Venue: Thu, May 30, 11h30. ICTP, Enrico Fermi Building - Meeting room 101.

Title:  Cohomological Hall algebras, their representations, and Nakajima operators

Abstract: In the first part of the talk, I will give a brief introduction to the theory of 2d cohomological Hall algebras (COHAs), focusing on the example of COHAs of zero-dimensional sheaves on smooth surfaces. Additionally, I will describe certain geometric representations of these COHAs and introduce Nakajima type operators. In the second part of the talk, I will discuss a generalization and categorification of this framework.

Venue: Wed, May 29, 15h00. UniTS, Seminar Room (H2bis, third floor).

Title:  Chebyshev varieties

Abstract: Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials or their multivariate generalizations. They play the role of toric varieties in sparse polynomial root finding, when monomials are replaced by Chebyshev polynomials. We introduce these objects and discuss their main properties, including dimension, degree, singular locus and defining equations, as well as some computational experiments.

Venue: Tue, May 28, 15h00. SISSA, Dubrovin Lecture Room (136, first floor).

Title:  Computing the base change conductor for Jacobians 

Abstract: Let K be a discretely valued field, with ring of integers R.The base change conductor of an abelian K-variety, denoted c(A),  is a numerical invariant which measures the failure of A to have semi-abelian reduction over R. It can be  difficult in general to compute c(A) explicitly. In this talk I will present an approach for Jacobians, using intersection theory and invariants of quotient singularities on certain normal R-models of the curve in question. Joint work with D. Eriksson and J. Nicaise.

Venue: Tue, May 14, 10h00. SISSA, Dubrovin Lecture Room (136, first floor).

Title:  The 3-fold K-theoretic DT/PT vertex correspondence

Abstract: Donaldson-Thomas (DT) and Pandharipande-Thomas (PT) invariants are two curve counting invariants for 3-folds. In the Calabi-Yau case, a correspondence between the numerical DT and PT invariants has been conjectured by Pandharipande and Thomas and proven by Bridgeland and Toda using wall-crossing. For equivariant K-theoretically refined invariants, the DT/PT correspondence reduces to a DT/PT correspondence of equivariant K-theoretic vertices. In this talk I will explain our proof of the equivariant K-theoretic DT/PT vertex correspondence using a K-theoretic version of Joyce's wall-crossing setup. An important technical tool is the construction of a symmetized pullback of a symmetric perfect obstruction theory on the original DT and PT moduli stacks to a symmetric almost perfect obstruction theory on auxiliary moduli stacks. This is joint work with Nick Kuhn and Henry Liu.

Venue: Mon, May 13, 16h00. SISSA, Dubrovin Lecture Room (136, first floor).

Title:  Deformations of stability conditions with applications to very general Hilbert schemes of points and abelian varieties

Abstract: The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem, especially when the canonical bundle

is trivial. In this talk, I will illustrate a new and very effective technique based on deformations.

A key ingredient is a general result about deformations of bounded t-structures (and with some additional and mild assumptions). Two remarkable applications are the construction of stability conditions for very general abelian varieties in any dimension and for irreducible holomorphic symplectic manifolds of Hilb^n-type, again in all possible dimensions. This is joint work with C. Li, E. Macrì, Alex Perry and X. Zhao.

Venue: Wed, Mar 20, 14h00 . SISSA, Dubrovin Lecture Room (136, first floor).

Title:  Moduli spaces of stability conditions and of quadratic differentials

Abstract: The space of Bridgeland stability conditions is a complex manifold attached to a triangulated category D, parametrizing some t-structures of the category. In some cases, when D is constructed from a Ginzburg algebra of a quiver, it is isomorphic to a moduli space of quadratic differentials on a Riemann surface. I will review this correspondence, which is due to Bridgeland-Smith in the simple zeroes case and was extended in [BMQS] to higher order zeroes, to motivate a tentative construction of a smooth compactification of the stability manifold. This is based on joint works with M.Moeller, Y.Qiu, and J.So.

Venue: Wed, Mar 13, 15h00. UniTS, MIGe, H2bis building,  Aula Seminari, third floor (mid-corridor).

Title:  Integrable systems and the Cremona-cubes group

Abstract: The standard Cremona transformation is a classical object in algebraic geometry. In a joint work with G. Gubbiotti (University of Milan), we studied the algebraic entropy and the invariants of birational maps of the projective 3-space defined as the composition of the standard Cremona transformation with some special projectivities. Precisely, we consider projectivities acting on 12 points in the Reye configuration. These kind of maps appear for instance as the Kahan–Hirota–Kimura discretisation of the Euler top. In my seminar I will explain how such results can be obtained using classical techniques from algebraic geometry. If time permits, I will discuss some higher dimensional generalisations of this construction (joint work with G. Gubbiotti and M. Weinreich (Harvard)). 

