Abstracts

Kevin Buzzard: C-groups

The Langlands Philosophy is sometimes a bit vague, because there are some subtleties in it that people don't quite understand, or only conjecturally understand. Here is one subtlety that a few years ago I didn't understand at all: (1) Given an "algebraic" automorphic form (for example a Dirichlet character, or a modular form which is an eigenform for the Hecke operators) there should surely be an associated Galois representation, probably into an "L-group". (2) Automorphic forms which contribute to the cohomology of Shimura varieties are "algebraic". However one can *prove* that the natural attempt to make all of this rigorous does not work. The reason is that 'half the sum of the positive roots is not always a weight'. Toby Gee and I have worked out a conjectural way to fix this up, involving an extension of Langlands' notion of an L-group to a "C-group". This C-group was independently constructed by Deligne in an unpublished letter to Serre from 2007. This talk will be a **non-technical introduction** to this area. I will explicitly explain what the problem is, by illustrating two special cases of all of this theory, namely the cases G=GL(1) and G=SL(2). Hence I will avoid all the technicalities of the general theory. The talk should be accessible to number theorists; I am aware the audience are not in general specialists.

Jens Marklof: From the Lorentz gas to random graphs: new applications of measure rigidity in statis- tical physics, number theory and combinatorics

Measure rigidity of flows on homogeneous spaces is a powerful tool that has recently seen many spectacular applications in number theory and mathematical physics. In this lecture I will discuss applications of measure rigidity to three seemingly unrelated problems: kinetic transport in the periodic Lorentz gas, diameters of random circulant graphs and Frobenius' coin exchange problem.

Catharina Stroppel: Fractional Euler characteristics and categorification

Categorification is a powerful tool which connects many different areas of mathematics and actually can be used to establish quite unexpected equivalences of categories. In this talk I will explain some basic ideas on categorification and sketch an application in knot theory. More precisely I will indicate who representation theory can be used to get finer knot invariants using categorification. This application on the other hand suggests generalizations to higher dimensional invariants which require the notion of fractional Euler characteristics. I will explain this notion and (if time allows) indicate possible applications of this concept.

Raf Bocklandt: Noncommutative mirror symmetry for punctured surfaces

A dimer model on a surface with punctures is an embedded quiver such that its vertices correspond to the punctures and the arrows circle round the faces they cut out. To any such dimer model we can associate 2 categories: A wrapped Fukaya category and a category of matrix factorizations. In both categories the objects are arrows, which are interpreted as Lagrangian subvarieties in the former and will give us matrix factorizations of a potential on the Jacobi algebra of the dimer in the latter. We show that there is a duality on a broad class of dimers that make the Fukaya category of a dimer isomorphic to the category of matrix factorizations of its dual dimer.

Stephen Harrap: Problems surrounding the mixed Littlewood conjecture for pseudo- absolute values

I will discuss a varient of the mixed Littlewood Conjecture, namely one relating to a `pseudo-absolute value' sequence, and how it can be solved under certain restrictions on this sequence. The method appeals to a measure rigidity theorem of Lindenstrauss and relies upon being able to prove the existance of some ergodic measure with positive entropy. Time permitting, I will also describe a metrical result complementing the mixed Littlewood Conjecture itself.

Alan Haynes: Equivalence classes for separated nets

A set Y in R^d is a separated net if there are constants r, R>0 such that every ball of radius R intersects Y and every ball of radius r contains at most one point of Y. One way to construct such sets is to let R^d act linearly on a k-dimensional torus and then take the collection of return times to some (k-d)-dimensional Poincare section. As long as the action of R^d is totally irrational we will get a separated net. In this talk we will discuss the problem of whether or not such separated nets can be deformed in a nice way to the integer lattice. This is joint work with Barak Weiss and Michael Kelly.

Florian Klein: A new approach to the Satake correspondence via twisted singular Soergel bimodules

Twisted singular Soergel bimodules are candidates for a geometry free categorification of the Satake isomorphism. Here, we explain this equivalence in both rank one cases. Along the way, we provide a presentation of the monoidal categories Rep(SL_2) and Rep(PSL_2) by generators and relations. If time permits, we will indicate how generators and relations for the 2-category of singular Soergel bimodules in the case of the infinite dihedral group can be deduce. The case of arbitrary ranks and types is work in progress.

Felipe Ramirez: Higher-degree cohomology for Anosov actions

There is a conjecture due to A. and S. Katok generalizing Livshitz's theorem to higher-rank Anosov actions.  Recall that Livshitz's theorem tells us that the obstructions to solving the coboundary equation for an Anosov flow are exactly those coming form periodic orbits.  It is conjectured that for an Anosov action by a rank-d abelian group, the obstructions to solving the degree-d coboundary equation also come from closed orbits of the action.  Furthermore, lower-degree cohomology should trivialize, as it is known to do in the first degree, by work of Katok and Spatzier.  I will discuss the conjecture, and some examples where it is known.

Xiuping Su: Filtrations in abelian categories with a tilting object of homological dimension two

In this talk I will discuss filtrations of objects in an abelian category $A$ induced by a tilting object $T$ of homological dimension at most two.

Anish Ghosh: Ergodic approach to Diophantine problems

Ergodic theory has had considerable success in addressing Diophantine problems in recent times. I will survey some of these successes and if time permits speak about recent developments.

Vanessa Miemietz: Representation theory of 2-categories

I will describe an intrinsic approach to the representation theory of finitary 2-categories, based on joint work with Volodymyr Mazorchuk.

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