We will look at the branching of irreducible representations of symmetric groups from the perspective of Okounkov-Vershik, and then look at Hecke algebras, affine Hecke algebras and cyclotomic Hecke algebras, in particular how the graded Grothendieck groups of their module categories “are” irreducible highest weight modules for affine $sl_l$, where $l$ is the “quantum characteristic”, and the branching graph is a highest weight crystal (for affine $sl_l$). The Fock space realisation of the highest weight crystal will get us back to  the Young graph for in the case of the symmetric group that we considered at the beginning.

We consider the four-point correlator of the stress-energy tensor in N=4 SYM, to leading order in inverse powers of the central charge, but including all order corrections in 1/lambda. This corresponds to the AdS version of the Virasoro-Shapiro amplitude to all orders in the small alpha'/low energy expansion. Using dispersion relations in Mellin space, we derive an infinite set of sum rules. These sum rules strongly constrain the form of the amplitude, and determine all coefficients in the low energy expansion in terms of the CFT data for heavy string operators, in principle available from integrability. For the first set of corrections to the flat space amplitude we find a unique solution consistent with the results from integrability and localisation.

The question of when you can quasiisometrically embed a solvable Baumslag-Solitar group into another turns out to be equivalent to the question of when you can (1,A)-quasiisometrically embed a rooted tree into another rooted tree. We will briefly describe the geometry of the solvable Baumslag-Solitar groups before attacking the problem of embedding trees. We will find that the existence of (1,A)-quasiisometric embeddings between trees is intimately related to the boundedness of a family of integer sequences. 

In this talk, I will investigate the origin of Euclidean wormholes in the gravitational part integral in the context of AdS/CFT. These geometries are confusing since they prevent products of partition functions to factorize, as they should in any quantum mechanical system. I will briefly review the different proposals for the origin of these wormholes, one of which is that one should consider ensemble of average of boundary systems instead of a fixed quantum system with a fixed Hamiltonian. I will explain that it seems unlikely that one can average over CFTs and present a new idea: averaging over approximate CFTs, which I will define. I will then study the variance of the crossing equation in an ensemble relevant for 3d gravity. Based on work in progress with de Boer, Jafferis, Nayak and Sonner.

In the first half of the talk, we will explore the concept of a characteristic class of a subvariety in a smooth ambient space. We will focus on the so-called stable envelope class,  in cohomology, K theory, and elliptic cohomology (due to Okoukov-Maulik-Aganagic). Stable envelopes have rich algebraic combinatorics, they are at the heart of enumerative geometry calculations, they show up in the study of associated (quantum) differential equations, and they are the main building blocks of constructing quantum group actions on the cohomology of moduli spaces.

In the second half of the talk, we will study a generalization of Nakajima quiver varieties called Cherkis’ bow varieties. These smooth spaces are endowed with familiar structures: holomorphic symplectic form, tautological bundles, torus action. Their algebraic combinatorics features a new powerful operation, the Hanany-Witten transition. Bow varieties come in natural pairs called 3d mirror symmetric pairs. A conjecture motivated by superstring theory predicts that stable envelopes on 3d mirror pairs are equal (in a sophisticated sense that involves switching equivariant and Kahler parameters). I will report on a work in progress, with T. Botta, to prove this conjecture.

Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on the non-empty open subset of algebraically smooth points, through which every invariant morphism factors uniquely. Some applications to quotients by connected groups will be discussed.