Venue: Thu, Feb 22, 15h00. SISSA, Room 134 (first floor).

Title:  Virtual class on a Quot scheme of points on a threefold

Abstract: We construct a semi-perfect obstruction theory on the Quot scheme of points corresponding to a vector bundle on a smooth projective threefold. We get a virtual class in the Chow group in degree zero and define the higher rank Donaldson Thomas invariants. All related definitions will be recalled.

Venue: Wed, Jan 24, 15h00. UniTS, H2bis building,  Aula Seminari, third floor (mid-corridor).

Title:  Recipes for exotic 2-links in 4-manifolds

Abstract: Two smoothly embedded surfaces in the same smooth 4-manifold are called exotic if they are C^0 ambient isotopic without being smoothly isotopic.

In this talk, I will briefly introduce this peculiar phenomenon and I will sketch a construction of Torres to produce infinite families of 2-spheres that are pairwise exotic and topologically unknotted (each 2-sphere bounds a locally flat embedded 3-disk).

I will then present a generalization of this construction based on a joint work with V. Bais, Y .Benyahia and R. Torres (see https://arxiv.org/abs/2206.09659). In particular we will produce exotic families of n-component 2-links (disjoint union of n embedded 2-spheres) for which the fundamental group of the complement is free with n generators.  

Lastly we will explore some of the proprieties of the constructed exotic families and we will compare them to proprieties of other families arising from different constructions.

Venue: Monday, Jan 15, 16h00. SISSA, Dubrovin Lecture Room (136), first floor.

Title: Reduced Gromov-Witten invariants in higher genus

Abstract: The Gromov-Witten invariants of projective spaces are not enumerative in positive genus. The reason is geometric: the moduli space of genus-g stable maps has several irreducible components, which contribute in the form of lower-genus GW invariants. In genus one, Vakil and Zinger constructed a blow-up of the moduli space of stable maps and used it to define reduced Gromov-Witten invariants, which correspond to curve-counts in the main component. I will present a new definition of reduced Gromov-Witten invariants of complete intersections in all genus using desingularizations of sheaves. This is joint work with E. Mann, C. Manolache and R. Picciotto and can be found in arXiv:2310.06727.

Venue: Wednesday December 13, at 15:00. UniTS, Seminar Room, third floor (mid corridor), H2 bis, Via Valerio 12/1.

Title: An exotic structure on a non-orientable 4-manifold via Gluck twisting

Abstract:  In this talk I will talk about a joint work with Rafael Torres, in which we create an exotic smooth structure on the non-orientable total space of a non-trivial 2-sphere bundle over the real projective plane, where by “exotic” we mean not equivalent to the standard smooth structure. Our procedure mimics Cappell-Shaneson’s construction of an exotic real projective 4-dimensional space, which I will sketch at the beginning of the talk as a warm up. The exotic structure we produce can also be obtained by applying a cut and paste procedure called Gluck twist to the non-orientable total space of the trivial 2-sphere bundle over the real projective plane.

Venue: Wednesday November 22, at 15:00. UniTS, Seminar Room, third floor (mid corridor), H2 bis, Via Valerio 12/1.

Title: Generalisations of the Hecke Algebras from loop braid groups

Abstract: This work takes inspiration by from the braid group revolution ignited by Jones in the early 80s, to study representations of the motion group of the free unlinked circles in the 3 dimensional space, the loop braid group LBn. Since LBn contains a copy of the braid group Bn as a subgroup, a natural approach to look for linear representations is to extend known representations of the braid group Bn. Another possible strategy is to look for finite dimensional quotients of the group algebra, mimicking the braid group / Iwahori-Hecke algebra / Temperlely-Lieb algebra paradigm. Here we combine the two in a hybrid approach: starting from the loop braid group LBn we quotient its group algebra by the ideal generated by (σi + 1)(σi − 1) as in classical Iwahori-Hecke algebras. We then add certain quadratic relations, satisfied by the extended Burau representation, to obtain a finite dimensional quo- tient that we denote by LHn. We proceed then to analyse this structure. Our hope is that this work could be one of the first steps to find invariants à la Jones for knotted objects related to loop braid groups.

