# Colloquium 2017-2018

The Colloquium is held on Mondays at 4pm in Deady 208.

### Spring Quarter, 2018

- April 2, No Colloquium – Moursund Lectures (Dusa McDuff)
- April 16, No Colloquium – Niven Lectures (Melanie Matchett Wood)
- April 23,
**Emmy Murphy** (Northwestern)

Contact geometry and knot theory
**Abstract**: associated to any smooth knot in R^3 we can define a torus in R^3 x S^2: the co-normal of the knot. The smooth isotopy type of the torus remembers nothing of the knot, however, the torus is compatible with a natural geometry on R^3 x S^2 called its contact structure. This additional geometry allows us to define a knot invariant, called knot contact homology, which are obtained from pseudo-holomorphic curve counts with boundary on our torus. We discuss some of the geometry underlying this theory, connections with other known knot invariants, completeness of this invariant, and other related topics.

- April 30,
**Stefan Steinerberger** (Yale)

Nice Sets of Points on Spheres, the Heat equation and Graph Theory
**Abstract**: Suppose we want to find a set of N points on the sphere S^2 that are regularly distributed. It’s pretty clear that we will go for the Dodecahedron if N=12 but it is less clear what to do if N = 11 or N = 13. I will discuss an old approach of Sobolev that gave rise to the study of a class of very regular points. This turns out to be related to Harmonic Analysis, Combinatorics and Partial Differential Equations. I will describe these connections and a new approach that generalizes to Graphs (many pretty pictures included) and might even have applications in medical sciences (admittedly, this came as a surprise to me as well).

- May 14,
**John Etnyre** (Georgia Tech)

Invariants of embeddings and immersions via contact geometry
**Abstract**: There is a beautiful idea that one can study topological spaces by studying associated geometric objects. More specifically one can associate to a manifold (that is a nice topological space) a symplectic or contact manifold (that is the geometric object). We will begin by looking at a classical and well-known theorem of Whitney concerning immersed curves in the plane and winding numbers. After an introduction to, and look back at the history of, contact geometry we will reinterpret Whitney’s result in terms of contact geometry and then vastly generalize it to come up with invariants of any submanifold in a manifold. Focusing on the case of knots in Euclidean 3-space we will describe recent work in knot contact homology and Ekholm, Ng, and Shende’s work showing that knot contact homology is (more or less) a complete invariant of knots.

- May 21,
**Akhil Mathew** (U. Chicago)

Integral p-adic Hodge theory (after Bhatt, Morrow, and Scholze)
**Abstract**: Given a smooth projective variety X over the complex numbers, one can study its topological singular cohomology. With complex coefficients, the singular cohomology is determined in terms of differential forms: essentially, by de Rham and Hodge theory. In particular, there is a purely algebraic description of its complex cohomology. Can one control the mod p cohomology in a similar fashion?

For a variety defined over Z (so can be reduced mod p), one can consider algebraic de Rham cohomology mod p as well. In general, this differs from the topological singular cohomology with F_p-coefficients. However, a recent result of Bhatt-Morrow-Scholze shows that (mod p) singular cohomology is always bounded by de Rham cohomology. This relies on the construction of a new cohomology theory, called A \Omega, which provides an interpolation between singular and de Rham cohomology. The cohomology theory is built using Scholze’s theory of perfectoid spaces.

In this talk, I will give an overview of the Bhatt-Morrow-Scholze cohomology theory and their semicontinuity result.

### Winter Quarter, 2018

- January 8,
**Jonathan Kujawa** (University of Oklahoma)

From UO to OU, or There and Back Again
**Abstract**: Support varieties were introduced in the 1980’s as a way to bring geometry to the representation theory of finite groups. It continues to be an active area of study and has inspired similar theories in a number of related settings. In this talk I’ll give an overview of support varieties and describe our ongoing work to bring this theory to the representations of Lie superalgebras. This is joint with Boe, Drupieski, Nakano, and others. No particular knowledge will be assumed and the talk will be aimed for a general math audience.

