Alberto Canonaco: Localizations of the categories of dg and of A∞ categories
I will report on a joint work with Mattia Ornaghi and Paolo Stellari, where we prove that, over an arbitrary commutative ring, the localizations with respect to quasi-equivalences of the categories of dg (differential graded) categories and of A∞ categories are equivalent. The equivalence holds not only as ordinary categories, but even as ∞-categories; moreover, one can replace (strictly unital) A∞ categories with unital or cohomologically unital ones. We also show that the internal Homs in the homotopy category of dg categories can be realized as suitable dg categories of A∞ functors.
Francesca Carocci: BPS invariants from p-adic integrals
We consider moduli spaces of one-dimensional semistable sheaves on del Pezzo and K3 surfaces supported on ample curve classes. Working over a non-archimedean local field, we define a canonical p-adic measure on such moduli spaces. We prove that the integral of a certain naturally defined gerbe on the moduli spaces with respect to this measure is independent of the Euler characteristic. Analogous statements hold for (meromorphic or not) Higgs bundles. Recent results of Maulik-Shen and Kinjo-Coseki imply that these integrals compute the BPS invariants for the del Pezzo case and for Higgs bundles. This is a joint work with Giulio Orecchia and Dimitri Wyss.
Alberto Cattaneo: An introduction to the Batalin-Vilkovisky formalism (mini course)
In these lectures, I will review the Batalin–Vilkovisky formalism (and its cognates) in which the spaces of fields of a physical theory are presented as complexes whose cohomology returns the physical content. Different but equivalent complexes may be used, which turns out to be important conceptually and in practice. A theory of integration (well-defined in the finite-dimensional case) is also available and is the starting point for the quantization of the theories.
Joana Cirici: Batalin-Vilkovisky and hypercommutative algebras in complex geometry
I will review some constructions of BV and hypercommutative algebras for manifolds with additional geometric structures, ranging from Poisson to Hermitian manifolds. Such algebra structures are related to the extended deformation theory introduced by Barannikov and Kontsevich for Calabi-Yau manifolds. I will explain how, using mixed Hodge theory at the homotopical level, one can prove hypercommutative formality of compact Kähler manifolds. This talk includes joint results with Geoffroy Horel and with Scott Wilson.
Andrea D'Agnolo: Irregular nearby cycles on the Betti side
Enhanced ind-sheaves describe the Betti side of the irregular Riemann-Hilbert correspondence, in a manner compatible with Grothendieck's operations. In this way, classical constructions on the de Rham side have their natural topological counterpart. In this talk we will illustrate some examples in complex dimension one. In particular, irregular nearby and vanishing cycles, and their behavior under the Fourier transform. This is from joint works with Masaki Kashiwara.
Chiara Esposito: Equivariant formality and reduction
In this talk, we discuss the reduction-quantization diagram in terms of formality. First, we propose a reduction scheme for multivector fields and multidifferential operators, phrased in terms of L∞ morphisms. This requires the introduction of equivariant multi- vector fields and equivariant multidifferential operator complexes, which encode the information of the Hamiltonian action, i.e., a G-invariant Poisson structure allowing for a momentum map. As a second step, we discuss an equivariant version of the formality theorem, conjectured by Tsygan and recently solved in a joint work with Nest, Schnitzer, and Tsygan. This result has immediate consequences in deformation quantization, since it allows for obtaining a quantum moment map from a classical momentum map with respect to a G-invariant Poisson structure.
Noriaki Ikeda: AKSZ theories and their generalizations
AKSZ sigma models are physical models constructed from dg manifolds (Q-manifolds). One of important AKSZ sigma models is the Poisson sigma model, which gives the Kontsevich formula in the deformation quantization and the formality by the quantization. We present generalizations of AKSZ sigma models as classical field theories, higher dimensional generalizations and deformations by pre-multisymplectic forms called 'fluxes'. Geometric structures of these theories including dg structures induced from BV formalisms are described by geometry of Lie algebroids and higher algebroids. We discuss problems of quantizations of AKSZ sigma models and generalized theories.
Emma Lepri: Ext algebras from the contraction algebra
By the work of Donovan and Wemyss, the functor of noncommutative deformations of a flopping irreducible rational curve C in a threefold X is representable by an algebra called the contraction algebra. This talk is based on a joint work in progress with Joseph Karmazyn and Michael Wemyss, where we construct a DG-algebra from the data of periodic projective resolution of the simple module on the contraction algebra, and prove that it reconstructs the A∞-algebra ExtX*(OC(-1), OC(-1)). We also discuss relations of this DG-algebra with Booth's derived contraction algebra, and the Donovan-Wemyss conjecture.
