L6

Fix a loop group LG, a level k∈ℕ, and let Repᵏ(LG) be corresponding category of positive energy representations.
For any pair of pants Σ (with complex structure in the interior and parametrized boundary), there is an associated functor Repᵏ(LG) × Repᵏ(LG) → Repᵏ(LG): (H,K) ↦ H⊠K, called the fusion product.

It had been widely expected (but never proven) that this operation should be unitary. Namely, that the choice of LG-invariant inner products on H and on K should induce an LG-invariant inner product on H⊠K. We show that this is not the case: there is an anomaly.
More precisely, there is an ℝ₊-torsor canonically associated to Σ. It is only after trivialising of this ℝ₊-torsor that the fusion product acquires an LG-invariant inner product. (The same statement applies when Σ is an arbitrary Riemann surface with boundary.)
Joint work with James Tener.

I will discuss ongoing work aimed at constructing higher categories of (enriched) higher categories. This should give the appropriate targets for many interesting examples of extended topological quantum field theories, including extended versions of the classical examples of TQFTs due to Turaev-Viro, Reshetikhin-Turaev, etc.

Klarrich showed that the Gromov boundary of the curve complex of a hyperbolic surface is homeomorphic to the space of ending laminations on that surface. Independent results of Bestvina-Reynolds and Hamenstädt give an analogous statement for the free factor graph of a free group, where the space of ending laminations is replaced with a space of equivalence classes of arational trees. I will give an introduction to these objects and describe some joint work with Bestvina and Horbez, where we show that the Gromov boundary of the free factor graph for a free group of rank N has topological dimension at most 2N-2.

For a smooth variety over a number field, one defines various different homology groups (Betti, de Rham, etale, log-crystalline), which carry various kinds of enriching structure and are thought of as a system of realisations for a putative underlying (mixed) motivic homology group. Following Deligne, one can study fundamental groups in the same way, and the study of specific realisations of the motivic fundamental group has already found Diophantine applications, for instance in the anabelian proof of Siegel's theorem by Kim.

It is hoped that study of fundamental groups should give one access to ``higher'' arithmetic information not visible in the first cohomology, for instance classical and p-adic heights. In this talk, we will discuss recent work making this hope concrete, by demonstrating how local components of canonical heights on abelian varieties admit a natural description in terms of fundamental groups.

Let k be a field of characteristic zero and K=k((t)). Semi-algebraic sets over K are boolean combinations of algebraic sets and sets defined by valuative inequalities. The associated Grothendieck ring has been studied by Hrushovski and Kazhdan who link it via motivic integration to the Grothendieck ring of varieties over k. I will present a morphism from the former to the Grothendieck ring of motives of rigid analytic varieties over K in the sense of Ayoub. This allows to refine the comparison by Ayoub, Ivorra and Sebag between motivic Milnor fibre and motivic nearby cycle functor.
 

Recall that an integer n is called y-smooth when each of its prime divisors is less than or equal to y. It is conjectured that, for any a>0,  any polynomial of positive degree having integral coefficients should possess infinitely many values at integral arguments n that are n^a-smooth. One could consider this problem to be morally “dual” to the cognate problem of establishing that irreducible polynomials assume prime values infinitely often, unless local conditions preclude this possibility. This smooth values conjecture is known to be true in several different ways for linear polynomials, but in general remains unproven for any degree exceeding 1. We will describe some limited progress in the direction of the conjecture, highlighting along the way analogous conclusions for polynomial smoothness. Despite being motivated by a problem in analytic number theory, most of the methods make use of little more than pre-Galois theory. A guest appearance will be made by several hyperelliptic curves. [This talk is based on work joint with Jonathan Bober, Dan Fretwell and Greg Martin].

This talk will report on the definition of some motivic cohomology classes and the proof that they satisfy the norm relations expected of Euler systems, emphasizing a connection with the local Gan-Gross-Prasad conjecture.

Let k be a field of characteristic zero and K=k((t)). Semi-algebraic sets over K are boolean combinations of algebraic sets and sets defined by valuative inequalities. The associated Grothendieck ring has been studied by Hrushovski and Kazhdan who link it via motivic integration to the Grothendieck ring of varieties over k. I will present a morphism from the former to the Grothendieck ring of motives of rigid analytic varieties over K in the sense of Ayoub. This allows to refine the comparison by Ayoub, Ivorra and Sebag between motivic Milnor fibre and motivic nearby cycle functor.
 

Abstract: If A is a finite dimensional algebra, and D(A) the unbounded
derived category of the full module category Mod-A, then it is
straightforward to see that D(A) is generated (as a "localizing
subcategory") by the indecomposable projectives, and by the simple 
modules. It is not so obvious whether it is generated by the 
indecomposable injectives. In 2001, Keller gave a talk in which he 
remarked that"injectives generate" would imply several of the well-known
homological conjectures, such as the Nunke condition and hence the 
generalized Nakayama
conjecture, and asked if there was any relation to the finitistic 
dimension conjecture. I'll show that an algebra that satisfies "injectives 
generate" also satisfies the finitistic dimension conjecture and discuss 
some examples. I'll present things in a fairly concrete way, so most of 
the talk won't assume much knowledge of derived categories.

 

Abstract: In this talk I will discuss the interplay between the local and
the global invariants in modular representation theory with a focus on the
first Hochschild cohomology $\mathrm{HH}^1(B)$ of a block algebra $B$. In
particular, I will show the compatibility between $r$-integrable 
derivations
and stable equivalences of Morita type. I will also show that if
$\mathrm{HH}^1(B)$ is a simple Lie algebra such that $B$ has a unique
isomorphism class of simple modules, then $B$ is nilpotent with an
elementary abelian defect group $P$ of order at least 3. The second part 
is joint work with M. Linckelmann.