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Bach Nguyen, Temple University
Cluster algebras was invented by Fomin and Zelevinsky around 2000 to study total positivity and canonical bases in Lie theory. Since then they have been applied to study many different subjects in mathematics such as commutative and noncommutative algebraic geometry, number theory, (quiver) representation theory, and mathematical physics.
This talk is a part of series of talks desired to give an inviting introduction to the theory of cluster algebras. We will discuss many motivating examples of cluster algebras and study properties and classification of cluster algebras. We also will cover some interesting connection between cluster algebras with number theory, Lie theory, dynamical system, and Poisson geometry if time permits.
Mihaela Ignatova, Temple University
I will describe results regarding the surface quasi-geostrophic equation (SQG) in bounded domains. The results concern global interior Lipschitz bounds for large data for the critical SQG in bounded domains. In order to obtain these, we establish nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. As an application, global existence of weak solutions of SQG are obtained. If time permits, I will also discuss an application to an electroconvection model.
Arjun Krishnan, University of Rochester
Consider a measurable dense family of semi-infinite nearest-neighbor paths on the integer lattice in d dimensions. If the measure on the paths is translation invariant, we completely classify their collective behavior in d=2 under mild assumptions. We use our theory to classify the behavior of semi-infinite geodesics in random translation invariant metrics on the lattice; it applies, in particular, to first- and last-passage percolation. We also construct several examples displaying unexpected behaviors. (Joint work with Jon Chaika.)
Bojana Gligorijevic, Temple University
Saul Schleimer, University of Warwick
Singular euclidean structures on surfaces are a key tool in the study of the mapping class group, of Teichmüller space, and of kleinian three-manifolds. François Guéritaud, while studying work of Ian Agol, gave a powerful technique for turning a singular euclidean structure (on a surface) into a triangulation (of a three-manifold). We will give an exposition of some of this work from the point of view of Delaunay triangulations for the L^\infty metric. We will review the definitions in a relaxed fashion, discuss the technique, and then present applications to the study of strata in the space of singular euclidean structures. If time permits, we will also discuss the naturally occurring algorithmic questions.
This is joint work with Mark Bell and Vaibhav Gadre. Some of our results are independently due to Ian Frankel, who has further applications.
Narek Hosyepyan, Temple University
We will discuss some interpolation formulae, such as Pick interpolation, recovery formulae for analytic functions from pieces of their boundary or interior data, and some aspects of the question of their extrapolation.
There are no conferences this week.