Title: Instanton and Ulrich bundles on cubics

Abstract: TBA

Title: Three lectures on vector bundles 

1. Lagrangian subspaces of the moduli space of simple sheaves on K3 surfaces. Applications 

2. Homogeneous and uniform vector bundles on projective spaces 

3. Ulrich bundles on general surfaces in a projective space 

Abstract: 

1. In the first lecture, we will present a new and useful method to construct isotropic and Lagrangian subspaces of the moduli space of simple sheaves on K3 surfaces.

2. In the second lecture, we will address the long standing open conjecture: Any uniform vector bundle of rank r<2n on a projective space of dimension n is homogeneous.

3. In the third lecture, we will discuss the problem of determining the pairs (r, d) such that a general surface of degree d in the 3-dimensional projective space supports an Ulrich bundle of rank r.

Title: Many facets of parabolic bundles

Abstract: A parabolic bundle is one of the simplest kinds of decorated vector bundles. It is a vector bundle with a collection of flags of subspaces on a fixed set of special points. In this talk, we review the basics of parabolic bundles, including several equivalent ways to understand them, the geometry of their moduli spaces, and some connections with other branches of mathematics, such as representation theory and conformal field theory. We also highlight some questions.

Title: Introduction to the theory of Ulrich, instanton and logarithmic bundles

Abstract: 

Lecture I (Ulrich bundles): Given a polarized projective variety (X, H), Ulrich bundles on it are the vector bundles with the simplest table of cohomology groups. In this lecture we are going to introduce the different approaches to the definition of Ulrich bundles, present different methods of construction and state the leading questions related to their existence. 

Lecture II (Instanton bundles): Instanton bundles on P^3 were defined as the algebraic geometric counterpart to SU(2)-connections with self-dual curvature on the real sphere S^4. Their study prompted the development of many techniques that have become central in algebraic geometry (monads, loci of jumping lines...). Recently its definition has been extended to arbitrary polarized projective varieties. In this lecture we are going to review the main points of this theory. 

Lecture III (Logarithmic sheaves on smooth projective varieties): Given a smooth projective variety X and a reduced divisor D on it,  the sheaf of logarithmic differential forms $\Omega^1_X(log D)$ is defined as the sheaf of one-forms having logarithmic poles along D. The sheaf was studied extensively by Deligne and Saito in connection with Hodge theory. On the other hand, this sheaf, or equivalently its dual sheaf $\mathcal{T}_X(- log D)$ of vector fields tangent along D, has been investigated from the point of view of stability theory, as well as related to Torelli problems. In this lecture we will introduce this objects and, time permitting, some generations.

Title: Hyperelliptic curves and Ulrich sheaves on the complete intersection of two quadrics

Subtitle 1: Vector bundles on hyperelliptic curves, Koszul algebras and BGG 

Subtitle 2: Clifford algebra, Morita equivalence and Reid's theorem 

Subtitle 3: Tate resolutions of O_X-modules and Ulrich bundles 

Abstract: Using the connection between hyperelliptic curves, Clifford algebras, and smooth complete intersections X of two quadrics, we describe Ulrich bundles on X and construct some of minimal possible rank. 

We will start out describing vector bundles on hyperelliptic curves via matrix factoriztions. The Jacobian of the hyperelliptic curve associated to X can be identified with the space of maximal isotropic subspaces of the pencil by Reid's theorem. We give a proof of Reid's theorem and apply these techniques to describe Ulrich bundles on X. Finally we will construct Ulrich bundles of smallest possible rank on X.

Title: Deformation of tangent bundles of projective hypersurfaces

Abstract: We provide a concrete birational model of the moduli space of sheaves on a projective hypersurface which contains the tangent bundle. It turns out to be an affine variety modeled after an irreducible representation associated to the ambient projective space. This is a joint work with Kiryong Chung and Jun-Muk Hwang.

