Title: On stratified vector bundles in characteristic p.

Abstract: This is a report on some work with Helene Esnault, motivated by a

conjecture of Gieseker, which was proved earlier by Esnault and Mehta.

For a smooth quasi-projective variety $X$ over $\bar{\F}_p$, with trivial

etale $\pi_1$, such that $X$ has a projective normal compactification with

codimension 2 boundary, we show that all stratified vector bundles on $X$

are trivial.

Another result of ours is the following: if a morphism of smooth

projective varieties in char. p induces the trivial map

on étale fundamental groups, then the pullback of any stratified vector

bundle is trivial, as a stratified bundle.

Title: Curvature, torsion and the quadrilateral gaps.

Abstract: If we travel on a flat plane by going straight for a distance s, then turn left in 90 degrees and go straight for the same distance s, and then repeat the turn and the straight travel twice more, then we complete a square and come back where we started. But if we do this journey on a curved surface by following geodesics, we will not come back exactly: the quadrilateral will not close but will have a gap between the starting point and the end point. Rajaram Nityananda asked for a quantification of this gap when s is small, and its relationship with curvature. We will answer this question, and prove a general higher dimensional theorem that characterizes the torsion and curvature of an affine connection in terms of such quadrilateral gaps.

Reference: arXiv:1910.06615

Title: Some characteristic p invariants

Abstract: Let R be a graded Noetherian ring with the graded maximal

ideal m and a graded ideal I of finite colength. Here we discuss some

char p invariants, namely the Hilbert-Kunz multiplicity $e^{HK}(R, I)$, the

Hilbert-Kunz density function $f_{R,I}$ and the F-thresholds $c^I(m)$.

As an application of a technique (taxi-cab distance) of Monsky for HK

multiplicity, we show that, for the set of syzygy vector bundles ${V_n}_{n∈\bN}

associated to any plane trinomial curve $k[x, y, z]/(h)$, the Frobenius

semistability behaviour of the reduction mod p of $V_n$ is a function of

the congruence class of p modulo $2λ_h$ (an integer invariant associated

to h).

On the other hand, computing related HK density functions for some

examples due to D. Gieseker (of vector bundles on projective curves),

we show that there is a two dimensional ring R where the set of F-

thresholds of m has limit points. This answers a question of Mustat ̧ ̆a-

Takagi-Watanabe (2005).

Title: Classification of obstructed bundles and Mestrano-Simpson Conjecture.

Abstract: Let S be a very general sextic surface over complex numbers. Let M(H, c2) be the moduli space of rank 2 stable bundles on S with fixed first Chern class H and second Chern class c2. In this talk, we will classify the obstructed bundles in M(H,c2) and apply this to give partial proof Mestrano -Simpson conjecture on the number of irreducible components of M(H, 11). We will also show that M(H, c2) is irreducible for c2 ≤ 10.

Title: Geometry of Witt vectors as per Borger

Abstract: This is a survey talk based on two papers of J. Borger

on geometry of Witt vectors.

Title: Vector bundles on curves and surfaces.

Abstract: In these talks we will present a survey of some of the

known results about vector bundles on curves and surfaces and the problem

of their classification.

These talks are meant for the students who are interested in

Algebraic Geometry.

Title: Vector bundles on curves and surfaces.

Abstract: In these talks we will present a survey of some of the

known results about vector bundles on curves and surfaces and the problem

of their classification.

These talks are meant for the students who are interested in

Algebraic Geometry.

Title: Vector bundles over complex projective hypersurfaces.

Abstract: A survey talk where I will discuss some results related to existence (or

rather non-existence) of indecomposable low-rank vector bundles over complex projective

space followed by similar questions for hypersurfaces. The second half will be devoted to

a generic version of the BGS conjecture for ACM bundles on hypersurfaces and a recent

joint work with Girivaru Ravindra.

Title: Cycles, K-theory and differential forms

Abstract: I shall show that the Zariski sheaves of differential

forms and certain K-groups on smooth schemes over a field are

motivic, i.e., can be described in terms of algebraic cycles.

I will also give some applications. This is based on a joint work

with Jinhyun Park.

Title: Etale motivic cohomology

Abstract: This is a survey talk which covers basic properties of

etale motivic cohomology and various applications.

Title: On stratified vector bundles in characteristic p.

Abstract: This is a report on some work with Helene Esnault, motivated by a

conjecture of Gieseker, which was proved earlier by Esnault and Mehta.

For a smooth quasi-projective variety $X$ over $\bar{\F}_p$, with trivial

etale $\pi_1$, such that $X$ has a projective normal compactification with

codimension 2 boundary, we show that all stratified vector bundles on $X$

are trivial.

