## Colloquium

To receive schedule updates by email, contact the Colloquium Chair, Prof. Marius Beceanu.

The colloquium usually meets on Friday at 3:00 pm in room ES-143.

The colloquium is partially supported from the Simons grant #584738 (Cristian Lenart)

John Meier (Lafayette College)

Topology at Infinity and Robots on Graphs

Friday, January 25, 2019

3:00 p.m. in ES-143

Abstract: The topology at infinity for an infinite, locally finite cell complex is the topology that persists in the complement of any finite subcomplex. For example, a graph is connected at infinity (more commonly called one-ended) if there is a single connected, unbounded component in the complement of any finite subgraph. We will focus on the properties of being connected at infinity and simply connected at infinity, in the context of infinite groups. The primary examples will come from braid groups on graphs, also thought of as robot motion planning where the robots move on fixed tracks.

Leonardo Mihalcea (Virginia Tech)

Cotangent Schubert Calculus

Friday, February 8, 2019

3:00 p.m. in ES-143

Abstract: Modern Schubert Calculus studies various intersection rings associated to flag manifolds. All these rings have several common features: they have a basis - the Schubert basis - indexed by a finite (permutation) group; the Schubert structure constants count points; the Schubert classes can be defined by equivariant localization; etc. A question with roots in representation theory and microlocal analysis is whether there are good analogues of Schubert classes to study intersection rings of the cotangent bundle of a flag manifold. One answer is given in terms of the characteristic classes of singular subvarieties in the flag manifold, such as the Chern-Schwartz-MacPherson classes. For flag manifolds, these classes are equivalent to stable envelopes defined recently by Maulik and Okounkov. I will explain these ideas, and draw parallels with the Schubert Calculus situation. For instance, instead of counting points in three Schubert cycles, in the cotangent situation one takes the Euler characteristic of the intersection of three Schubert cycles.

Jae-Hoon Kwon (Seoul National University)

Representations of classical Lie algebras and combinatorics of their branching rules

Friday, February 15, 2019

3:00 p.m. in ES-143

Abstract: The theory of crystal basis provides a powerful tool to study combinatorial structure of representations of quantum groups. In this talk, we introduce a combinatorial model for irreducible characters of classical Lie algebras of type BCD, which is obtained by applying the crystal basis theory to higher level fermionic Fock space. As one of its applications, we explain how classical branching rules for irreducible characters of classical Lie groups in a stable range of highest weights, including the well-known Littlewood restriction rule, can be extended to arbitrary range in a bijective way.

Michael Lesnick (University at Albany)

Quantifying Genetic Innovation: Mathematical Foundations for the Topological Study of Reticulate Evolution

Friday, February 22, 2019

3:00 p.m. in ES-143

Abstract: A topological approach to the study of genetic recombination, based on persistent homology, was introduced by Chan, Carlsson, and Rabadán in 2013. This associates a sequence of signatures called barcodes to genomic data sampled from an evolutionary history. In this talk, I will discuss work which develops a theoretical foundation for this approach. We introduce and study the novelty profile, a simple, stable statistic of an evolutionary history which not only counts recombination events but also quantifies how recombination creates genetic diversity. We then study the problem of lower-bounding the novelty profile using barcodes. We focus on a low-recombination regime, where the evolutionary history can be described by a directed acyclic graph called a galled tree, which differs from a tree only by isolated topological defects. This is joint work with Raúl Rabadán and Daniel Rosenbloom.

Ralf Schiffler (UConn)

An introduction to cluster algebras

Friday, March 1, 2019

3:00 p.m. in ES-143

Abstract: Cluster algebras are commutative algebras with a special combinatorial structure. They originated in Lie theory in the context of canonical bases and total positivity, but are now connected to a number of areas including combinatorics, representation theory, knot theory, integrable systems and algebraic geometry. In this talk, we give an overview of the theory and highlight some of its applications.

Israel Michael Sigal (University of Toronto)

Magnetic vortex lattices

Friday, March 15, 2019

3:00 p.m. in ES-143

Abstract: The Ginzburg - Landau equations play a fundamental role in various areas of physics, from superconductivity to elementary particles. They present the natural and simplest extension of the Laplace equation to line bundles. Their non-abelian generalizations - Yang-Mills-Higgs and Seiberg-Witten equations have applications in geometry and topology.

Of a special interest are the least energy (per unit volume) solutions of the Ginzburg - Landau equations. Though the equations are translation invariant, these turned out to have a beautiful structure of (magnetic) vortex lattices discovered by A.A. Abrikosov. (Their discovery was recognized by a Nobel prize. Finite energy excitations are magnetic vortices, called Nielsen-Olesen or Nambu strings, in particle physics.)

I will review recent results about the vortex lattice solutions and their relation to the energy minimizing solutions on Riemann surfaces and, if time permits, to the microscopic (BCS) theory.

Matthew Zaremsky (University at Albany)

Discrete Morse theory on Vietoris-Rips complexes

Friday, March 29, 2019

3:00 p.m. in ES-143

Abstract: Discrete Morse theory is a powerful tool for leveraging "local" topological information about a cell complex to make "global" topological conclusions. One incarnation is Forman's discrete Morse theory, which is popular in topological data analysis (TDA). Another is Bestvina-Brady discrete Morse theory, which has proven to be a fundamental tool in geometric group theory (GGT). In this talk I will discuss a new generalization that encompasses both, and explain how one can view Forman's theory as a special case of Bestvina-Brady's. I will also discuss some applications, both to TDA and to GGT, involving Vietoris-Rips complexes of metric spaces.

**Maheshwari Colloquium:** Ken Ono (Emory University)

Why does Ramanujan, "The Man Who Knew Infinity,” matter?

Friday, April 12, 2019

4:00 p.m. in Lecture Center 23.

Reception in the lobby at 3:15 p.m.

Abstract: Srinivasa Ramanujan (1887-1920), one of the most inspirational figures in the history of mathematics, was a poor Brahmin from lush south India who left behind three shabby notebooks that engineers, mathematicians and physicists continue to mine today. His ideas came to him as visions from a Hindu goddess. As a two-time college dropout, he could have easily been lost to the world, a thought that scientists cannot begin to absorb. Ono will explain why Ramanujan matters today, and he will share several clips from the film "The Man Who Knew Infinity" (starring Dev Patel and Jeremy Irons) about Ramanujan. He shall promote the “Idea of Ramanujan”: talent is often found in the most unpromising circumstances. It must be located and then nurtured.

Rongwei Yang (University at Albany)

TBD

Friday, April 26, 2019

3:00 p.m. in ES-143