- January 15, No Colloquium – MLK Holiday
- January 22,
**David Pengelley** (Oregon State University)

“Voici ce que j’ai trouvé”: Did Sophie Germain (almost) prove Fermat’s Last Theorem?
**Abstract**: Recent discoveries in Sophie Germain’s manuscripts from around 1819 reveal a multifaceted attack on numerous different results about the Fermat equation, and even other Diophantine equations. Germain’s approaches are characterized by unfailing emphasis on theoretical techniques of broad applicability, and include considerable adumbration of a group-theoretic point of view.

Germain’s papers present a striking image of original ideas focused on ambitious results, but in great isolation and independence, even from her mentors Legendre and Gauss, which left her vulnerable to undetected errors. Her manuscripts suggest that she was polishing her work for submission to the French Academy’s prize competition on Fermat’s Last Theorem, even though she never made a submission, and the work remains unpublished. Her work has likely lain unread for nigh 200 years. We decipher and analyze her mathematical approaches, which argue for a substantial elevation of her stature as a number theorist, and we mention how her manuscripts have been used to teach a beginning number theory course.

- February 5,
**Benson Farb** (University of Chicago)

Braids, polynomials and Hilbert’s 13th problem
**Abstract**: There are still completely open fundamental questions about polynomials in one variable. One example is Hilbert’s 13th Problem, a conjecture going back long before Hilbert; indeed, the invention of algebraic topology grew out of an attempt to understand how the roots of a polynomial depend on the coefficients. The point of this talk is to explain part of the circle of ideas surrounding these questions.

Along the way we will see some beautiful classical objects – the space of monic, degree d square-free polynomials, hyperplane complements, algebraic functions, discriminants, braid groups, Galois groups, and configuration spaces – all intimately related to each other, all with mysteries still to reveal.

- February 19,
**Nikhil Srivastava** (UC Berkeley)

Interlacing Families
**Abstract**: Eigenvalues of random matrices play a central role in many areas of applied mathematics and computer science. Asymptotic random matrix theory has been immensely successful at precisely explaining the limiting spectra of large random matrices with independent entries (or other symmetries). For more general models in finite dimensions, the picture is less crystalline but tools such as the “Matrix Chernoff Bound” give useful coarse bounds on the extreme eigenvalues.

I will describe an object which shares features of both these regimes — the expected characteristic polynomials of finite random matrices — and which can be used to show that some of the sharp bounds from the former setting hold with non-negligible probability in the latter. The technique is based on certain interlacing relations between polynomials with all real roots, and is elementary and should be accessible to a general audience.

Based on joint work with Adam Marcus and Daniel Spielman.

- February 26,
**Emilie Purvine** (Pacific Northwest National Laboratory)

Applied category theory – Oxymoron or surprisingly natural?
**Abstract**: Category theory was developed in the 1940s by Eilenberg and Mac Lane as a way to unify their work in geometry and topology. The basics are simple: categories are collections of objects and maps between those objects (called morphisms). But when studying category theory one can quickly be pulled into the theoretical weeds. For many decades category theory remained mostly in the realm of pure mathematics due to this seeming unapproachability. There were some specific exceptions of motivated researchers finding applications in their own domains, but this was not widespread. Recent developments since the 2000s, however, have shown that principles from category theory can be broadly useful in the applied sciences. Category theory researchers have realized this applicability and made a point to educate non-mathematicians. In particular, these perspectives and approaches are illustrated in a book by David Spivak (Category Theory for the Sciences), and numerous papers and blog posts by others in the area. In this talk I will introduce the necessary category theory and survey the work of Spivak which studies databases through the lens of category theory, both theoretically and through practical examples.

**Special Time:** 3pm in Deady 208

- March 5,
**Henry Cohn** (Microsoft Research)

The sphere packing problem in dimensions 8 and 24
**Abstract**: What is the densest packing of congruent spheres in Euclidean space? This problem arises naturally in geometry, number theory, and information theory, but it is notoriously difficult to solve, and until recently no sharp bounds were known above three dimensions. In 2016 Maryna Viazovska found a remarkable solution of the sphere packing problem in eight dimensions, which is much simpler than the proof in three dimensions but tells us nothing about dimensions four through seven. In this talk I’ll describe how her breakthrough works and where it comes from, as well as follow-up work extending it to twenty-four dimensions (joint work with Kumar, Miller, Radchenko, and Viazovska).

