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). Friday 5th June, 1:30-2:30pm. Huxley 130.
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Federico Ardila (Queen Mary). The Combinatorics of CAT(0) Cube Complexes. Friday 12th June, 1:30-2:30pm. Huxley 130.
Abstract: There are numerous contexts where a discrete system moves according to local, reversible moves. The configuration space, which contains all possible states of the system, is often a CAT(0) (i.e. a globally non-positively curved) cube complex. When this is the case, we can use techniques from geometric group theory to understand, measure, and navigate these spaces. I will present a self-contained introduction to these ideas, and discuss some applications to robotic motion planning, phylogenetics, probability, and enumerative combinatorics.
The talk will assume no previous knowledge of CAT(0) cube complexes. It will include joint work with many people, including Tia Baker, Naya Banerjee, Hanner Bastidas, César Ceballos, John Guo, Megan Owen, Seth Sullivant, Coleson Weir, and Rika Yatchak.
Sebastian Opper (Charles University, Prague). Friday 19th June, 1:30-2:30pm. Huxley 130.
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Holly Krieger (Cambridge). Friday 26th June, 1:30-2:30pm. Huxley 130.
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