Departmental Colloquium

	Date	Speaker	Talk Information
View details	September 23	David Constantine Wesleyan University	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.
View details	September 30	James Murphy Tufts University	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.
View details	October 7	Jennifer Balakrishnan Boston University	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.
View details	October 21	Alan Reid Rice University	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.
View details	October 26	Deanna Needell University of California Los Angeles	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.
View details	November 11	Thomas Hales University of Pittsburgh	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.
View details	November 18	Charles Smart Yale University	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.
View details	December 2	Mason Porter University of California Los Angeles	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.
View details	December 14	Rachel Skipper Ohio State University	We will take a tour through some families of groups of historic importance in geometric group theory, including self-similar groups and Thompson’s groups. We will discuss the rich, continually developing theory of these groups which act as symmetries of the Cantor space, and how they can be used to understand the variety of infinite simple groups. Finally, we will discuss how these groups are serving an important role in the newly developing field of big mapping class groups which are used to describe symmetries of surfaces. Summer 2021 - past

Date

Speaker

Talk Information

September 23

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.

View details

September 30

James Murphy
Tufts University

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.

View details

October 7

Jennifer Balakrishnan
Boston University

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.

View details

October 21

Alan Reid
Rice University

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.

View details

October 26

Deanna Needell
University of California Los Angeles

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.

View details

November 11

Thomas Hales
University of Pittsburgh

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.

View details

November 18

Charles Smart
Yale University

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.

View details

December 2

Mason Porter
University of California Los Angeles

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.

View details

December 14

Rachel Skipper
Ohio State University

We will take a tour through some families of groups of historic importance in geometric group theory, including self-similar groups and Thompson’s groups. We will discuss the rich, continually developing theory of these groups which act as symmetries of the Cantor space, and how they can be used to understand the variety of infinite simple groups. Finally, we will discuss how these groups are serving an important role in the newly developing field of big mapping class groups which are used to describe symmetries of surfaces. Summer 2021 - past

Department of Mathematics

College of Science

Fall 2021