Title: Moduli Spaces of Sheaves on Complete Intersections via Net Logarithmic Tangent Sheaves

Abstract: A subspace of the moduli space of semistable torsion-free sheaves on a smooth projective variety with fixed Chern classes can be realized as a locus of logarithmic sheaves associated with an arrangement of divisors on the variety. In this talk, I will introduce a method to describe the moduli space of stable sheaves on complete intersections via a generalized logarithmic tangent sheaf, termed the net logarithmic tangent sheaf. This approach allows us to describe non-generic boundary points that are not realized as logarithmic sheaves in the classical sense. This talk is based on joint work with Sukmoon Huh.

Title: Moduli spaces of rational curves in moduli spaces of vector bundles over curves

Abstract: Let M be the moduli space of stable vector bundles of rank r and degree d over a smooth projective curve of genus >1, where r and d are coprime. When r>1, the moduli space R of rational curves in M of fixed degree c is non-compact for c>1. In this talk, I will discuss the birational geometry of R and compare various moduli theoretic compactifications by Hilbert scheme, stable maps and quasi-maps. Of particular interest is the case of conics in M with r=2 where the Hecke correspondence plays a key role. I will compare the moduli theoretic compactifications with those birational models from geometric invariant theory. Based on an ongoing project with Sanghyeon Lee.

Title: Resonance, Syzygies, and Rank-3 Ulrich bundles on V_5

Abstract: An Ulrich bundle E on an n-dimensional projective variety (X, O(1)) is a vector bundle whose module of twisted global sections is a maximal Cohen-Macaulay module having the maximal number of generators in degree 0. It was once studied by commutative algebraists, but after Eisenbud and Schreyer introduced its geometric viewpoint, many people discovered several important applications in wide areas of mathematics. In this pioneering paper they asked whether every projective variety supports an Ulrich sheaf, and if yes, then what is the smallest possible rank. Thanks to a number of studies, the answer for the both question is now well-understood for del Pezzo threefolds. In particular, a del Pezzo threefold V_d of (degree d≥3) has an Ulrich bundle of rank r for every r at least 2. The Hartshorne-Serre correspondence translates the existence of rank-3 Ulrich bundle into the existence of an ACM curve C in V_d of genus g=2d+4 and degree 3d+3. The construction of rank-3 Ulrich bundle on a (general) cubic threefold is first suggested by Arrondo-Costa and then by Geiss-Schreyer, by showing that a "random" curve of given genus and degree lies in a cubic threefold and satisfies the whole conditions we needed. We discuss how this problem is related to the unirationality of the Hurwitz space H(k, 2g+2k-2) and the moduli of curves M_g. An analogous construction works for d=4, however, for d=5 a general curve of genus 14 and degree 18 in P^6 is not contained in V_5, observed in a work of Verra on the unirationality of M_14, and also pointed out recently by Ciliberto-Flamini-Knutsen. We characterize geometric conditions when does such a curve can be embedded into V_5 using the vanishing resonance. This is a joint work with Marian Aprodu.

Title: Positivity of the (co)tangent bundle of smooth projective varieties

Abstract: The positivity of (co)tangent bundle has profound implications for its geometry, as various conditions like ampleness, bigness, semi-ampleness, nefness, or pseudo-effectivity restrict its structure. In this surveying talk, I will discuss some geometric restrictions and applications on smooth projective manifolds due to the positivity of the (co)tangent bundle.

Title: Curious Chern symmetry of moduli space of stable bundles on curves

Abstract: The cohomology ring of the moduli space of bundles on curves has long been studied, leading to an almost complete understanding of its structure. This cohomology ring admits a refined structure called the Chern filtration. Numerical experiments suggest that this refinement exhibits a certain symmetry. In this talk, I will explain a conjectural symmetry of the Chern filtration of the moduli space of bundles on curves, motivated by the curious Hard Lefschetz symmetry of character varieties. This is a joint work with M. Moreira and W. Pi.