Colloquia

In 2022-2023, our colloquia will be in-person talks in B239 unless otherwise stated.

September 9 , 2022, Friday at 4pm Jing Tao (University of Oklahoma)

(host: Dymarz, Uyanik, WIMAW)

On surface homeomorphisms

In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, thus completing the work initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. Over the years, many alternative proofs have been obtained, using different aspects of surface theory. In this talk, I will overview the classical theory and sketch the ideas of a new proof, one that offers new insights into the hyperbolic geometry of surfaces. This is joint work with Camille Horbez.

September 23, 2022, Friday at 4pm Pablo Shmerkin (University of British Columbia)

(host: Guo, Seeger)

Incidences and line counting: from the discrete to the fractal setting

How many lines are spanned by a set of planar points?. If the points are collinear, then the answer is clearly "one". If they are not collinear, however, several different answers exist when sets are finite and "how many" is measured by cardinality. I will discuss a bit of the history of this problem and present a recent extension to the continuum setting, obtained in collaboration with T. Orponen and H. Wang. No specialized background will be assumed.

September 30, 2022, Friday at 4pm Alejandra Quintos (University of Wisconsin-Madison, Statistics)

(host: Stovall)

Dependent Stopping Times and an Application to Credit Risk Theory

Stopping times are used in applications to model random arrivals. A standard assumption in many models is that the stopping times are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. In the first part of the talk, we use a modified Cox construction, along with the bivariate exponential introduced by Marshall & Olkin (1967), to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We also present a series of results exploring the special properties of this construction.

In the second part of the talk, we present an application of our model to Credit Risk. We characterize the probability of a market failure which is defined as the default of two or more globally systemically important banks (G-SIBs) in a small interval of time. The default probabilities of the G-SIBs are correlated through the possible existence of a market-wide stress event. We derive various theorems related to market failure probabilities, such as the probability of a catastrophic market failure, the impact of increasing the number of G-SIBs in an economy, and the impact of changing the initial conditions of the economy's state variables. We also show that if there are too many G-SIBs, a market failure is inevitable, i.e., the probability of a market failure tends to one as the number of G-SIBs tends to infinity.

October 7, 2022, Friday at 4pm Daniel Litt (University of Toronto)

(host: Ananth Shankar)

The search for special symmetries

What are the canonical sets of symmetries of n-dimensional space? I'll describe the history of this question, going back to Schwarz, Fuchs, Painlevé, and others, and some new answers to it, obtained jointly with Aaron Landesman. While our results rely on low-dimensional topology, Hodge theory, and the Langlands program, and we'll get a peek into how these areas come into play, no knowledge of them will be assumed.

October 14, 2022, Friday at 4pm Andrew Sageman-Furnas (North Carolina State)

(host: Mari-Beffa)

Constructing isometric tori with the same curvatures

Which data determine an immersed surface in Euclidean three-space up to rigid motion? A generic surface is locally determined by only an intrinsic metric and extrinsic mean curvature function. However, there are exceptions. These may arise in a family like the isometric family of vanishing mean curvature surfaces transforming a catenoid into a helicoid.

For compact surfaces, Lawson and Tribuzy proved in 1981 that a metric and non-constant mean curvature function determine at most one immersion with genus zero, but at most two compact immersions (compact Bonnet pairs) for higher genus. In this talk, we discuss our recent construction of the first examples of compact Bonnet pairs. It uses a local classification by Kamberov, Pedit, and Pinkall in terms of isothermic surfaces. Moreover, we describe how a structure-preserving discrete theory for isothermic surfaces and Bonnet pairs led to this discovery.

The smooth theory is joint work with Alexander Bobenko and Tim Hoffmann and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.

October 20, 2022, Thursday at 4pm, VV911 Simon Tavaré (Columbia University)

(host: Kurtz, Roch)

Note the unusual time and room!

An introduction to counts-of-counts data

Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers C₁, C₂, … of species represented once, twice, … in a sample of size

N = C₁ + 2 C₂ + 3 C₃ + ^…. containing S = C₁ + C₂ + ^… species; the vector C = (C₁, C₂, …) gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.

In this talk I will outline some of the stochastic models used to model the distribution of C, and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of S in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.

References

[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943

[2] Arratia R, Barbour AD & Tavaré S. Logarithmic Combinatorial Structures, EMS, 2002

[3] Ewens WJ. Theoret Popul Biol, 3, 1972

[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)