The Sheffield AGMP seminar - spring 2025

Weinstein Neighborhoods under HMS

Monday 24 March 2025

Speaker: Jeff Hicks (St. Andrews)
Location: J11
Time: 2.00pm

Abstract:  The Weinstein neighborhood theorem states that every Lagrangian submanifold L of a symplectic manifold X has a symplectic neighborhood modeled on the cotangent bundle of L. A consequence of this is that, in some nice cases, we have a Viterbo restriction functor from the Fukaya category of X to the Fukaya category of T*L. In this talk, I'll discuss some parallels between this restriction functor and the restriction to an affine neighborhood of a point in an algebraic variety, and use this to provide a definition of a "noncommutative affine neighborhood" of an object in a triangulated category. We'll then construct a few examples of these noncommutative affine neighborhoods and discuss applications. While we will use some intuition from symplectic geometry in this talk, we won't use any symplectic techniques in our examples or computations.

Linearization problem for finite subgroups of the plane Cremona group

Tuesday 22 April 2025

Speaker: Antoine Pinardin (Edinburgh)
Location: J11
Time: 2.00pm

Abstract:  The plane Cremona group is the group of birational self-maps of the projective plane. Over the field of complex numbers, its subgroups have been extensively studied, and the most complete classification dates back to 2006, with the work of  Dolgachev and Iskovskikh. They outline a question yet to be answered, which consists in describing the linearizable subgroups of the plane Cremona group, those which are conjugated to a subgroup of linear automorphisms of the projective plane. This problem is of particular importance, because it is equivalent to the question of G-equivariant rationality. We give a complete answer over the field of complex numbers.
Joint work with Arman Sarikyan and Egor Yasinsky.

Categorical resolutions of A_2 singularities 

Monday 28 April 2025

Speaker: Celine Fietz (Leiden)
Location: J11
Time: 2.00pm

Abstract:  In this talk I will present the results of my recent paper “Categorical resolutions of cuspidal singularities”, see https://arxiv.org/abs/2411.19380. I showed that there exists a particularly small (“crepant”) categorical resolution of the derived category of a projective variety with an isolated A_2/cuspidal singularity.  More importantly, I explicitly described generators of its kernel: In the case of an even dimensional variety with an isolated A_2 singularity, the kernel can be generated by two 2-spherical objects, which are related to spinor sheaves on a nodal quadric and induce autoequivalences on the categorical resolution. 

Refined BPS numbers on compact Calabi-Yau 3-folds from Wilson loops

Monday 12 May 2025

Speaker: Albrecht Klemm (Sheffield)
Location: J11
Time: 2.00pm

Abstract:  We relate the counting of  refined BPS numbers  on compact elliptically fibred Calabi-Yau 3-folds $\hat X$ to Wilson loop expectations values in the gauge theories that emerge in various rigid local limits  of the 5d supergravity theory defined by M-theory compactification on $\hat X$. In these local limits $X_*$ the volumes of curves in certain classes go to infinity, the corresponding very massive M2-brane states can be treated as Wilson loop particles and the refined topological string partition function on $\hat X$ becomes a sum of terms proportional to associated refined Wilson loop expectation values. The resulting ansatz for the complete refined topological partition function on $\hat X$ is written in terms  of the proportionality coefficients which depend only on the $\epsilon$ deformations and the Wilson loop expectations values which satisfy holomorphic anomaly equations. Since the ansatz is quite restrictive and can be further constrained by the one-form symmetries and $E$-string type limits for large base curves, we can efficiently evaluate the refined BPS numbers on $\hat X$,  which we  do explicitly for local gauge groups up to rank three and $h_{11}(\hat X)=5$. These refined BPS numbers pass an impressive number of consistency checks imposed by the direct counting of these numbers using the moduli space of one dimensional stable sheaves on $\hat X$ and give us numerical predictions for the complex structure dependency of the refined BPS numbers.

Spherical twists on K3 surfaces and Bondal-Polishchuk’s conjecture for Fano threefolds

Monday 19 May 2025

Speaker: Anya Nordskova (Hasselt)
Location: J11
Time: 2.00pm

Abstract:  Let Y be a smooth K3 surface of Picard rank 1. We prove that the subgroup of Aut(D^b(Y)) generated by all spherical twists is a (not necessarily finitely generated) free group and provide an explicit recipe to find its (free) generators. This statement essentially makes the description already obtained by Bayer and Bridgeland more precise and the proof uses their deep results concerning the space of stability conditions on such Y. 

As an application, which I will focus on in this talk, we prove an instance of a conjecture by Bondal and Polishchuk (1993) suggesting that the braid group action on the set of full exceptional collections in a triangulated category is always transitive. In this generality the conjecture has been recently disproved by Chang, Haiden and Schroll. However, if one restricts to derived categories of smooth projective varieties, the question is still widely open. We prove that Bondal-Polishchuk’s conjecture holds for Fano threefolds of Picard rank 1 (in particular, the projective space P^3). This is the first 3-dimensional case where the transitivity has been verified. 

The talk is based on joint work with Michel Van den Bergh. 

Quantum S^2 and H^2 are the same: Morita theory of topological quantum spaces (and a bit of 2D quantum gravity)

Monday 2 June 2025

Speaker: Kuba Krawczyk (Sheffield)
Location: J11
Time: 2.00pm

Abstract:  Locally compact quantum groups are the most satisfactory notion of quantum groups we have so far. Expressed in operator-algebraic language, the theory is particularly rich, but this comes at the cost of significant computational complexity. Nonetheless, homogeneous spaces of such quantum groups have recently appeared, for the first time, in a physical context -- as the natural arena for the holographic quantum gravity dual to the SYK model.

In this talk, I will first briefly review and compare locally compact quantum groups with other well-known noncommutative frameworks (such as Drinfeld–Jimbo quantum groups and Connes’ spectral triples). Then, inspired by the above physical context, I will define Morita theory for quantum spaces and develop a method for producing pairs of Morita equivalent quantum homogeneous spaces. As a special case, I will describe the equivalence between the deformation of the 2-sphere and the hyperbolic plane.