Mathematics Colloquium
The Mathematics Department Colloquium is a quarterly series of invited speakers from all mathematical fields and geared toward a broad audience. Archives can be found here.
The talks are generally every Tuesday from 4-5 PM, with an informal tea beforehand. For the duration of COVID-19, it will be held via Zoom, with a meet-and-greet 30m before the talk-- bring your own beverage.
Schedule (click dates for title and abstract)
| Tues 1/19 | @ 4pm PST | Emily Clader | SF State University |
| Thurs 1/21 | @ 9am PST | Tam Nguyen Phan | Karlsruhe Institute of Technology |
| Fri 1/22 | @ 2pm PST | Nick Salter | Columbia University |
| Tues 1/26 | @ 2pm PST | Robin Neumayer | Northwestern University |
| Thurs 1/28 | @ 4pm PST | Jiayin Pan | UC Santa Barbara |
| Fri 1/29 | @ 2pm PST | Yiming Zhao | MIT |
| Tues 2/2 | @ 2pm PST | Kasia Jankiewicz | University of Chicago |
| Thur 2/4 | @ 9am PST | Mehdi Yazdi | Oxford University |
| Fri 2/5 | @ 9am PST | Luca Asselle | Justus Liebig Universität Gießen |
Emily Clader, SF State University
Permutohedral Complexes and Curves With Cyclic Action
There is a beautiful combinatorial story connecting a polytope known as the permutohedron, the algebra of the symmetric group, and the geometry of a particular moduli space of curves first studied by Losev and Manin. I will describe these three seemingly disparate worlds and their connection to one another, and then I will discuss joint work with C. Damiolini, D. Huang, S. Li, and R. Ramadas that generalizes the story to a family of "permutohedral complexes", a family of complex reflection groups, and a new family of moduli spaces.
Thursday, January 21, at 9am
Tam Nguyen Phan, Karlsruhe Institute of Technology, Institute of Algebra and Geometry
Flat cycles in the homology of congruence covers of SL(n,ℤ)\SL(n,ℝ)/SO(n)
The locally symmetric space SL(n,ℤ)\SL(n,ℝ)/SO(n), or the space of flat n-tori of unit volume, has immersed, totally geodesic, flat tori of dimension (n − 1). These tori are natural candidates for nontrivial homology cycles of manifold covers of SL(n,ℤ)\SL(n,ℝ)/SO(n). In joint work with Grigori Avramidi, we show that some of these tori give nontrivial rational homology cycles in congruence covers of SL(n,ℤ)\SL(n,ℝ)/SO(n). We also show that the dimension of the subspace of the (n − 1)-homology group spanned by flat (n − 1)-tori grows as one goes up in congruence covers. The prerequisite for this talk is very basic linear algebra.
Nick Salter, Columbia University
Families of Riemann surfaces and higher spin structures
Riemann surfaces are central objects in mathematics, bringing complex analysis, algebraic geometry, topology, group theory, dynamics (and more) into close conversation. In many situations, Riemann surfaces occur in families that parameterize some additional algebraic or geometric structure that can be placed on a fixed underlying topological surface. The first part of this talk will be an introduction to families of Riemann surfaces, with an emphasis on the topological aspects of the theory. In the second part, I will discuss some of my own contributions (in collaboration with Aaron Calderon and Pablo Portilla Cuadrado), concerning families of Riemann surfaces equipped with so-called higher spin structures, which arise in a surprising diversity of settings (linear systems on algebraic surfaces, singularity theory, Teichmüller dynamics).
Robin Neumayer, Northwestern University
Convergence and Regularity in Geometric Analysis
A broad theme in geometric analysis aims to understand the geometric structure of Riemannian manifolds that satisfy constraints on curvature. The notion of scalar curvature describes how a manifold is curved on average at each point and is of fundamental importance in general relativity and differential geometry. We will present recent developments in the study of scalar curvature, including convergence theorems for Riemannian manifolds with uniform scalar curvature lower bounds and quantitative stability estimates for the Yamabe problem. This talk is based on joint work with several collaborators.
Jiayin Pan, UC Santa Barbara
Fundamental groups of open manifolds with nonnegative Ricci curvature
In this talk, I will give a survey about my recent progress on the fundamental groups of open manifolds with nonnegative Ricci curvature. This includes finite generation and virtual abelianness of fundamental groups under certain geometric conditions. The main methods are asymptotic geometry and the structure theory of Ricci limit spaces.
Yiming Zhao, MIT
Recovering the shapes of convex bodies
Convex geometric analysis is the study of convex bodies from a geometric and analytic point of view. Examples of convex bodies include ellipsoids and polytopes. The boundary of a convex body is rarely smooth. In fact, the boundary of a convex body may well contain fractals. This talk will survey problems associated with shape recovery of convex bodies via isoperimetric inequalities (that date back to at least the ancient Greeks) and via inverse problems (known as Minkowski-type problems). Also presented will be connections of these problems with differential geometry, nonlinear PDEs, optimal mass transport, functional analysis, algebraic geometry, and, of course, applied math. No specific background is needed to follow this talk.
Kasia Jankiewicz, University of Chicago
Artin groups and their geometry
Artin groups form a broad family of infinite groups that includes free groups, free abelian groups, and braid groups. They arise as the fundamental groups of complex hyperplane arrangements, and are closely related to Coxeter groups. They are defined by simple-looking presentations, but their geometry is mysterious, and many basic questions about them remain open. I will discuss some of my contributions to understanding Artin groups.
Mehdi Yazdi, Oxford University
The fully marked surface theorem
In his seminal 1976 paper, Bill Thurston observed that a closed leaf S of a codimension-1 foliation of a compact 3-manifold has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. We give a converse for taut foliations: if the Euler class of a taut foliation F evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation G such that S is homologous to a union of compact leaves and such that the plane field of G is homotopic to that of F. In particular, F and G have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. My previous work, together with our main result, gives a negative answer to Thurston's conjecture. We mention how Thurston's conjecture leads to natural open questions on contact structures, flows, as well as representations into the group of homeomorphisms of the circle. This is joint work with David Gabai.
Luca Asselle, Justus Liebig Universität Gießen
The motion of a charged particle in a magnetic field
Understanding the motion of charged particles in a magnetic field is a classical problem which is still ubiquitous in contemporary physics: From collider accelerators and plasma fusion, where the magnetic field confines the particles in a desired region of space, to Earth’s magnetosphere and mass spectrometry, where the magnetic field acts as a mirror or prism. From a mathematical perspective, such a problem has been put into the context of Hamiltonian systems and symplectic geometry by Arnol’d in the early 1960s and has been since then object of extensive study. In this talk, after discussing the state of the art about existence and multiplicity of periodic orbits with prescribed energy, we will show how to apply an (improved) Hamiltonian version of the so-called “guiding center approximation” in classical electrodynamics to: i) show the existence of trapping regions via the KAM theorem; ii) prove the existence of periodic orbits; iii) show that the motion is periodic for every initial condition if and only if the magnetic field is constant and the surface has constant curvature. This is based on joint work with Gabriele Benedetti.
