# Departmental Colloquium 2021-2022

*The schedule for last year, 2020-2021, can be found here.*

## Fall 2021 - Upcoming

__ September 23 (Thursday), 4:00pm__ -

*Virtual*

**Speaker:**David Constantine, Wesleyan University

**Title:**

*Geodesic flows on locally CAT(-1) spaces*

**Abstract:**

*Geodesic flows on compact, negatively curved Riemannian manifolds famously have lots of extremely nice dynamical properties. To what extent do those properties hold for geodesic flows on metric spaces that are negatively curved? In this talk I'll discuss how we can consider geodesic flows on general metric spaces, and then discuss some results on the geodesic flow of a compact, locally CAT(-1) space. It turns out that the CAT(-1) condition is sufficient for us to recover many nice properties. This is joint work with Jean-Francois Lafont and Daniel Thompson.*

__ September 30 (Thursday), 4:00pm__ -

*In person*

**Speaker:**James Murphy, Tufts University

**Title:**

*Geometric and Statistical Approaches to Shallow and Deep Clustering*

**Abstract:**

*We propose approaches to unsupervised clustering based on data-dependent distances and dictionary learning. By considering metrics derived from data-driven graphs, robustness to noise and ambient dimensionality is achieved. Connections to geometric analysis, stochastic processes, and deep learning are emphasized. The proposed algorithms enjoy theoretical performance guarantees on flexible data models and in some cases guarantees ensuring quasilinear scaling in the number of data points. Applications to image processing and bioinformatics will be shown, demonstrating state-of-the-art empirical performance. Extensions to active learning, generative modeling, and computational geometry will be discussed.*

__ October 7 (Thursday), 4:00pm__ -

*Virtual*

**Speaker:**Jennifer Balakrishnan, Boston University

**Title:**

*Rational points on curves and quadratic Chabauty*

**Abstract:**

*Let C be a smooth projective curve defined over the rational numbers with genus at least 2. It was conjectured by Mordell in 1922 and proved by Faltings in 1983 that C has finitely many rational points. However, Faltings' proof does not give an algorithm for finding these points, and in practice, given a curve, provably finding its set of rational points can be quite difficult.*

In the case when the Mordell--Weil rank of the Jacobian of C is less than the genus, the Chabauty--Coleman method can be used to find rational points, using the construction of certain p-adic line integrals. Nevertheless, the situation in higher rank is still rather mysterious. I will describe the quadratic Chabauty method (developed in joint work with N. Dogra, S. Müller, J. Tuitman, and J. Vonk), which can apply when the rank is equal to the genus. I will also highlight some examples of interest, from the time of Diophantus to the present day.

In the case when the Mordell--Weil rank of the Jacobian of C is less than the genus, the Chabauty--Coleman method can be used to find rational points, using the construction of certain p-adic line integrals. Nevertheless, the situation in higher rank is still rather mysterious. I will describe the quadratic Chabauty method (developed in joint work with N. Dogra, S. Müller, J. Tuitman, and J. Vonk), which can apply when the rank is equal to the genus. I will also highlight some examples of interest, from the time of Diophantus to the present day.

__ October 21 (Thursday), 4:00pm__ -

*In person*

**Speaker:**Alan Reid, Rice University

**Title:**

*The geometry, topology and arithmetic of Bianchi orbifolds and their finite covers*

**Abstract:**

*Let d be a square-free positive integer, and let O*

_{d }denote the ring of integers in Q(sqrt(−d)) . Then the groups PSL(2,O_{d}) are known as the Bianchi groups, and are a natural generalization of the modular group PSL(2,Z). The Bianchi groups are discrete subgroups of PSL(2,C), and as such act discontinuously on H^{3}. The quotient Bianchi orbifolds, Q_{d }= H^{3}/PSL(2,O_{d}) are non-compact finite volume hyperbolic 3-orbifolds with h_{d}(class number of O_{d }) cusps. These represent the totality of commensurability classes of non-compact arithmetic hyperbolic 3-orbifolds. This talk will survey some recent and not so recent work on understanding the geometry and topology of these orbifolds, their covers and connections to number theory.
__ October 26 (Tuesday), 4:00pm__ -

*Virtual,*AWM Speaker Series

**Speaker:**Deanna Needell, University of California Los Angeles

**Title:**

*On the topic of topic modeling: enhancing machine learning approaches with topic features*

**Abstract:**

*In this talk we touch on several problems in machine learning that can benefit from the use of topic modeling. We present topicmodeling based approaches for online prediction problems, computer vision, text generation, and others. While these problems have classical machine learning approaches that work well, we show that by incorporating contextual information via topic features, we obtain enhanced and more realistic results. These classical methods include non-negative matrix and tensor factorization, generative adversarial networks, and even traditional epidemiological SIR models for prediction. In this talk we provide a brief overview of these problems and show how topic features can be used in these settings. We include supporting theoretical and experimental evidence that showcases the broad use of our approaches.*

__ November 11 (Thursday), 4:00pm__ -

*In person, Distinguished Lecturer Series*

**Speaker:**Thomas Hales, University of Pittsburgh

**Title:**

*Formal Proof*

**Abstract:**

*A formally verified proof is a proof that has been checked by software to be free of errors. The software is based on the foundational axioms and rules of logic of mathematics. This talk will give a survey of this field of research, including some of the recent theorems that have been formally verified (such as the Kepler conjecture about sphere packings and the Continuum Hypothesis). I will describe some recent advances in the technology aimed at bringing these tools to a larger mathematical audience.*

__ November 18 (Thursday), 4:00pm__ -

*Virtual*

**Speaker:**Charles Smart, Yale University

**Title:**

*Localization and unique continuation on the integer lattice*

**Abstract:**

*I will discuss results on localization for the Anderson-Bernoulli model. This will include my work with Ding as well as work by Li-Zhang. Both develop new unique continuation results for the Laplacian on the integer lattice.*

__ December 2 (Thursday), 4:00pm__ -

*In person*

**Speaker:**Mason Porter, University of California Los Angeles

**Title:**

*Topological Data Analysis of Spatial Complex Systems*

**Abstract:**

*From the venation patterns of leaves to spider webs, roads in cities, social networks, and the spread of COVID-19 infections and vaccinations, the structure of many systems is influenced significantly by space. In this talk, I'll discuss the application of topological data analysis (specifically, persistent homology) to spatial systems. I'll discuss a few examples, such as voting in presidential elections, city street networks, spatiotemporal dynamics of COVID-19 infections and vaccinations, and webs that were spun by spiders under the influence of various drugs.*

## Summer 2021

__ July 6 (Tuesday), 4:00pm__ · (Online)

**Speaker:**Li-Cheng Tsai, Rutgers University

**Title:**

*When particle systems meet PDEs*

**Abstract:**

*Interacting particle systems are models that involve many randomly evolving agents or particles. These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems: the law of large numbers, random fluctuations, and large deviations (the study of rare events). Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.*