Speaker: E. Mehmet Kiral. TBA.
Canonical bases first appeared as Kazhdan-Lusztig bases in the study of Hecke
algebras which can be thought of as "q-deformations" of symmetric groups (or,
more generally, Coxeter groups). These are distinguished bases of an algebra
with prescribed invariance properties when q is replaced by its inverse. They
are also triangular with respect to some natural initial basis. Such bases
play an important role in algebra since Lusztig constructed them in quantum
groups in early 90-ies using geometric methods.
Dual graph diagrams are an alternate way to present oriented knots and links in $R^3$. In this talk we will see how to turn dual graph Reidemeister moves into an algebraic structure known as biquasiles and use this structure to define new integer-valued counting invariants of oriented knots and links.
Planar algebras were first introduced in the late 90's by Vaughan Jones as an axiomatization of the standard invariant of a subfactor. Jones' idea was that the structure of standard invariants had a description in terms of planar diagrams, and that one could compute things about the subfactor by manipulating the pictures. Since then, planar algebras have been used extensively as a framework for performing rigorous calculations by manipulating diagrams. In this talk I will give an example-driven introduction to planar algebras and diagrammatic calculation, and demonstrate some of the features of working with pictures. If time permits, I will also discuss the state of the on-going project to classify `small' planar algebras, as well as the role played by planar algebras in constructive quantum field theory.
Diffusion is a fundamental transport mechanism whereby spatial paths are determined from probabilistic distributions. In examples such as the pollination of a flower or immune response to infection, the arrival of a single particle can initiate a cascade of events. The movement of these particles is driven by random motions, yet these systems largely function in an ordered and predictable way. This process, and others like it, can be modeled as a problem for the arrival time of a diffusing molecule to hit a small absorbing target. In this talk I will discuss the mathematical models of this phenomenon and introduce the governing equations which are PDEs of parabolic and elliptic type with a mix of Dirichlet (Absorbing) and Neumann (Reflecting) boundary conditions. A particular feature of cellular problems is that the absorbing set has a large number of very small sites. I will present a new homogenized theory which replaces the heterogeneous configuration of boundary conditions with a uniform Robin type condition. To verify this limit, I will describe a novel spectral boundary element method for the exterior mixed Neumann-Dirichlet boundary value problem. The numerical formulation reduces the problem to a linear integral equation supported on each of the absorbing sites. Real biological systems feature thousands of absorbing sites and our numerical method can rapidly resolve this realistic limit to high accuracy.
Lattices are discrete periodic structures in Euclidean spaces, which are essential in many areas of mathematics and their applications. In particular, lattices are vital in number theory, discrete geometry, discrete optimization, digital communication, and other areas. Some of the most interesting lattice constructions come from various algebraic contexts, resulting in lattices with remarkable geometric properties: it appears that algebraic structure often informs the geometry. One of such properties is extremality, which ensures certain strong optimization potential, however extremal lattices are not so easy to construct. In this talk we will give a basic overview of lattice theory with a view towards optimization properties, and will exhibit some new systematic constructions of extremal lattices from surprisingly simple algebraic objects.
We begin with a brief discussion of the classical hyperbolic metric on the unit disk and present several of its generalizations, commonly known as hyperbolic-type metrics. In some details we will discuss two particular metrics, Scale-Invariant Cassinian metric and Mobius-Invariant Cassinian metric. We end the talk with a discussion of averaging hyperbolic-type metrics and its potential applications in geometric function theory.