C5

The existence of minimizers in the calculus of variations is closely tied to various notions of convexity. A fundamental question in this regard is how these notions relate to each other. 

In this talk, we study several such generalized notions of convexity for continuous integrands on the space of symmetric $3\times 3$ matrices and establish their mutual relationships.

In particular, we construct a symmetric quasiconvex integrand that is not polyconvex at symmetric matrices, and an integrand that is both symmetric rank-one convex and Hessian quasiconvex but not symmetric quasiconvex. We also establish a transfer principle that converts smooth rank-one convex functions that fail to be Hessian quasiconvex into symmetric rank-one convex integrands while preserving this failure. Applying this scheme to existing counterexamples, we obtain symmetric rank-one convex integrands that are not Hessian quasiconvex. Together with previous results, this settles all pairwise implications among the notions considered here.

TBA

Stallings core graphs are a powerful graphical representation of the finitely generated subgroups of a free group. They bring geometric insight to the study of free groups, make many properties and algorithmic tasks transparent, and are a ubiquitous tool in combinatorial and geometric group theory.
Free group algebras are strikingly analogous to free groups. Most notably, just as the Nielsen-Schreier theorem says that every subgroup of a free group is free, the Cohn-Lewin theorem says that every right (or left) ideal of a free group algebra is free. Yet free group algebras have had no counterpart to core graphs.
In this talk I will present one. We define the core of a module over a free group algebra, and show that it is unique whenever the module is algebraic (also known as bound). Many of the concepts of Stallings' theory, such as folding and pullbacks, generalize to these new objects, and this yields simple proofs and algorithms for many natural problems about free group algebras. For example, I will present an algorithm for testing primitivity in free modules which generalizes (Wilton's simplification of) Whitehead's classical algorithm for testing primitivity in free groups.
This is joint work with Doron Puder.
 

Concordances between knots have provided much framework for the study and classification. The slice-ribbon conjecture posed by Fox in 1961 continues to be one of the largest open problems in the field. In this talk, I will provide background on concordances and the slice-ribbon conjecture, as well as explaining how concordances can induce maps on both Khovanov and Knot-Floer homology. I will then cover how this ties into my own current research and future directions.

Using the techniques of arc presentations and open book foliations, we prove new lower bounds on the crossing number and arc index of a satellite knot in terms of it's companion, pattern, and wrapping number.

In this talk I will discuss homological and homotopical isoperimetric inequalities, including those of higher rank.  I will emphasise what is known in the non-positive curvature case.