Title: Fourier-Mukai transformation and logarithmic Higgs bundles on punctual Hilbert schemes.

Abstract: Given a vector bundle $E$ on a smooth projective curve or surface $X$ carrying the structure of a $V$-twisted Hitchin pair for some vector bundle $V$, we observe that the associated tautological bundle $E^{[n]}$ on the punctual Hilbert scheme of points $X^{[n]}$ has an induced structure of a $((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a vector bundle on $X^{[n]}$ constructed using the dual $V^\vee$ of $V$. In particular, a Higgs bundle on $X$ induces a logarithmic Higgs bundle on the Hilbert scheme $X^{[n]}$. We then show that the known results on stability of tautological bundles and reconstruction from tautological bundles generalize to tautological Hitchin pairs.

Title: $\mathbb{A}^1$ connected components of ruled surfaces

Abstract: A conjecture of Morel states that the sheaf of $\mathbb{A}^1$-connected components of any space is $\mathbb{A}^1$-invariant. We will prove this result for birationally ruled surfaces over an algebraically closed field. This is joint work with Anand Sawant.

Title: Moduli of semiorthogonal decompositions in families

Abstract: In a joint work with Shinnosuke Okawa and Andrea Ricolfi we have constructed a moduli space of semiorthogonal decompositions, and described some of its geometric properties. I will introduce semiorthogonal decompositions, and explain how they behave in families of smooth projective varieties. As an application I will discuss how its geometry can be used to show how certain derived categories of smooth projective varieties are indecomposable by studying indecomposability in families. This is joint work with the previous co-authors and Francesco Bastianelli.

Title : Hurewicz map in unstable homotivic homotopy theory.

Abstract : In classical algebraic topology, we have two kinds of invariants : homotopy groups and homology groups. The first one is more complicated compared to the second one and Hurewicz map together with Hurewicz theorem gives a way to compare the homotopy groups with homology groups. In my talk we will see the relation between unstable $\mathbb{A}^1$ homotopy groups and $\mathbb{A}^1$-homology groups using the Hurewicz map in $\mathbb{A}^1$-homotopy theory. We will show that the Hurewicz map in degree 1, i.e the map from $\mathbb{A}^1$-fundamental group to the first $\mathbb{A}^1$-homology is surjective. In classical algebraic topology the Hurewicz map in degree 1 is just abelianization, therefore surjective. We will show unlike the classical setting the Hurewicz map in degree 1 in $\mathbb{A}^1$-homotopy theory is not abelianization in general. This is a joint work with Amit Hogadi.

Title: On a degeneration of moduli of bundles on a smooth curve.

Abstract: Given a flat degeneration of a smooth curve to a nodal curve, Gieseker and Nagaraj-Seshadri constructed a flat degeneration of moduli of vector bundles. The objects of the spacial fibre are certain vector bundles over semi-stable models of the nodal curve. We will present a computation of cohomology of the special fibre when the rank is 2. We will also prove a Torelli type theorem for a nodal curve using a stratification on the special fibre of the degeneration.

Title: DG monad and TT-Geometry

Abstract: The subject Tensor Triangular (TT-) Geometry, introduced by P. Balmer, studies tensor triangulated categories using Algebraic Geometry tools. In the process of developing etale morphism in TT-Geometry, P. Balmer used the Eilenberg-Moore (EM) construction. He raised a question on the existence of canonical triangulated structure on EM construction and also the existence of exact adjunction realizing an exact monad.

In joint work with Vivek M. Mallick, we noticed that the Eilenberg-Moore construction in the enriched setting can be used to partially answer the question of Balmer. In this talk, we will discuss these results and some examples coming from derived categories.

Title: Gamma conjecture III

Abstract: I will formulate a Gamma conjecture III that relates some distinguished semi-orthogonal decompositions of the derived category of coherent sheaves to the asymptotics of the solutions of the quantum connection - this is a joint work with Hiroshi Iritani heavily based on our earlier work with Vasily Golyshev and Sanda-Shamoto’s “Dubrovin-style conjecture”. Then I will show some examples and corollaries.

