Cluster algebras and monoidal categories

A monoidal category is called a monoidal categorification of a cluster algebra if its Grothendieck ring is isomorphic to the cluster algebra and the cluster monomials correspond to simple objects. In this talk, we will review the definition of cluster algebras and look at some examples of monoidal categorifications that arise in the representation theory of quantum affine algebras and quiver Hecke algebras.

Translation lengths in RAAGs

When a group G acts on a metric space (X,d) by isometries, the translation length of an element g of G  is defined as the limit of d(x, g^n x)/n as n tends to the infinity, where x is any point of X.

In the talk, I will give a brief overview of the translation lengths  for the action of a right-angled Artin group on the extension graph.  We will focus on the minimum nonzero translation length  and the rationality of the translation lengths.

Local and Nonlocal equations with degenerate coefficients

I will provide a brief overview of the history of regularity theory for local and nonlocal elliptic equations. In particular, I will focus on equations with degenerate or singular coefficients and introduce a recent work with Linus Behn, Lars Diening and Julian Rolfes from Bielefeld (Germany) on fractional, nonlocal equations with degenerate or singular coefficients and related function spaces.

Recent progress on the 1-loop conjecture

Motivated by the volume conjecture and its generalizations, Dimofte and Garoufalidis introduced a power series associated with an ideal triangulation of a cusped hyperbolic 3-manifold. This power series is conjectured to be equal to the asymptotic expansion of the Jones polynomial. In particular, its constant (1-loop) term is conjectured to correspond to the adjoint Reidemeister torsion. In this talk, I will introduce this 1-loop equal torsion conjecture and discuss recent progress.