Calendar of Events

We propose a new framework of monoidal categorifications of finite ADE types cluster algebras involving triangulated categories instead of abelian categories. Let Q be any Dynkin quiver and let A_Q denote the corresponding finite type cluster algebra (with a suitable choice of frozen variables). We define a certain additive symmetric monoidal category out of the Auslander-Reiten theory of Q and consider its (bounded) homotopy category K_Q. Using some iterated mapping cone technique, we construct a family of chain complexes in bijection with almost positive roots that are characterized by natural exactness conditions. We then prove that the Euler charasteristics of the chain complex associated to any positive root β coincides with the truncated q-character of the simple module categorifying the cluster variable x[β] in A_Q via Hernandez-Leclerc’s monoidal categorification. We also conjecture that all exchange relations in A_Q are categorified by distinguished triangles in K_Q. If time allows, we will discuss potential extensions of this construction to other cluster structures.

Understanding when stable vector bundles with fixed topological invariants exist---or fail to exist---is a longstanding and difficult problem. While there are many topological obstructions, algebraic ones are rarer: beyond the classical Bogomolov inequality, which gives a quadratic bound in terms of the rank and the first two Chern characters, few general tools are available. In higher dimensions, one expects…

Over the last twenty years, many people have contributed to a large-scale
programme to study invariants of size across mathematics. At its heart is
the definition of magnitude, a numerical invariant of enriched categories
(including metric spaces and graphs). It unifies size-like invariants such
as groupoid cardinality, Euclidean measure, Euler characteristic, metric
dimension, and entropy. It also comes with its own homology theory that
categorifies it (à la Khovanov), and even a spectral sequence.

The magnitude bibliography currently lists 141 works by 130 authors from
vastly different parts of mathematics and beyond. That's way too much to
cover in one talk, so I'll concentrate on two things: what I view as the
outstanding achievements and the outstanding open questions.

Symplectic singularities, representation theory and Integrability

Symplectic singularities, representation theory and Integrability

Symplectic singularities, representation theory and Integrability

Symplectic singularities, representation theory and Integrability