Title: Incompressibility for some covers of moduli spaces

Abstract: Recently Farb, Kisin and Wolfson (FKW) have proved the incompressibility of congruence covers for a large collection of Shimura varieties and their subvarieties using mixed characteristic methods. In this talk, based on joint work with Patrick Brosnan, I will explain a different criterion for incompressibility which allows us to recover some of the results of FKW and is also applicable in some cases where their method does not apply, e.g., certain "quantum" covers of the moduli space of curves.

Title:The Hodge conjecture for moduli spaces of stable sheaves over a nodal curve

Abstract: The Hodge conjecture is known for the Jacobian variety of a general, smooth, projective curve. Balaji-King-Newstead used this to prove the conjecture for the moduli space of rank 2 stable sheaves with fixed odd degree determinant over a general, smooth, projective curve of genus g ≥ 2. In this talk I will discuss an analogous result when the underlying curve is general, irreducible nodal and show why techniques from the smooth case fail. This is joint work with A. Dan.

Title: Equivariant tt-Chow groups

Abstract: In this talk, based on joint work with Dr Umesh Dubey, we study some properties of Chow groups of tensor triangulated categories, as defined by Balmer, and studied by Klein, especially the relationship between the Chow group of a tensor triangulated category and the Chow group of the G-equivariant tensor triangulated category for a finite group G. On the way, prove some similar results for K_0. The talk will end with some applications.

Title: Projective bundles and blow-ups of Projective spaces.

Abstract: In this talk we discuss about blow ups of Projecive spaces which admits projective bundle structure.

Title: Arc spaces and the topology of (some) singular algebraic varieties.

Abstract: Arc spaces of algebraic varieties were introduced by Nash in a paper written in 1968 and published in 1995. They were studied by algebraic geometers in the late 1990s and early 2000s, but more recently they have become of interest in representation theory and automorphic forms. I will discuss some facts about arc spaces and give an application to some singular varieties of interest in representation theory.

Title: Exceptional collections on the Hirzebruch surface of degree 2

Abstract: The classification of exceptional collections of the derived category of coherent sheaves on del Pezzo surfaces is understood well by a work by Kuleshov and Orlov in 1994. The problem becomes much more difficult on weak del Pezzo surfaces, since exceptional objects are not necessarily sheaves any more due to the extra symmetry of the category called spherical twists. I will briefly review the work of Kuleshov-Orlov and then explain how one can manage to generalize their results to the Hirzebruch surface of degree 2, the simplest weak del Pezzo surface. This is a joint work with Akira Ishii and Hokuto Uehara.

Title: Calabi-Yau structures

Abstract: Calabi-Yau structures on categories arise naturally in the study of topological field theories, and there is a sense in which they can be thought of as noncommutative/categorical analogues of symplectic structures. A relative Calabi-Yau structure on a functor is the noncommutative analogue of a Lagrangian structure in symplectic geometry. I will introduce and motivate these structures, and relate them to the theory of spherical functors. Using these ideas, I will describe a local-to-global principle for constructing Calabi-Yau structures on certain Fukaya-type categories arising in symplectic geometry. This talk is based on joint work with Ludmil Katzarkov and Ted Spaide.

Title: Tannakian Categories in AG

Abstract: Tannakian categories are modeled the category of finite dimensional representations of a group. These occur in at least two ways in Algebraic Geometry---while studying vector bundles over a variety and associated principle bundles and while studying motives. We will introduce the basic notions of Abelian and Tannakian categories and exhibit various examples and counter-examples.

Title: Introduction to triangular Grothendieck-Witt groups, after Balmer.

Abstract: This talk will give an exposition of Balmer's work on Witt groups and Grothendieck-Witt groups for triangulated categories which are equipped with a good notion of duality.