Xiaobo Liu: Intersection numbers on moduli space of curves and symmetric polynomials
Generating functions of intersection numbers on moduli spaces of curves provide geometric solutions to integrable systems. Notable examples are the Kontsevich-Witten tau function and Brezin-Gross-Witten tau function. In this talk I will first describe how to use Schur's Q-polynomials to obtain simple formulas for these functions. I will then discuss possible extensions for more general geometric models using Hall-Littlewood polynomials. This talk is based on joint works with Chenglang Yang.
Emanuele Macrì : Deformations of stability conditions
Bridgeland stability conditions have been introduced about 20 years ago, with motivations both from algebraic geometry, representation theory and physics. One of the fundamental problems is that we currently lack methods to construct and study such stability conditions in full generality. In this talk I would present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari and Zhao. As application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces, and we prove a conjecture by Kuznetsov and Shinder on quartic double solids.
Claudia Rella : Strong-weak symmetry and quantum modularity of resurgent topological strings
Quantizing the mirror curve to a toric Calabi-Yau threefold gives rise to quantum operators whose fermionic spectral traces produce factorially divergent series in the Planck constant and its inverse. These are captured by the Nekrasov-Shatashvili and standard topological strings via the TS/ST correspondence. In this talk, I will discuss the resurgence of these dual asymptotic series and present an exact solution for the spectral trace of local P^2. A full-fledged strong-weak symmetry is at play, exchanging the perturbative/nonperturbative contributions to the holomorphic and anti-holomorphic blocks in the factorization of the spectral trace. This relies on a network of relations connecting the dual regimes and building upon the analytic properties of the L-functions with coefficients given by the Stokes constants and the q-series acting as their generating functions. Finally, I will mention how these results fit into a broader paradigm linking resurgence and quantum modularity. This talk is based on arXiv:2212.10606, 2404.10695, and 2404.11550.
Nicolò Sibilla: Elliptic cohomology and mapping stacks
In this talk I will report on work on elliptic cohomology with Tomasini and Scherotzke. With Tomasini we give a new construction of equivariant elliptic cohomology in terms of a stack of maps out of the elliptic curve. This construction generalizes beautiful recent work of Moulinos-Robalo-Toen to the equivariant setting, and brings elliptic cohomology closer to well known constructions in algebraic geometry such as (secondary) Hochschild homology. Also it opens the way to possible non-commutative generalizations. With Scherotzke we show that equivariant elliptic cohomology is not a derived invariant, thus confirming the heuristics that elliptic cohomology captures higher categorical information. This depends on a careful analysis of the equivariant elliptic cohomology of toric varieties.
Paolo Stellari: Geometric triangulated categories: enhancements and weak approximation (mini course)
I will illustrate the most recent results on how to enhance triangulated categories (and exact functors) of geometric nature. We will then move to the problem of lifting equivalences between various triangulated categories and illustrate the new interplay between the theory of weakly approximable triangulated categories and the existing results about the uniqueness of enhancements. Applications to a generalization of a classical result by Rickard and to derived invariants of schemes will be discussed. The new results are joint works (partly in progress) with Alberto Canonaco, Amnon Neeman and Mattia Ornaghi.
Yukinobu Toda: Quasi-BPS categories for Higgs bundles (mini course)
The Donaldson-Thomas invariants count stable coherent sheaves on Calabi-Yau 3-folds which were introduced by Thomas around 1998. Later Joyce-Song, Kontsevich-Soibelman and Davison-Meinhardt introduced integer valued invariants, called BPS invariants, which also take account of strictly semistable sheaves. The BPS invariants play important roles in modern enumerative geometry. In this talk, I will introduce (quasi-)BPS categories for Higgs bundles. They are regarded as categorifications of BPS invariants of local curves (which are non-compact Calabi-Yau 3-folds), and are regarded as non-commutative analogue of Hitchin integrable systems. I will propose a conjectural symmetry of BPS categories which swaps Euler characteristic and weight, inspired by Dolbeaut Geometric Langlands equivalence of Donagi-Pantev, by the Hausel-Thaddeus mirror symmetry for Higgs bundles and χ-independence phenomena for BPS invariants of curves on Calabi-Yau 3-folds. I will give some evidence of the above conjecture for rank two cases and for topological K-theories. This is a joint work with Tudor Padurariu.
Nicolò Bignami, José María Cantarero López, Anunoy Chakraborty, Christian David Forero Pulido, Sangjun Ko, Alessandro Lehmann, Ziqi Liu, Nivedita, Ville Nordstrom, Jae-Suk Park, Kaichuan Qi, Nicholas Rekuski, Catherine Elizabeth Rust, Seokbong Seol, Noé Sotto