Click on seminar heading to go to seminar page.
Vasily Dolgushev, Temple University
The talk will be devoted to open questions of the theory of cluster algebras: the positivity conjecture, its strong version and the conjecture on cluster monomials. I will follow closely the paper by Lauren Williams "Cluster algebras: an introduction".
Rafe Mazzeo, Stanford University
The notion of a moduli space (of geometric objects, or solutions to a problem) goes back (at least) to the 19th century and captures the idea of "fully describing" the family of all solutions to a given equation. This plays a central role in algebraic geometry. However, moduli spaces of "geometric structures", originating in the work of Riemann, have come to play an increasingly important role in many other parts of mathematics and physics. While this subject is far too large to do much justice to in an hour, I will describe some interesting settings, both old and new, concerning geometric features of the totality of solutions to a given partial differential equation This will be a light-hearted survey with a lot of pictures.
Rafe Mazzeo, Stanford University
In this and the next lecture I will describe a collection of related results by several authors, obtained over the past several years, concerning the geometry of the so-called Hitchin moduli space, and its relationship with the emerging Kapustin-Witten theory. The Hitchin moduli space is a rather fundamental object, with deep connections to several fields, and I will describe some of its basic features, assuming you have never seen it before and have no idea why it might be interesting. This will lead to newer work on its asymptotic geometry. I will then connect this to Kapustin-Witten theory, conjectured by Witten to lead to a new analytic way to study certain knot invariants. The connection between these different fields is that solutions to the Hitchin equations serve as boundary conditions for this higher dimensional theory.
Emily Stark, Technion - Israel Institute of Technology
Two far-reaching methods for studying the geometry of a finitely generated group with non-positive curvature are (1) to study the structure of the boundaries of the group, and (2) to study the structure of its finitely generated subgroups. Cannon--Thurston maps, named after foundational work of Cannon and Thurston in the setting of fibered hyperbolic 3-manifolds, allow one to combine these approaches. Mitra (Mj) generalized work of Cannon and Thurston to prove the existence of Cannon--Thurston maps for normal hyperbolic subgroups of a hyperbolic group. I will explain why a similar theorem fails for certain CAT(0) groups. I will also explain how we use Cannon--Thurston maps to obtain structure on the boundary of certain hyperbolic groups. This is joint work with Algom-Kfir--Hilion and Beeker--Cordes--Gardham--Gupta.
Rafe Mazzeo, Stanford University
In this lecture I continue with the description of a collection of related results by several authors, obtained over the past several years, concerning the geometry of the so-called Hitchin moduli space, and its relationship with the emerging Kapustin-Witten theory. The Hitchin moduli space is a rather fundamental object, with deep connections to several fields, and I will describe some of its basic features, assuming you have never seen it before and have no idea why it might be interesting. This will lead to newer work on its asymptotic geometry. I will then connect this to Kapustin-Witten theory, conjectured by Witten to lead to a new analytic way to study certain knot invariants. The connection between these different fields is that solutions to the Hitchin equations serve as boundary conditions for this higher dimensional theory.
Nordine Mir, Texas A&M University-Qatar
We discuss recent joint results with B. Lamel on the $C^\infty$ regularity of CR maps between smooth CR submanifolds embedded in complex spaces of possibly different dimensions. Applications to the boundary regularity of proper holomorphic maps of positive codimension between domains with smooth boundaries of Dâ€™Angelo finite type will be given.
Michael Morabito, Lehigh University
von Willebrand Factor (vWF) is a large multimeric protein found in blood plasma. vWF plays an indispensable role in the blood clotting process by initiation of clot formation that stops bleeding due to vascular damage. vWF is able to sense elevated hydrodynamic force in blood flow at the site of vessel hemorrhage, and respond by undergoing conformational changes. Understanding the functionality of this flow-sensitive biological polymer requires interdisciplinary collaboration. The dynamics and mechanical response behavior of vWF can be probed using coarse-grained Brownian molecular dynamics simulations. The mathematical foundations of this method will be presented, and simulation results for vWF in shearing flows will be discussed. Simulation and experimental results are also used as input to machine learning algorithms, which have proven to be powerful data-driven analysis tools for this bioinformatics application.
There are no conferences this week.