Instanton bundles on $\mathbb{P}^3$ emerged from the twistor program as the algebraic counterpart of classical problems in Yang-Mills theory. Inspired by this connection, the study and generalization of instantons have garnered significant interest in recent years. In this talk, I will focus on the point-line flag threefold $F(0,1,2)$, which serves as the twistor space of the complex projective plane. In particular, I will introduce the notion of ’t Hooft instanton bundles on $F(0,1,2)$ and discuss both their existence and moduli spaces. This is joint work with F. Malaspina, S. Marchesi, and J. Pons-Llopis. 

S. Ramanan proved that a universal family (also called a Poincar´e bundle) exists for the moduli problem of vector bundles on a smooth curve if and only if the rank and degree are coprime. One of the key elements in the proof is the computation of the Picard group of the moduli space. First, we prove the non-existence of a Poincar´e bundle for the moduli problem of vector bundles on nodal curves when the degree and rank are not coprime closely following that of Ramanan.

 When the degree is sufficiently high, the pushforward of a Poincar´e bundle to the moduli space is a vector bundle, called the Picard bundle. Although the existence of Poincar´e bundles (hence Picard bundles) depend on the rank and degree being relatively prime, there always exists a Poincar´e family of projective bundles called the projective Poincar´e bundle. Similarly, there is a projective Picard bundle. Next, we discuss the stability of these bundles. 

On the way to achieve these goals, we compute the codimension of a few closed subsets of the moduli spaces. Using these results on codimension, we show that the stable vector bundles on nodal curves, which arise from representations, form a big open set of the moduli space. We also use them to compute Picard groups of the moduli spaces. 

This is a joint work with Prof. Usha N. Bhosle and Dr. Sanjay Singh. 

In this talk, I will survey aspects of Brill-Noether theory for moduli spaces of sheaves on surfaces. I will first discuss the cohomology of the general stable sheaf on surfaces such as Hirzebruch, K3 and abelian surfaces. I will then describe some recent results on the cohomology jumping loci. This talk is based on joint work with Jack Huizenga, Howard Nuer, Neelarnab Raha and Kota Yoshioka. 

The Weierstrass semigroup of pole orders of meromorphic functions in a point p of a smooth algebraic curve C is a classical object of study; a celebrated problem of Hurwitz is to characterize which subsemigroups S of the natural numbers with finite complement are realizable as Weierstrass semigroups S=S(C,p). We establish realizability results for cyclic covers of hyperelliptic targets marked in hyperelliptic Weierstrass points; and we show that realizability is dictated by the behavior under j-fold multiplication of certain divisor classes in hyperelliptic Jacobians naturally associated to our cyclic covers, as j ranges over all natural numbers. 

In this talk, we describe different compactifications of the moduli space of n distinct weighted labeled points in a flag of affine spaces. These spaces are constructed via generalizations of the Fulton-MacPherson compactification. For specific weight choices, we show that our moduli problem admits toric compactifications that coincide with the polypermutohedral variety of Crowly-Huh-Larson-Simpson-Wang and with the polystellahedral variety of Eur-Larson. This is joint work with J. Gonzalez-Anaya and J.L. Gonzalez.  

A $\mu$-stable vector bundle $\mathcal{E}$ of rank 2 with $c_1 (\mathcal{E})=0$ on $\mathbb{P}_{\mathbb{C}}^{3}$ is called a mathematical instanton bundle if $\mathrm{H}^1 (\mathbb{P}^{3}, \mathcal{E}(-2))=0$. This type of bundle has been generalized to other varieties in various ways. First, it has been generalized to odd-dimensional projective spaces by M. M. Capria and S. M. Salamon, then to non-locally free sheaves of any rank on arbitrary projective spaces by M. Jardim. Then, D. Faenzi and A. Kuznetsov extended the definition to other Fano threefolds, and later, V. Antonelli and F. Malaspina modified the definition to apply to any polarization of Fano threefolds, introducing the concept of an $h$-instanton bundle. Finally, V. Antonelli and G. Casnati further broadened the definition to cover any polarized variety $(X,h)$.

In this talk, we will focus on rank 2 $h$-instanton sheaves on ruled Fano threefolds with Picard rank 2 and index 1. This is a joint work with Marcos Jardim.

For n greater or equal than 1 we show that the length 1 nested Hilbert scheme of the total space Xn of the line bundle OP1(-n), parameterizing pairs of nested 0-cycles in Xn, is a quiver variety associated with a suitable quiver with relations. This generalizes previous work about nested Hilbert schemes on C2 in one direction, and about the Hilbert schemes of points of Xn in another direction. 

