetime trajectories in which physics calculations are rigorously done by drawing intuitive pictures. Not only is this mathematical description of topological phases beautiful, it should also be useful! Topological phases are an active area of interdisciplinary research and development in both the public and private sectors motivated in part by the potential application of topological phases to quantum computing hardware and software, with national labs and companies like Microsoft and Google leading projects integrating theory and experiment.

Triple Symmetric Functions

Yegor Zenkevich
October 23

Abstract: Symmetric functions (e.g. Schur polynomials) play a major role in representation theory of general linear group and symmetric groups, a classical subjects. Recently it has been found that a mysterious generalization these functions which may be called triple symmetric functions seems to exist, but it is not clear what are the corresponding groups and representaitons. We will review the basics of Schur functions, then introduce the triple symmetric functions and play a bit with them.

Moduli spaces of curves

Hannah Larson
October 9

Abstract: I'll introduce the concept of moduli spaces in algebraic geometry with the example of the moduli space of circles. This is an example of a moduli space of "embedded curves." However, as I'll explain, the associated "abstract curves" are all the same. I'll finish by talking about moduli spaces of abstract curves and share some recent results, which are joint work with Samir Canning.
Prerequisites: The only requirements are complex numbers, polynomials, and a willingness to visualize!

Falconer's distance set conjecture and its many faces

Ruixiang Zhang
October 2

Abstract: In 1985, Falconer made the conjecture that whenever \(E \subset \mathbb{R}^2\) has dimension \(>1\), its distance set \(\Delta(E) = \{|x-y|: x, y \in E\}\) will have positive measure. Despite its innocent appearance, this conjecture remains unsolved as of today. We will introduce this conjecture and three interesting results towards it. The methods to prove these results are related to Fourier analysis, classical algebraic geometry/topology and geometric measure theory.
Prerequisites: Math 104 and 110

Large cardinal hypotheses and new axioms for set theory

Gabriel Goldberg
September 25

Abstract: In this talk I will introduce the ZFC axioms of set theory, which serve as an adequate foundation for almost all of mathematics but are inadequate to resolve many fundamental problems in set theory itself. I'll then turn to extensions of the ZFC axioms obtained by adding large cardinal hypotheses, also known as strong axioms of infinity, and finally I'll discuss axioms that are inspired by the ongoing search for canonical models of set theory.

The spectrum of an infinite product and the Stone-Čech compactification of \(\mathbb{N}\)

Owen Barrett
September 18

Abstract: Consider the spectrum of a countable product of a fixed field \(k\). Mumford wrote the following in his Red Book: ‘Those familiar with ultrafilters and similar far-out mysteries will have no problem proving that this topological space is the Stone-Čech compactification of \(\mathbb{N}\). Logicians assure us that we can prove more theorems if we use these outrageous spaces.’ This talk is about these outrageous spaces and their natural appearance in the study of commutative rings.

Crystals!

Nicole Gonzales
September 11

Abstract: A basic question in linear algebra is finding the coefficients of a vector in terms of a given basis. A fundamental goal in combinatorics is finding formulas for these coefficients when our vector spaces are certain polynomial rings. It turns out questions like these can be encoded and answered using certain directed graphs known as crystals. However, these crystals contain much more information than the original polynomials themselves. Algebraic properties like symmetries of the polynomials and multiplication rules are all encoded as symmetries and tensor product structures on these graphs. Hence, the polynomials can be seen as "shadows" of the crystals. In fact, the crystals themselves are also shadows of more complicated algebraic objects known as representations. In this talk we will introduce crystals from scratch and with lots of examples. We will show how the product of symmetric polynomials follows from tensoring certain symmetric crystal graphs. We will also discuss how certain bases in the full polynomial ring correspond to truncations of these symmetric crystals and how multiplying these truncated graphs turns out to be unexpectedly mysterious.

Monstrous Moonshine

Richard Borcherds
August 28

Abstract: This talk will be about the monster group of order 808,017,424,794,512,875,886,459,904, 961,710,757,005,754,368,000,000,000 and its relation to the elliptic modular function \(1/q + 744 + 196884q + 21493760q^2+ \ldots\)