Abstract: Braids are familiar objects that appear when we twist and weave strands, but they also have a deep mathematical structure. In this talk, we will start by understanding what braid groups are, how they can be described geometrically and algebraically. From there, we’ll explore Artin groups, a natural generalization of braid groups, and look at their defining properties and some examples. 

Artin groups are at the centre of many exciting questions in modern mathematics. We will discuss some of the main open problems about these groups and highlight the wide range of techniques—spanning geometry, topology, and algebra—that mathematicians use to tackle them. The talk will be accessible to anyone with basic knowledge of algebra and topology. 

Abstract: In this talk, I will talk about the joint work with Dr. Dheeraj Kulkarni. The goal of this talk is to define a new invariant of Legendrian knots in R³ with the standard tight contact structure.

We will start by recalling some background on Legendrian knots and the Jones polynomial invariant for smooth knots. I will define two invariants of the Legendrian knot type, namely, the Legendrian Jones polynomial and the Legendrian Khovanov homology. We will see that both of these invariants are natural generalizations of the Jones polynomial and the Khovanov homology for smooth knots to the setting of Legendrian knots. In other words, for a Legendrian knot K, the Legendrian Jones polynomial PK(A, r) reduces to the Jones polynomial of the underlying smooth knot after substituting r = 1. We will also see that the Thurston-Bennequin number which is a classical invariant of the Legendrian knot type, occurs as a grade shift in the Legendrian Khovanov homology.

Abstract:  The unknotting number is one of the most elementary knot invariants, but in general also one of the hardest to compute.  I will describe a new lower bound on the unknotting number that can be extracted from Khovanov homology.  It subsumes all previously known lower bounds from Khovanov homology and offers some unexpected applications.  This is joint work with Lukas Lewark and Laura Marino.

Abstract: I am interested in optimal tube shapes; that is the shape of a tube, whose centreline is given as a curve and the curve trajectory minimises a certain energy function. The energy functions are inspired by the morphometric approach to modelling solvation and measure the change in thermodynamic potential energy when dissolving the tube in a fluid.— So I am thinking of a really really tiny tube that fits between the molecules of the fluid, I think of adding the tube to a beaker of water say, and see what shape the curve, and so tube, adopts within the fluid environment, understanding that the tube will relax into a thermodynamically favourable configuration. The knotting comes into to the picture because the entanglement of the curve, when thickened into a tube, constrains the shape the curve can adopt. This sort of investigation is an effort to understand the physical ramifications of knotting i.e. physical knot theory. The (nice) thing about physical knot theory is that, whilst you may begin with a bit of topology and geometric analysis, you soon get caught up in all sorts of other mathematics, along with needing to know something about optimisation algorithms and suddenly physics on top. The work I will present in this talk is no different; we will be meandering through a few topics; from biomolecules in fluids to integral geometry, in particular the curvature measures of tubular neighbourboods of curves, to shape optimisation, including the algorithmic implementation of thickening constraints of curves and simulated annealing optimisation of curve shape.

This is joint work with Prof. Myf Evans of the University of Potsdam.