Solomon Lefschetz showed that the Picard group of a general surface in P3 of degree greater than three is ZZ. That is, the vast majority of surfaces in P3 have the smallest possible Picard group. The set of surfaces of degree greater than 3 on which this theorem fails is called the Noether-Lefschetz locus. This locus has infinite components and their dimensions are somehow mysterious.

In this talk, I will describe infinitely many Noether-Lefschetz components and compute their dimension. This will exhibit infinitely many components that were previously unknown. This is joint work with Montserrat Vite and Manuel Leal. 

In an important paper, Eisenbud--Schreyer posed the question of whether every projective variety has an Ulrich sheaf. This question has turned out to be complicated to be answered in general, although many attempts to attack the problem have been made in the last 20 years. In this presentation, we show a positive response to a specialization of this problem to the case of $d$-Veronese embedding of $\mathbb{P}^3$, first conjectured by Costa--Miró-Roig. Surprisingly, the proof involves our understanding of the moduli of instanton bundles and the existence of a SL_2-invariant instanton bundle. This is the result of a joint work with Daniele Faenzi. 

Compact Hyperkähler manifolds (also known as IHS) are notable Kähler manifolds renowned for their rich and intriguing geometry. Using classical results from quadratic forms and methods from Bridgeland stability conditions, we will show that any projective Hyperkähler manifold (deforming to the Hilbert scheme of n points on a K3 surface) with a Picard number at least 4 is always isomorphic to a moduli space of (twisted) stable sheaves on a K3 surface. We will also discuss examples with Picard numbers less than 4, including cases that are not birational to a moduli space of twisted sheaves on a K3 surface. These examples reveal connections to an open problem in algebraic geometry regarding the rationality of cubic fourfolds. 

In the setting of a variety X with a tilting bundle T, we investigate how X can be reconstructed as a quiver GIT quotient of the algebra defined by the endomorphism ring of T on X. Examples are provided where this algebra is expressed using a quiver with relations, enabling a detailed understanding of the derived category of coherent sheaves on X. The talk will primarily focus on extending these methods to the framework of supervarieties. 

Let X be a smooth irreducible projective surface.  A coherent system on X is a pair (E, V ) where E is a coherent sheaf on X, and V ⊆ H^0 (X, E). Associated to the coherent systems there is a notion of stability which depends on a parameter α ∈ Q[m]. This notion allows the construction of the moduli space of α−stable coherent systems and thus leads to a finite family of moduli spaces. In this talk, we establish a bound for the maximum value of dim(V ) for which there exists stable coherent systems. Also, We will study the moduli space of coherent systems in P^2 and we will show necessary conditions for the existence of α-semistable coherent systems (E, V ) with rank E = 2 and dim(V ) ≥ 2. Afterwards, we give numerical conditions to the nonemptiness of the moduli space 

We find all the nonsingular quartics in P^3(C) which are invariant under a finite primitive subgroup of PGL_4(C). Given the well-known classification of such groups, we will describe an algorithm that effectively computes such quartics. For example, we will prove that (x_0)^4+(x_1)^4+(x_2)^4+(x_3)^4+12(x_0)(x_1)(x_2)(x_3)=0 corresponds to the biggest primitive finite group, and that its automorphism group is (Z_2)^4\times S_5. The algorithm can be used for any hypersurface of any degree.

I this talk I will present the K-theoretical invariants appearing in the classification of the electronic properties of crystals. 

Let ( m ) be a square-free positive integer, and let ( A ) be an ( n \times n ) integer matrix of order ( m ). The matrix ( A ) defines an action of the cyclic group of order ( m ) on the ( n )-dimensional torus. In this talk, we describe a method for computing the topological K-theory of the resulting toroidal orbifold for any matrix ( A ). This is achieved by considering the associated semidirect product of a free abelian group of rank ( n ) with a cyclic group of order ( m ), where the conjugacy action is defined using the matrix ( A ) and its classifying space for proper actions. This work is a collaboration with Luis Jorge Sánchez.

Understanding moduli spaces of some objects is a task in algebraic geometry, and there have been different methods of approaching it. In this seminar, we expect to explore some methods of describing the Bridgeland moduli space (which is a generalization of the usual Mumford moduli space) of complexes of sheaves that sit in P3. In more detail,  we will describe how some objects behave when crossing particular walls, some of them in terms of quiver representations and some particular types of instantons in terms of monadic-type representations.