# Colloquium 2019-2020

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

### Winter Quarter, 2020

- January 27,
**Marina Guenza** (University of Oregon, Chemistry)

Bridging length scales in molecular liquids
- February 10,
**Ailana Fraser** (UBC)

Extremal eigenvalue problems and minimal surfaces
**Abstract**: When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. For surfaces, the critical metrics turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and Euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. We will give an overview of progress that has been made for surfaces with boundary, and in higher dimensions.

- March 2,
**Renate Scheidler** (Calgary)

A journey of cryptography in class groups of quadratic fields
**Abstract**: Cryptography in class groups of quadratic fields dates back to 1988, with the advent of the first Diffe-Hellman key style agreement protocol whose security resides in the intractability of extracting discrete logarithms in the class group of an imaginary quadratic field. Since then, the area has undergone a turbulent history. A host of other class group cryptosystems were put forward, founded on both the discrete logarithm problem and the integer factorization problem. Following devastating breaks of the factoring-based schemes in 2009, the topic made a come-back in 2015 with the advent of linearly homomorphic encryption in class groups of imaginary quadratic fields, which revived research in this area. Class groups also feature prominently in elliptic curve isogeny based cryptographic protocols. This talk tells the tumultuous story of class group based cryptography, from its beginnings some 30 years ago to ongoing research on quantum resistant schemes. No background preparation beyond first undergrad courses in abstract algebra and number theory as well as general public knowledge of cryptography is required.

### Spring Quarter, 2020

- May 4,
**Jonathan Kadmon** (Stanford, Neuroscience)
- May 11,
**Andy Putman** (Notre Dame)

### Fall Quarter, 2019

- October 21,
**Brittany Erickson** (University of Oregon, CS)

A Computational Framework for Earthquake Sequences in Nonlinear Media
**Abstract**: Deficient understanding of the earthquake cycle is the single greatest barrier to minimizing the devastating effects of earthquakes on society and the human environment. An ongoing goal of earthquake science is to synthesize physical models with observational data in order to assess seismic hazards and promote societal resilience. In this talk I will share my recent work in the development of a new computational framework for physically robust models of the earthquake cycle. One of the greatest challenges in earthquake modeling stems from the fact that earthquakes are not stand-alone, independent events, but rather are influenced by a millenial-long history of plate motion and a cycle of past earthquakes. Computations are further complicated because faults contain geometric and frictional complexities, and the solid Earth is heterogeneous and behaves nonlinearly. I will discuss details of the applied mathematical techniques that form the computational methods, while providing a context within the larger scope of earthquake science.

- October 28,
**Phillip Wood** (UC Berkeley)

Singularity of Discrete Random Matrices
**Abstract**: Consider an n by n square matrix where n is large. For each entry, flip a fair coin, making the entry +1 if the coin comes up heads, and -1 if the coin comes up tails. What is the probability that the matrix has determinant equal to zero? This talk will highlight work of Terence Tao and Van Vu, Rudelson and Vershynin, joint work with Jean Bourgain and Van Vu, and a recent breakthrough by Konstantin Tikhomirov. One idea that runs through all the work is quantifying structure in random objects in order to understand their behavior.

- November 11,
**Matthew Gursky** (Notre Dame)

Asymptotically hyperbolic Einstein metrics: existence, obstructions, and generalizations
**Abstract**: In this talk I will describe an interesting singular boundary value problem in differential geometry that has important connections to theoretical physics. The basic example of a solution is the Poincare ball model of hyperbolic space, so I will begin by reviewing some well known properties of hyperbolic space as a way of introducing the general boundary value problem. After describing some important existence results, I will spend some time at the end talking about obstructions to solving the problem, and some generalizations of the existence question that have been the focus of considerable attention lately.

- November 25,
**Aaron Lauda** (USC)

A new look at quantum knot invariants
**Abstract**:

In this talk we will explain how Lie theory leads to interesting families of invariants for knots and links that can all be defined in an elementary diagrammatic fashion.

The Reshetikhin-Turaev construction associated knot invariants to the data of a simple Lie algebra and a choice of irreducible representation. The Jones polynomial is the most famous example coming from the Lie algebra sl(2) and its two-dimensional representation. In this talk we will explain Cautis-Kamnitzer-Morrison’s novel new approach to studying RT invariants associated to the Lie algebra sl(n). Rather than delving into a morass of representation theory, we will show how two relatively simple Lie theoretic ingredients can be combined with a powerful duality (Howe duality) to give an elementary and diagrammatic construction of these invariants. We will explain how this new framework solved an important open problem in representation theory, proves the q-holonomic conjecture in knot theory (joint with Garoufalidis and LĂȘ), and leads to a new elementary approach to `categorifying’ these knots invariants to link homology theories.

- December 2,
**Julie Rowlett** (Chalmers University)

The mathematics of “hearing the shape of a drum”
**Abstract**: Have you heard the question, “Can one hear the shape of a drum?” Do you know the answer? In 1966, M. Kac’s article of the same title popularized the inverse isospectral problem for planar domains. Twenty six years later, Gordon, Webb, and Wolpert demonstrated the answer, but many naturally related problems remain open today. We will discuss old and new results inspired by “hearing the shape of a drum.”

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