TITLES AND ABSTRACTS
I will introduce a set of equations on a principal bundle over a compact complex manifold coupling a connection on the principal bundle, a section of an associated bundle with Kähler fibres and a Kähler structure on the base manifold. These equations are a generalization of the constant scalar curvature condition for a Kähler metric studied by Donaldson, Tian, Yau and others, and the Yang--Mills--Higgs equations for Kähler fibrations studied by Mundet i Riera. I will provide a moment map interpretation of the equations, construct some examples, and study obstructions to the existence of solutions. I will also explain a surprising relation with the physics of cosmic strings.
Based on joint work with Mario Garcia-Fernandez, Oscar García-Prada, Vamsi Pingali and Chengjian Yao (Geom. Topol. 2013, Comm. Math. Phys. 2017, Pure Appl. Math. Q. 2019, Math. Ann. 2021, arXiv:2201.03455 and further work in progress).
Marcel Bökstedt (Aarhus U) [video]
Chain complexes of graphs and generalized configuration spaces
I will discuss a generalization of configuration spaces, and a version of graph complexes. The conjecture is that the graph complex determines the real homotopy type of these configuration spaces. I will give some preliminary results, and some evidence for this conjecture.
Alexander Braverman (U Toronto) [video]
Derived geometric Satake equivalence and geometric constructions
In this talk I plan to do the following: (1) Review the (derived) geometric Satake equivalence (which gives an algebraic description of certain derived category of sheaves on the affine Grassmannian of a reductive group G). (2) Explain how to construct ring objects in the derived Satake category (in several different ways). (3) Combine 1 and 2 and get a construction of many (often new) complex Poisson varieties starting with elementary data (e.g. a representation of G). Examples might include Coulomb branches of 3d gauge theories, Moore-Tachikawa varieties, S-dual varieties of smooth affine G-varieties, etc. If time permits, I will discuss the relation to the work of Ben-Zvi, Sakellaridis and Venkatesh.
Mykola Dedushenko (SCGP Stony Brook) [video]
Three roles of the (K-theoretic) vortex partition function
The vortex partition function in 3d N=2 theories, or a closely related quantity known as "half-index" in physics, is a rich object that lies at the intersection of several research programs in mathematical physics. One can, in particular, single out three roles for it: (1) as character of certain boundary vertex algebras; (2) as an enumerative quantity counting quasimaps and also probing the symplectic duality between Higgs and Coulomb branches; (3) as a device which (in the presence of special insertions) produces Bethe eigenstates in the context of relation to quantum integrable systems. I will give a non-technical overview of the latter two roles.
Tudor Dimofte (UC Davis + U Edinburgh) [video]
Vortices, affine Springer fibers, and link homology
I will begin by reviewing the role of vortex equations (and their quantization) in constructing line operators and state spaces in A-type topological twist of 3d N=4 gauge theories. From an algebraic perspective/limit, this connects with the "BFN Springer theory" of Kamnitzer, Hilburn, and Weekes. I will then explain a particularly interesting application of vortex equation constructions to HOMFLY-PT link homology, following recent work with Garner, Hilburn, Oblomkov, and Rozansky. I aim to highlight both some well-understood aspects of this connection (which are mainly formulated algebraically) and some more conjectural ones (formulated analytically).
Camilla Felisetti (U Trento) [video]
A support theorem for nested Hilbert schemes of planar curves
Consider a family of integral complex locally planar curves. We show that under some assumptions on the base, the relative nested Hilbert scheme is smooth. In this case, the decomposition theorem of Beilinson, Bernstein and Deligne asserts that the pushforward of the constant sheaf on the relative nested Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. During the talk, we will introduce a refined notion of discriminant locus, namely the “higher discriminants”, and we will use such tool to show that no summand is supported in positive codimension, i.e. that the cohomology of the total spaces of the family is uniquely determined by smooth fibers.
