Chunyi Li (University of Warwick). Bridgeland stability conditions on projective varieties. Friday 1st May, 1:30-2:30pm. Huxley 130.

Abstract: Slope stability for vector bundles on curves was introduced by Mumford in the 1960s, providing a rigorous foundation for the construction of moduli spaces. In higher dimensions, the notion of stability splits into slope stability and Gieseker stability. While both retain many of the desirable properties as in the curve case, each also presents subtle technical limitations.
The Bridgeland stability condition can be viewed as a generalized slope stability on curves to higher-dimensional varieties in a more unified and robust way, combining advantages of both slope and Gieseker stability. A central question in the theory has been to determine which varieties admit Bridgeland stability conditions. In this talk, I will discuss recent progress on the existence of stability conditions on all projective varieties.

Matthew Habermann (Imperial). Dubrovin’s conjectures in the Landau—Ginzburg setting. Friday 8th May, 1:30-2:30pm. Huxley 130.

Abstract: Landau–Ginzburg (LG) models are the natural mirror candidates to Fano manifolds and have an enumerative theory analogous to quantum cohomology, known as FJRW theory. In this talk, I will begin by giving an overview of these ideas, keeping a running comparison to quantum cohomology in order to explain some of the similarities and subtleties. I will then briefly explain the notion of Frobenius manifolds, how they arise in FJRW theory and how Dubrovin’s conjectures for Fano varieties lead to natural analogues in this LG setting. Finally, I would like to discuss work-in-progress in collaboration with Yefeng Shen and Weiqiang He. 

Qaasim Shafi (Heidelberg University). Hilbert schemes of points, quantum cohomology and tropical
curves. Friday 15th May, 1:30-2:30pm. Huxley 130.

Abstract: For a smooth surface S, the Hilbert scheme of points on S gives a smooth compactification of the configuration space of n distinct points on S. Its cohomology is by now well understood and exhibits deep connections with representation theory. Understanding its quantum cohomology, a deformation of ordinary cohomology involving curve counting invariants, has since received a lot of attention. I will explain joint work with Georg Oberdieck and Aaron Pixton about how to compute this ring for an elliptic surface, with the help of tropical geometry.

Yanki Lekili (Imperial). Curves on surfaces and moduli of associative algebras. Friday 22nd May, 1:30-2:30pm. Huxley 130.

Abstract: A signed Gauss word determines an immersion of a circle in a surface. We can view such an immersion as an object in a Fukaya category of any partial compactification of the surface. We will explain how to efficiently calculate the corresponding A_infty structures, and use this to construct explicit flat families of finite-dimensional associative algebras. It turns out this construction realizes essentially all associative algebras of rank less than or equal to 4, and all radical square zero algebras (of any rank).

Bogdan Simeonov (Imperial). The special McKay correspondence and homological mirror symmetry for orbifold log CY surfaces. Friday 29th May, 1:30-2:30pm. Huxley 130.

Abstract: Given a finite cyclic subgroup G of GL(2,C) acting on C^2, it was first noticed by Wunram in the 80s that there is a correspondence between certain special representations of G and the exceptional curves appearing in the minimal resolution Y of the surface singularity C^2/G. In modern terms, this was reformulated by Ishii and Ueda as the existence of a fully faithful functor from the derived category of the minimal resolution Y of a surface with cyclic quotient singularities X to the derived category of X (considered as an orbifold). In this talk, I will describe a mirror symmetric interpretation of this which exhibits the fully faithful inclusion in algebraic geometry as a sequence of Lefschetz stabilizations in symplectic geometry. 

Aline Zanardini (EPFL). A tale of three GIT problems. Friday 5th June, 1:30-2:30pm. Huxley 130.

Abstract: A general net of quadric surfaces, together with a choice of a base point, defines a net of plane cubics via Gale duality. To both nets, one can also naturally associate the same smooth plane quartic. In this talk, I will report on joint work with M. Hattori and T. Papazachariou, concerning a generalisation of this classical threefold cycle of correspondences. I will explain how, by extending these correspondences, one can obtain a complete criterion for GIT stability of the three underlying geometric objects using a birational-geometric method.

Federico Ardila (Queen Mary). Polytopes from amplitudes. Friday 12th June, 1:30-2:30pm. Huxley 130.

Abstract: Scattering amplitudes and other quantities in physics are given by enormous, intricate sums that are very challenging to compute in practice, and often involve mysterious, extensive cancellations. A powerful technique to explain this phenomenon is to encode the combinatorial complexity of these sums in a geometric object. I will introduce some of the beautiful polytopes that arise and discuss their rich combinatorial structure. Our two central examples will be the associahedron (first discovered in homotopy theory and rediscovered in scattering amplitudes) and the cosmohedron (first discovered in cosmology and seeking a homotopy theoretic interpretation). 
My talk will discuss joint work with Marcelo Aguiar and {Nima Arkani-Hamed, Carolina Figueiredo, and Francisco Vazão}, and will not assume previous knowledge of this topic.

Sebastian Opper (Charles University, Prague). Lie theory for autoequivalence groups of triangulated categories. Friday 19th June, 1:30-2:30pm. Huxley 130.

Abstract: One of the cornerstones of classical Lie theory is the exponential map with which one can reduce many questions about  `global’ Lie groups to problems about their `local’ Lie algebras. This allows one to study symmetries of geometric objects in terms of algebraic data. In a similar way, one is often interested in describing the symmetry groups of triangulated categories and the talk will focus on the first stages to develop a Lie theoretic approach for them. I will explain how one can construct an exponential map which relates autoequivalences of triangulated categories with Hochschild cohomology and how this can be put to use to the compute derived Picard groups of Fukaya categories of surfaces in the sense of Bocklandt and Haiden-Katzarkov-Kontsevich and derived categories of stacky nodal curves of Lekili-Polishchuk.

Richard Thomas (Imperial). Localisation and commuting vector fields. Friday 26th June, 1:30-2:30pm. Huxley 130.

Abstract: If a connected abelian Lie group A acts on a compact manifold M, we can sometimes localise the characteristic classes of M – particularly its Euler class – to the fixed locus of the action.
• If dim A=1 we can apply the Poincaré-Hopf theorem to the resulting vector field on M.
• If A is compact (i.e. a torus) we can use Atiyah-Bott-Berline-Vergne localisation in equivariant cohomology.
• If M is complex Baum-Bott-Illusie localise to the locus in M where the A-action drops rank, rather than the smaller locus where the rank drops to zero.
• Bonatti has solved the problem for real analytic actions in dimensions ≤4.
The general case seems to be underappreciated and wide open. I will explain two approaches in the complex case, and some virtual motivation from the MNOP conjecture in enumerative algebraic geometry.
Different parts of this project are joint work with Francesca Carocci, Yifan Zhao and Maurico Correa.