### Fall Quarter, 2017

- September 25, No Colloquium
- October 2, No Colloquium
- October 9,
**Samuel Coskey** (Boise State University)

Borel complexity theory and classification problems
**Abstract**: Borel complexity theory is the study of the relative complexity of classification problems in mathematics. At the heart of this subject is invariant descriptive set theory, which is the study of equivalence relations on standard Borel spaces and their invariant mappings. The key notion is that of Borel reducibility, which identifies when one classification is just as hard as another. Though the Borel reducibility ordering is wild, there are a number of well-studied benchmarks against which to compare a given classification problem. In this talk we will introduce Borel complexity theory, present several concrete examples, and explore techniques and recent developments surrounding each.

- October 23,
**Kirsten Eisentraeger** (Pennsylvania State University)

Quantum algorithms and classical cryptography
**Abstract**: Computational problems that can be solved exponentially faster on a quantum computer than on a classical computer have mostly been number theoretic. It turns out that some of these problems, like factoring and the discrete log problem, are also required to be computationally difficult for certain cryptosystems to be secure. Hence RSA and Elliptic Curve Cryptography, that are based on the hardness of these problems, are not secure against quantum computers. In this talk I will discuss some recently proposed cryptosystems that have been suggested as alternatives to RSA and Elliptic Curve Cryptography. These fall into two categories, lattice-based systems and systems based on supersingular isogenies. We will discuss their security, both classically and against quantum computers.

- October 30,
**Mark Rudelson** (University of Michigan and MSRI)

Non-asymptotic approach in random matrix theory
- November 13,
**Dev Sinha** (UO)

Topological Data Analysis (&…)
**Abstract**: Topological data analysis aims to understand the shape of data sets, especially “large” ones, using tools from the relatively modern field of algebraic topology. We highlight two tools: the Reeb graph, which is organizes data graphically (as in, “with a graph,” but also as in “producing a picture”) with respect to a reference function, and persistent homology, which counts cycles in the data. We illustrate the effectiveness of these tools in three different settings: breast cancer, nanoporous materials, and neuroscience, drawing from papers from PNAS and Nature Comm. We will then also have some sociological discussion of the groups which have effectively done this work and musings about how such work – broadly defined to include the many areas of modern mathematics currently interfacing with science – could happen on our campus.

- November 27,
**Rafal Latała** (Warsaw/MSRI)

Upper and lower bounds for suprema of stochastic processes
**Abstract**: One of the fundamental questions of probability theory is the investigation of suprema of stochastic processes. Besides various practical motivations it is closely related to such important theoretical problems as boundedness and continuity of sample paths of stochastic processes, convergence of orthogonal series, random series and stochastic integrals, estimates of norms of random vectors and random matrices, limit theorems for random vectors and empirical processes, combinatorial matching theorems and many others.

The modern approach to this issue is based on the chaining methods. During the talk I will review several classical and more recent estimates for suprema of stochastic processes, discussing both lower and upper bounds. In particular I will present Dudley-Sudakov entropy-based bounds and Fernique-Talagrand generic chaining technique and will try to explore some of their applications and extensions.

- December 4,
**Kathryn Hess** (EPFL)

Topological vistas in neuroscience
**Abstract**: I will describe results obtained in collaboration with the Blue Brain Project on the topological analysis of the structure and function of digitally reconstructed microcircuits of neurons in the rat cortex and outline our on-going work on topology and synaptic plasticity. The talk will include an overview of the Blue Brain Project and a brief introduction to the topological tools that we use. If time allows, I will also briefly sketch other collaborations with neuroscientists in which my group is involved.

**Previous years:** 2016 2015 2013 2012 2011 2010