## Weekly Bulletin (it)

**Notiziario dei seminari di carattere matematico**

a cura del Dipartimento di Matematica

*Guido Castelnuovo*, Sapienza Università di Roma

Settimana dal 09-09-2024 al 15-09-2024

**Lunedì 09 settembre 2024**

Aula I, Dipartimento di Chimica, Edificio Caglioti

Convegno GAP XIX: *Moduli spaces and higher structures*

Programma:

- 9:00 Yukinobu Toda (Kavli IPMU)
*Quasi-BPS categories for Higgs bundles, I*

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. - 10:30 Nicolò Sibilla (SISSA)
*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. - 11:30 Noriaki Ikeda (Ritsumeikan University)
*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. - 14:10 Paolo Stellari (Università degli Studi di Milano)
*Geometric triangulated categories: enhancements and weak approximation, I*

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. - 15:10 Emanuele Macrì (Université Paris-Saclay)
*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.

*fiorenza@mat.uniroma1.it*

**Martedì 10 settembre 2024**

Aula I, Dipartimento di Chimica, Edificio Caglioti

Convegno GAP XIX: *Moduli spaces and higher structures*

Programma:

- 9:00 Paolo Stellari (Università degli Studi di Milano)
*Geometric triangulated categories: enhancements and weak approximation, II* - 10:30 Yukinobu Toda (Kavli IPMU)
*Quasi-BPS categories for Higgs bundles* - 11:30 Chiara Esposito (Università di Salerno)
*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. - 14:10 Xiaobo Liu (Beijing International Center for Mathematical Research)
*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.

*fiorenza@mat.uniroma1.it*

**Mercoledì 11 settembre 2024**

Aula I, Dipartimento di Chimica, Edificio Caglioti

Convegno GAP XIX: *Moduli spaces and higher structures*

Programma:

- 9:00 Yukinobu Toda (Kavli IPMU)
*Quasi-BPS categories for Higgs bundles* - 10:30 Paolo Stellari (Università degli Studi di Milano)
*Geometric triangulated categories: enhancements and weak approximation, III* - 11:30 Joana Cirici (Universitat de Barcelona)
*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.

*fiorenza@mat.uniroma1.it*

**Mercoledì 11 settembre 2024**

Ore 16:00, Aula Dal Passo, Dipartimento di Matematica, Università di Roma Tor Vergata

Seminario Algebre di Operatori

Karl-Hermann Neeb (FAU Erlangen-Nürnberg)
*Local nets on causal flag manifolds*

We are interested in obtaining local nets in the sense of Haag--Kastler from unitary representations of a connected Lie group G. A natural sets of axioms naturally leads to a causal structure on M. We focus on the case where M = G/P is a flag manifold of a simple Lie group G, or a covering space thereof. Then G must be hermitian Lie group and M a conformal compactification of a Euclidean Jordan algebra V. It's simply connected covering is a simple space-time manifold in the sense of Mack--de Riese. We show that the unitary representations permitting non-trivial nets are the positive energy representations (direct integrals of lowest weight representatiosn). These nets have several interesting features. One is that the ``wedge regions'' that link the geometry of M to the modular theory of the algebras involved are given by the intervals (double cones) of W.~Bertram's cyclic order on M. Another is that locality properties of the net can be specified in terms of open G-orbits in the space of pairs, which is most interesting for covering spaces because the number of these orbits corresponds to the number of sheets in the covering. Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

**Giovedì 12 settembre 2024**

Aula I, Dipartimento di Chimica, Edificio Caglioti

Convegno GAP XIX: *Moduli spaces and higher structures*

Programma:

- 9:00 Alberto Cattaneo (Universität Zürich)
*An introduction to the Batalin-Vilkovisky formalism, I*

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. - 10:30 Alberto Canonaco (Università di Pavia)
*Localizations of the categories of dg and of \(A_\infty\) 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 \(A_\infty\) categories are equivalent. The equivalence holds not only as ordinary categories, but even as \(\infty\)-categories; moreover, one can replace (strictly unital) \(A_\infty\) 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 \(A_\infty\) functors. - 11:30 Emma Lepri (University of Glasgow)
*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 \(A_\infty\)-algebra \(Ext_X^\ast(O_C(-1), O_C(-1))\). We also discuss relations of this DG-algebra with Booth's derived contraction algebra, and the Donovan-Wemyss conjecture. - 14:10 Alberto Cattaneo (Universität Zürich)
*An introduction to the Batalin-Vilkovisky formalism, II* - 15:10 Andrea D'Agnolo (Università degli Studi di Padova)
*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.

