Bernard Leclerc: Cluster algebras and quantum loop algebras

16th August, 2016

In 2012, Hernandez and Jimbo introduced a new tensor category of representations of a Borel subalgebra of a quantum loop algebra, and classified its simple objects. This category contains the finite-dimensional representations of the quantum loop algebra, together with some new infinite dimensional representations. The motivation of Hernandez and Jimbo came from mathematical physics, in particular from papers of Bazhanov et al. where some examples of these new representations were used to define analogues of Baxter’s Q-operators in conformal field theory. Recently, using this new category, Frenkel and Hernandez were able to prove a long-standing conjecture of Frenkel and Reshetikhin on the spectra of the transfer matrices of some quantum integrable systems associated with quantum loop algebras. In this talk, I will explain that the new category of Hernandez and Jimbo fits very well with cluster algebras. More precisely I will show that cluster structures occur naturally in its Grothendieck ring, and can be helpful in finding new interesting functional relations. This is a joint work with David Hernandez.

Antoine Touzé: Stabilization and cup products for polynomial representations of GLn(k)

15th August, 2016

It is known for a long time that polynomial representations of GLn(k) stabilize when n grows, i.e. Schur algebras S(n, d) are all Morita equivalent when n ≥ d. A model of the category of stable polynomial representations is given by the strict polynomial functors of Friedlander and Suslin. Using the formalism of strict polynomial functors, we prove a rather counter-intuitive results on cup products, namely that the cup product

Ext∗(M, N) ⊗ Ext∗(P(r), Q(r)) → Ext∗(M ⊗ P(r), N ⊗ Q(r))

induces an isomorphism in low degrees when M, N, P, Q are stable polynomial representations. We shall explain some consequences of these results (including a new proof of the Steinberg tensor product theorem, as well as more general structure theorems which generalize it) and connections with the cohomology of the symmetric group.

Hugh Thomas: Semistable subcategories for preprojective algebras

15th August, 2016

A linear form φ on the Grothendieck group of an algebra determines a abelian, extension-closed subcategory of its finite length modules: the φ-semistable subcategory (in the sense of King). This subcategory is abelian and extension-closed. As φ varies, the subcategories picked out exhibit a wall-and-chamber structure. If the algebra is hereditary and finite type, we recover the combinatorics of Igusa-Orr-Weyman-Todorov pictures, or, equivalently, of the cluster complex. It turns out that for finite-type preprojective algebras, we obtain combinatorics described by Nathan Reading's 'shards' (originally introduced by Reading to study the combinatorics of weak order on the associated Coxeter group). Shards provide a beautiful picture from which we can recover the combinatorics for any quotient of the preprojective algebra, including the hereditary cases. Time permitting, I will also say something about affine type. This project is joint work with David Speyer, and also draws on previous joint work with Osamu Iyama, Nathan Reading, and Idun Reiten.

Rosanna Laking: Indecomposable objects in the homotopy category of a derived-discrete algebra

15th August, 2016

In this talk I will present joint work with K. Arnesen, D. Pauksztello and M. Prest. We classify the indecomposable pure-injective complexes in the homotopy category of projective modules K(ProjΛ) over a derived-discrete algebra Λ. The set of indecomposable pure-injective complexes are the points of a topological space known as the Ziegler spectrum. We give a complete description of the Ziegler topology and, making use of the interactions between this space and categories of functors, we prove that every indecomposable object in K(ProjΛ) is pure-injective.

Osamu Iyama: Finiteness of global dimension of endomorphism algebras

15th August, 2016

In representation theory, it is basic to study modules whose endomorphism algebras have finite global dimension. They appear naturally in many situations, e.g. Auslander correspondence and representation dimension, Dlab-Ringel's approach to quasi-hereditary algebras of Cline-Parshall-Scott, Rouquier’s dimensions of triangulated categories, and cluster tilting in higher-dimensional Auslander-Reiten theory. Recently such modules are called non-commutative resolutions, and studied in commutative ring theory and algebraic geometry after Van den Bergh's work in birational geometry. In this talk, I will show some of typical examples of non-commutative resolutions, including rings with Krull-dimension at most one, certain hypersurface singularities and Stanley-Reisner rings.

Douglas Ulmer: Rational curves on elliptic surfaces

5th May, 2016

Given a non-isotrivial elliptic curve E over K=𝔽q(t), there is always a finite extension L of K which is itself a rational function field such that E(L) has large rank. The situation is completely different over complex function fields: For "most" E over K=ℂ(t), the rank E(L) is zero for any rational function field L=ℂ(u). The yoga that suggests this theorem leads to other remarkable statements about rational curves on surfaces generalizing a conjecture of Lang.

Cagatay Kutluhan: A Heegaard Floer analogue of algebraic torsion

21st April, 2016

The dichotomy between overtwisted and tight contact structures has been central to the classification of contact structures in dimension 3. Ozsvath-Szabo's contact invariant in Heegaard Floer homology proved to be an efficient tool to distinguish tight contact structures from overtwisted ones. In this talk, I will motivate, define, and discuss some properties of a refinement of the contact invariant in Heegaard Floer homology.

Junecue Suh: Standard conjecture of Künneth type with torsion coefficients

21st April, 2016

p>A.Venkatesh asked us the question, in the context of torsion automorphic forms: Does the Standard Conjecture (of Grothendieck's) of Künneth type hold with mod p coefficients? We first review the geometric and number-theoretic contexts in which this question becomes interesting, and provide answers: No in general (even for Shimura varieties) but yes in special cases.