Another result of ours is the following: if a morphism of smooth

projective varieties in char. p induces the trivial map

on étale fundamental groups, then the pullback of any stratified vector

bundle is trivial, as a stratified bundle.

Title: Some characteristic p invariants

Abstract: Let R be a graded Noetherian ring with the graded maximal

ideal m and a graded ideal I of finite colength. Here we discuss some

char p invariants, namely the Hilbert-Kunz multiplicity $e^{HK}(R, I)$, the

Hilbert-Kunz density function $f_{R,I}$ and the F-thresholds $c^I(m)$.

As an application of a technique (taxi-cab distance) of Monsky for HK

multiplicity, we show that, for the set of syzygy vector bundles ${V_n}_{n∈\bN}

associated to any plane trinomial curve $k[x, y, z]/(h)$, the Frobenius

semistability behaviour of the reduction mod p of $V_n$ is a function of

the congruence class of p modulo $2λ_h$ (an integer invariant associated

to h).

On the other hand, computing related HK density functions for some

examples due to D. Gieseker (of vector bundles on projective curves),

we show that there is a two dimensional ring R where the set of F-

thresholds of m has limit points. This answers a question of Mustat ̧ ̆a-

Takagi-Watanabe (2005).

Title: Graph potentials and Moduli spaces of vector bundles of curves

Abstract: We construct and study graph potentials, a collection of Laurent polynomials

associated with colored graphs of small valency. The

potentials we construct are related to the moduli spaces of vector bundles of rank

two with fixed determinant on algebraic curves. We will discuss relations between these graph potentials and Gromov-Witten invariants of the moduli spaces. This is joint work with P. Belmans and S. Galkin.

Title: Algebraic Cycles and Modular Forms

Abstract: There are many instances when special sub-varieties of Shimura varieties give rise to modular forms. One such is the theo- rem of Gross and Zagier linking Heegner divisors with coefficients of modular forms. We discuss a generalisation of this theorem to higher codimensional cycles and an approach to proving it along the lines of Borcherds’ proof of the Gross-Zagier theorem. As an application we obtain algebraicity results for values of higher Greens functions.

Title: Ulrich Bundles on Double Planes.

Abstract: Consider a smooth complex surface X which is a double cover of the projective plane $P^2 $ branched along a smooth curve of degree $ 2s $. .

In this article, we study the geometric conditions which are equivalent to the existence of Ulrich line bundles on X with respect to this double covering.

Also, for every s≥1, we describe the classes of such surfaces which admit Ulrich line bundles and give examples.

Title: K3 surfaces over finite fields

Abstract: The Tate Conjecture for K3 surfaces over finite fields was proved a few years ago by Maulik, Charles, and Madapusi-Pera. Subsequently, Charles gave a more geometric "second generation" proof of the conjecture. In this talk, we shall discuss some finiteness results leading up to it, and, time permitting, the proof itself.

Title: Primitive extensions of local fields.

Abstract: A finite separable extension of a field is called primitive if there are no intermediate extensions. We give a natural parametrisation of the set of primitive extensions of local fields with finite residue field.

Title: On Double Danielewski Surfaces: A counter-example to the Cancellation Problem

Abstract: A version of the famous Cancellation Problem asks:

If $A$ and $B$ are two finitely generated algebras over a field $k$ such that the polynomial rings $A[X]$ and $B[X]$ are $k$-isomorphic, does this necessarily imply that $A$ and $B$ are $k$-isomorphic?

We present a two-dimensional family of affine surfaces which provide new counter-examples to the Cancellation Problem. We describe a certain invariant of $\bG_a$-action of these surfaces called Makar-Limanov invariant. Using the Makar-Limanov invariant we describe the isomorphism classes of these surfaces and a characterization of their automorphisms. We also determine all the locally nilpotent derivations of a double Danielewski surface.

This talk is based on joint works with Neena Gupta.

Title: Functorial moduli construction of parabolic sheaves

Abstract: The functorial construction of moduli spaces of sheaves on projective varieties was given by A ^́lvarez-C ́onsul and A. King using the moduli of Kronecker quiver representations. The aim of this talk is to give similar construction for moduli of parabolic sheaves using the moduli of filtered Kronecker representations.

This is based on joint work in progress with Sanjay Amrutiya.

Title: On the relative holomorphic connections

Abstract: We investigate relative connections on a sheaf of modules. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic vector bundle over a complex analytic family. We show that the relative Chern classes of a holomorphic vector bundle admitting relative holomorphic connection vanish, if each of the fiber of the complex analytic family is compact and Khler. This is based on joint work with Prof. I. Biswas.