*fiorenza@mat.uniroma1.it*

**Giovedì 12 settembre 2024**

Ore 14:15, Aula M1, Dipartimento di Matematica e Fisica, Università Roma Tre

Seminario di Geometria

Gavril Farkas (Humboldt)
*The Green-Lazarsfeld Secant Conjecture*

The Green-Lazarsfeld Secant Conjecture is a generalization of Green's Conjecture on syzygies of canonical curves to the cases of arbitrary line bundles. It predicts that on a curve embedded by a line bundle of sufficiently high degree, the existence of a p-th syzygy is equivalent to the existence of a certain secant to the curve. I will discuss the history of this problem, then establish the Green-Lazarsfeld Secant Conjecture for curves of genus g in all the divisorial cases, that is, when the line bundles that satisfy the corresponding secant condition form a divisor in the Jacobian of the curve.

Per informazioni, rivolgersi a: * amos.turchet@uniroma3.it*

**Giovedì 12 settembre 2024**

Ore 15:00, Sala Consiglio, Dipartimento di Matematica, Sapienza Università di Roma

Seminario di presentazione della tesi di dottorato

Massimiliano Puglisi (Dipartimento di Matematica, Sapienza Università di Roma)
*The Stolz' positive scalar curvature sequence for G-proper manifolds and depth-1 pseudomanifolds*

We will introduce the Stolz sequence and explain how it plays a role in the study of metrics with positive scalar curvature. We shall then extend it to two different contexts: that of (G, F)-spaces, i.e., proper G-spaces with isotropy groups belonging to a family F of subgroups of G, and that of manifolds with non-isolated singularities. In both cases, the sequence is studied for appropriate classes of metrics with positive scalar curvature, and it is shown, in particular, how the Stolz R-groups have a strict dependence on the 2-skeleton. Subsequently, the mapping of the Stolz sequence to the Higson-Roe analytic surgery sequence in the singular setting with (L,G)-singularities will be explained. This will then be used in order to provide a lower bound on the rank of the bordism group of these psc metrics.

Per informazioni, rivolgersi a: * paolo.piazza@uniroma1.it*

**Venerdì 13 settembre 2024**

Aula I, Dipartimento di Chimica, Edificio Caglioti

Convegno GAP XIX: *Moduli spaces and higher structures*

Programma:

- 9:00 Alberto Cattaneo (Universität Zürich)
*An introduction to the Batalin-Vilkovisky formalism, III* - 10:30 Francesca Carocci (Université de Genève)
*Logarithmic Linear series*

Linear series on smooth curves parametrize invertible sheaves together with linear subspaces of their global sections. This has been generalized to nodal curve of compact type by Eisebud-Harris and Ossermann. I will present an approach to the problem using logarithmic geometry which allows us to extend the theory of linear series to arbitrary nodal curves. A prominent role is played by vector bundles on Olsson fans. This is joint work with Luca Battistella and Jonathan Wise. - 11:30 Claudia Rella (Université de Genève)
*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.

Per informazioni, rivolgersi a:

*fiorenza@mat.uniroma1.it*

**Venerdì 13 settembre 2024**

Ore 14:30, Aula Dal Passo, Dipartimento di Matematica, U Roma Tor Vergata

Algebra and Representation Theory Seminar

Gastón Andrés García (U Nacional de La Plata)
*On the representation theory of generalized small quantum groups*

The small quantum groups u_q(g) are finite-dimensional quotients of quantum universal enveloping algebras U_q(g) at a root of unity q for g a semisimple complex Lie algebra. After the work of Lusztig, the representation theory of these quantum objects was intensively studied because of its relation with the representation theory of semisimple algebraic groups in positive characteristic. In this talk, I will present some results on the representation theory of what we call "generalized" small quantum groups. A particular feature of these objects is that the role of the corresponding Cartan subalgebra is played by a finite non-abelian group. Nevertheless, they still admit a triangular decomposition and share similar properties with the standard quantum groups, like the existence of weights (that are no longer one-dimensional) and Verma modules. This talk is based on a joint work with Cristian Vay [Simple modules of small quantum groups at dihedral groups, Doc. Math. 29 (2024), 1–38].

**Venerdì 13 settembre 2024**

Ore 16:00, Aula Dal Passo, Dipartimento di Matematica, U Roma Tor Vergata

Algebra and Representation Theory Seminar

Sabino di Trani (U Roma Sapienza)
*Graph cohomologies, matroids and colorings.*

A celebrated result in graph theory links the chromatic polynomial of a graph to the Tutte polynomial of the associated graphic matroid. In 2005, Helme-Guizon and Rong proved that the chromatic polynomial is categorified by a cohomological theory called chromatic cohomology. In this talk, I will describe how to associate a matroid to a directed graph G, called the multipath matroid of G, which encodes relevant combinatorial information about edge orientation. We also show that a specialization of the Tutte polynomial of the multipath matroid of G provides the number of certain "good" digraph colorings. Finally, analogously to the relationship between the chromatic polynomial and chromatic cohomology, I will show how the polynomial expressing the number of "good" digraph colorings is linked to multipath cohomology, introduced in a work with Caputi and Collari in 2021.

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