Andres Fernandez Herrero (Columbia U) [video]
Compatifications of the moduli of vector bundles on stable curves
In this talk I would like to explain a new perspective on the modular compactifications of the moduli of vector bundles over stable nodal curves considered by Gieseker, Nagaraj-Seshadri, Schmitt, Kausz and others. We interpret the moduli problem in terms of Kontsevich stable maps into a "degenerate" universal quot scheme. This yields an unbounded smooth stack, which is analogous to the stack of vector bundles on a smooth curve. Recent stack-theoretic techniques ("infinite dimensional GIT") lead to a Θ-stratification of the unbounded stack, and a conceptually simple construction of the "semistable" moduli space that bypasses the previous analysis of semistability and "filling of families". One advantage of this approach is that it directly generalizes to obtain compactifications of stacks of maps from marked curves into a given quotient stack of the form [X/GLn] for some projective GLn scheme X. This can be thought of as a generalization of the work of Frenkel-Teleman-Tolland, who considered the case of GL1. We define a version of K-theoretic virtual counts of curves in [X/GLn] via these compactifications.
This talk is based on ongoing work joint with Daniel Halpern-Leistner.
Andrea Ferrari (Duham U) [video]
State spaces of 3d supersymmetric gauge theories and the geometry of vortex moduli spaces
Generalised vortex equations appear in the context of 3d supersymmetric gauge theories, and in particular supersymmetric localisation computations, as one subset of the possible BPS equations of a theory. In this talk, I will first explain how some physical observables such as partition functions/state spaces on a closed Riemann surface times S1/R can consequently be calculated in terms of the geometry of generalised vortex moduli spaces. I will then emphasise mathematical implications of this fact such as symplectic duality statements, algebra actions on the cohomology of the spaces, and if time will permit, relations to moduli spaces of equations that are not of vortex type.
Michael McBreen (Chinese University of Hong Kong)
Twisted quasimaps and symplectic duality of hypertoric varieties
Hypertoric varieties are a family of symplectic resolutions associated with hyperplane arrangements. Starting from a hypertoric X, I will describe a pair of infinite type symplectic spaces. One is a simplified model of the loop space of X, the other is a periodisation of X which arises in 2d mirror symmetry. I will explain how certain quasimap invariants of X can be understood in terms of symplectic duality for this pair, yielding a surprising relationship between Koszul duality and enumerative geometry.
Joint with Artan Sheshmani and Shing-Tung Yau.
I will discuss how the equivariant quantum cohomology of a monotone symplectic manifold with Hamiltonian G-action defines a (holonomic) module over the Coulomb branch of the Langlands dual group. I will then discuss a conjectural formula for the quantum cohomology of "anti-canonically polarized" symplectic quotients in terms of the geometry of these Coulomb branches and sketch the proof of this formula when G is abelian.
The talk is based on joint work with Eduardo González and Cheuk Yu Mak and work in progress with Constantin Teleman.
There is an upper bound on how many abelian Higgs vortices a compact Riemann surface can accommodate, proportional to its area. In the limit that the area shrinks to the minimum allowed (for a given vortex number), the vortices spread out and delocalize completely. Hence this is sometimes called the "dissolving limit". For vortices on a two-sphere, the moduli space Mn of n-vortices is biholomorphic to CPn . This space carries a natural metric, the L2 metric, which controls the classical and quantum dynamics of vortices. In 2002, Baptista and Manton conjectured that, in the dissolving limit, this metric converges to the Fubini-Study metric on CPn. I will describe how this conjecture can be made precise and proved.
Joint work with René García Lara.
Chris Woodward (Rutgers U)
Some remarks on open symplectic field theory in the Morse model
Witten noted an interesting relationship between holomorphic curves in cotangent bundles and knot homology. I will report on results in progress with Yuhan Sun, Soham Chanda and Kenneth Blakey, in which we construct a version of contact homology for Legendrians in circle-fibered manifolds generalizing knot contact homology of Ekholm-Ng and others. In the case of lifts of monotone tori we prove part of a conjecture of Dimitroglou-Rizell-Golovko on the relation of the augmentation variety with the disk potential, and generalize to higher dimensions Dimitroglou-Rizell and Treumann-Zaslow's result that the standard Legendrian torus in S 5 is not exactly fillable.