Venue: Thursday November 16, at 15:15. SISSA, Dubrovin Lecture Room 136 

Title: Chiral Gauss-Manin connection 

Abstract: We will explain how to construct the Gauss-Manin connection on the relative chiral de Rham complex. The construction uses (a chiral version of) the "calculus" formalism introduced  by Boris Tsygan. This is a joint work with Fyodor Malikov and Boris Tsygan. 

Venue: Tuesday October 17, at 15:00. ICTP, Leonardo Building, Stasi Room

Title: Curve counting on the Enriques surface and orthogonal modular forms 

Abstract: An Enriques surface is the quotient of a K3 surface by a fixed-point free involution. Klemm and Marino conjectured that the Gromov-Witten invariants (which are virtual counts of curves) of the local Enriques surface are certain orthogonal modular forms. In particular the genus 1 series recovers Borcherds famous automorphic form on the moduli space of Enriques surface. However, not much is known about the more general descendent Gromov-Witten theory of the Enriques surface. In this talk I will first discuss and sketch the proof of the Klemm-Marino formula and then explain a conjecture which links the descendent Gromov-Witten invariants of the Enriques to orthogonal modular forms in general. If time permits I will give some explicit formulas for point constraints and their descendents. 

Venue: Tuesday July 18, at 16:00. ICTP, Leonardo Building, Stasi Room

Title: Non-commutative abelian surfaces and generalized Kummer varieties.

Abstract: Examples of non-commutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and Gushel-Mukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkähler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these non-commutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkähler manifolds deformation equivalent to a generalized Kummer variety is not yet available.

In this talk we will construct families of non-commutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkähler manifolds of generalized Kummer deformation type. This is a joint work in progress with Arend Bayer, Alex Perry and Xiaolei Zhao.

Venue: Leonardo Building, Stasi Room  at 17:00. ICTP

Title: Flags on Fano 3-fold hypersurfaces

Abstract: The existence of Kaehler-Einstein metrics on Fano 3-folds can be determined by studying some positive numbers called stability thresholds. K-stability is ensured if appropriate bounds can be found for these thresholds. An effective way to verify such bounds is to construct flags of point-curve-surface inside the Fano 3-folds. This approach was initiated by Abban-Zhuang, and allows us to restrict the computation of bounds for stability thresholds only on flags. We employ this machinery to prove K-stability of terminal quasi-smooth Fano 3-fold hypersurfaces. Many of these varieties had been attacked by Kim-Okada-Won using log canonical thresholds. In this talk I will tackle the remaining Fano hypersurfaces via Abban-Zhuang Theory. 

Venue: UniTS, H2bis building in via Valerio 12/1, Aula Seminari, third floor at 15:00

Title: Khovanov homology of positive links and of L-space knots

Abstract: A knot is called positive if it admits a diagram with only positive crossings. We will discuss that such knots enjoy many interesting properties and are closely related to other fields, such as algebraic geometry, symplectic and contact topology, gauge theory, and smooth 4-manifold topology. Then we will introduce a new obstruction for positivity of knots in terms of Khovanov homology. This is based on joint work with Naageswaran Manikandan, Leo Mousseau, and Marithania Silvero.

Venue: UniTS, H2bis building in via Valerio 12/1, Aula Seminari, third floor at 15.00

Title: O’Grady tenfolds as moduli spaces of sheaves

Abstract: Knowing birational models of irreducible holomorphic symplectic (ihs) manifolds is a significant stage towards the full comprehension of their geometry. Many examples of them can be realized as moduli spaces of sheaves on abelian and K3 surfaces, or in the case of O’Grady type manifolds, as symplectic resolutions of some singular moduli spaces of sheaves with specific numerical invariants. The latter form a codimension 3 locus in the moduli space of ihs manifolds of OG10 type: it is then natural to wonder when a manifold of OG10 type is birational to a moduli space of sheaves on a K3. In the talk, I will provide a lattice theoretic answer to this question and present a criterion to determine when a birational transformation of an OG10 is induced by an automorphism of the K3 surface. If time permits, I will show applications to the Li–Pertusi–Zhao variety of OG10 type associated to any smooth cubic fourfold. The talk is based on a joint work with Franco Giovenzana and Annalisa Grossi. 

Venue: UniTS, H2bis building, Aula Seminari - third floor at 15.00

Title: Alexander polynomials and symplectic curves in CP^2

Abstract: Libgober defined Alexander polynomials of (complex) plane projective curves and showed that it detects Zariski pairs of curves: these are curves with the same singularities but with non-homeomorphic complements. He also proved that the Alexander polynomial of a curve divides the Alexander polynomial of its link at infinity and the product of Alexander polynomials of the links of its singularities. We extend Libgober's definition to the symplectic case and prove that the divisibility relations also hold in this context. This is work in progress with Hanine Awada.