Ana Rita Pires: Symplectic embeddings and infinite staircases

15th April, 2016

McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a 4-ball. The graph of this function includes an infinite staircase determined by the odd index Fibonacci numbers. Infinite staircases have also been shown to exist in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid E(2,3). This talk describes joint work with Dan Cristofaro-Gardiner, Tara Holm, and Alessia Mandini, in which we use ECH capacities to show that infinite staircases exist for these and a few other target manifolds. I will also explain why we conjecture that these are the only such twelve.

Georgios Dimitroglou-Rizell: Classification results for two-dimensional Lagrangian tori

7th April, 2016

We present several classification results for Lagrangian tori, all proven using the splitting construction from symplectic field theory. Notably, we classify Lagrangian tori in the symplectic vector space up to Hamiltonian isotopy; they are either product tori or rescalings of the Chekanov torus. The proof uses the following results established in a recent joint work with E. Goodman and A. Ivrii. First, there is a unique torus up to Lagrangian isotopy inside the symplectic vector space, the projective plane, as well as the monotone S2×S2. Second, the nearby Lagrangian conjecture holds for the cotangent bundle of the torus.

Alex Kontorovich: Low-lying, fundamental, reciprocal geodesics

24th March, 2016

Markoff numbers give rise to extremely low-lying reciprocal geodesics on the modular surface, but it is unknown whether infinitely many of these are fundamental, that is, the corresponding binary quadratic form has fundamental discriminant. In joint work with Jean Bourgain, we unconditionally produce infinitely many low-lying (though not "extremely" so), reciprocal geodesics on the modular surface, settling a question of Einsiedler-Lindenstrauss-Michel-Venkatesh. Some ingredients are expander graphs, sieving on thin orbits, and producing a level of distribution "beyond expansion". No prior knowledge of these topics will be assumed.

Ailsa Keating: Homological Mirror Symmetry for singularities of type Tpqr

10th March, 2016

We present some homological mirror symmetry statements for the singularities of type Tp,q,r. Loosely, these are one level of complexity up from so-called 'simple' singularities, of types A, D and E. We will consider some symplectic invariants of the real 4-dimensional Milnor fibres of these singularities, and explain how they correspond to coherent sheaves on certain blow-ups of the projective space P2, as suggested notably by Gross-Hacking-Keel. We hope to emphasize how the relations between different 'flavours' of invariants (e.g., versions of the Fukaya category) match up on both sides.

Kyler Siegel: Subflexible symplectic manifolds

3rd March, 2016

After recalling some recent developments in symplectic flexibility, I will introduce a class of open symplectic manifolds, called 'subflexible', which are not flexible but become so after attaching some Weinstein handles. For example, the standard symplectic ball has a Weinstein subdomain with non-trivial symplectic topology. These are exotic symplectic manifolds with vanishing symplectic cohomology. I will explain how to study them using a deformed version of symplectic cohomology, and how this invariant can computed using the machinery of Fukaya categories and Lefschetz fibrations. This is partly based on joint work with Emmy Murphy.

Peter Albers: Positive loops-on a question by Eliashberg-Polterovich and a contact systolic inequality

25th February, 2016

In 2000 Eliashberg-Polterovich introduced the concept of positivity in contact geometry. The notion of a positive loop of contactomorphisms is central. A question of Eliashberg-Polterovich is whether C0-small positive loops exist. We give a negative answer to this question. Moreover we give sharp lower bounds for the size which, in turn, gives rise to a L∞-contact systolic inequality. This should be contrasted with a recent result by Abbondandolo et. al. that on the standard contact 3-sphere no L2-contact systolic inequality exists. The choice of L2 is motivated by systolic inequalities in Riemannian geometry.

Will Sawin: Vanishing cycles and bilinear forms

18th February, 2016

In joint work with Emmanuel Kowalski and Philippe Michel, we prove two different estimates on sums of coefficients of modular forms - one related to L-functions and another to the level of distribution. A key step in the argument is a careful analysis of vanishing cycles, a tool originally developed by Lefschetz to study the topology of algebraic varieties. We will explain why this is helpful for these problems.

Mohammed Abouzaid: Floer theory revisited

4th February, 2016

I will describe a formalism for (Lagrangian) Floer theory wherein the output is not a deformation of the cohomology ring, but of the Pontryagin algebra of based loops, or of the analogous algebra of based discs (with boundary on the Lagrangian). I will explain the consequences of quantum cohomology, and the expected applications of this theory.

Jean Bourgain: Decoupling in harmonic analysis and the Vinogradov mean value theorem

17th December, 2015

Based on a new decoupling inequality for curves in ℝd, we obtain the essentially optimal form of Vinogradov's mean value theorem in all dimensions (the case d=3 is due to T. Wooley). Various consequences will be mentioned and we will also indicate the main elements in the proof (joint work with C. Demeter and L. Guth).

Michael Harris: Modularity and potential modularity theorems in the function field setting

17th December, 2015

Let G be a reductive group over a global field of positive characteristic. In a major breakthrough, Vincent Lafforgue has recently shown how to assign a Langlands parameter to a cuspidal automorphic representation of G. The parameter is a homomorphism of the global Galois group into the Langlands L-group LG of G. I will report on my joint work in progress with Böckle, Khare, and Thorne on the Taylor-Wiles-Kisin method in the setting of Lafforgue's correspondence. New (representation-theoretic and Galois-theoretic) issues arise when we seek to extend the earlier work of Böckle and Khare on the case of GLn to general reductive groups. I describe hypotheses on the Langlands parameter that allow us to apply modularity arguments unconditionally, and I will state a potential modularity theorem for a general split adjoint group.