Venue: SISSA, Room 134

Title: BPS invariant from p-adic integrals

Abstract: 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 F, we define a natural measure on the F-points of 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.

Venue: SISSA, Room 133 at 15.00

Title:: Double nested Hilbert schemes and stable pair invariants

Abstract: Hilbert schemes of points on a smooth projective curve are symmetric powers of the curve itself; they are smooth and we know essentially everything about them. We propose a variation by studying double nested Hilbert schemes of points, which parametrize flags of 0-dimensional subschemes satisfying certain nesting conditions dictated by Young diagrams. These moduli spaces are almost never smooth but admit a virtual structure à la Behrend-Fantechi. We explain how this virtual structure plays a key role in (re)proving the correspondence between Gromov-Witten invariants and stable pair invariants for local curves.

Venue: SISSA, Room 133 at 15.00

Title:: The hierarchy of topological type and the equivalence conjecture

Abstract: Cohomological field theories (CohFTs) were introduced by Kontsevich and Manin in 1994. In 2001 Dubrovin and Zhang defined an integrable hierarchy starting from a semisimple CohFT. In 2015 a new construction of an integrable hierarchy by Buryak appeared. It is conjectured that these two constructions are related by a so-called Miura transformation.

In the first part of this seminar, we will define the hierarchy introduced by Dubrovin and Zhang and we discuss the above equivalence conjecture. In the second part, we will explicitly show the equivalence in some specific examples

Venue: SISSA, Room 133 at 15.00

Title: The E3-structure on the spherical category of a reductive group

Abstract: Let G be a reductive group over the complex numbers, e.g. GL_n. The notion of affine Grassmannian associated to G leads to the 

introduction of a monoidal triangulated/dg-category Sph(G), called the spherical category of G, which plays an important role in the Geometric 

Langlands program. For example, its behaviour provides important constraints in the formulation of the Geometric Langlands Conjecture. This monoidal category is not symmetric monoidal, but it admits a t-structure whose heart is symmetric monoidal: more precisely, by the Geometric Satake Theorem (Ginzburg, Mirkovic-Vilonen) the heart is monoidal-equivalent to a category of representations of a group (the Langlands dual of G) with its (symmetric monoidal) tensor product. In this talk, I will present how to upgrade the existing E1-monoidal structure on Sph(G) to an E3-monoidal one, which formally recovers the symmetric monoidal structure of the heart. The construction implements ideas of Jacob Lurie and uses a strongly topologically-flavoured presentation of Sph(G), namely as a category of constructible sheaves over a stratified space. Part of this work is joint with Morena Porzio.

Venue: ICTP, Room D (old SISSA building) at 16.00

Title: Double EPW-sextics with actions of A7 and irrational GM threefolds

Abstract:  We construct two examples of projective hyper-Kähler fourfolds of K3[2]-type with an action of the alternating group A7, making them some of the most symmetric hyper-Kähler fourfolds. They are realized as so-called double EPW sextics and this allows us to construct an explicit family of irrational Gushel-Mukai threefolds.

Venue: ICTP, Budinich Lecture Hall 14.00

Abstract:  Starting with an ADE singularity C^2/Gamma for Gamma a finite subgroup of SL(2,C), one can build various moduli spaces of geometric and representation-theoretic interest as Nakajima quiver varieties. These spaces depend in particular on a stability parameter; quiver varieties at both generic and non-generic stability are of geometric interest. Generating functions of Euler characteristics at different points in stability space are controlled by specialisation formulae of characters of Lie algebra representations. We will explain some of these connections, focusing in particular on the abelian case. Based on joint papers and projects with Craw, Gammelgaard, Gyenge, and Nemethi.

Venue: SISSA, Room 134 at 10:00.

Title: Derived algebraic geometry and the filtered circle

Abstract: In this talk I will describe my work with Robalo and Toën on the filtered circle. This is an algebro-geometric object of a homotopical nature (more precisely, it is a stack) whose cohomology admits a natural filtration. Using basic constructions in derived algebraic geometry, one then forms mapping spaces with this object as a source and an arbitrary (derived) scheme X as the target. The cohomology of these mapping spaces recovers the Hochschild homology of X, together with its functorially defined HKR filtration. I will of course describe some key ideas from derived algebraic geometry necessary to understand the construction.