Nitin Nitsure

Fundamental algebraic geometry : Grothendieck's FGA explained

Grothendieck topologies, fibered categories and descent theory: Introduction Preliminary notions Contravariant functors Fibered categories Stacks Construction of Hilbert and Quot schemes: Construction of Hilbert and Quot schemes Local properties and Hilbert schemes of points: Introduction Elementary deformation theory Hilbert schemes of points Grothendiecks existence theorem in formal geometry with a letter of Jean-Pierre Serre: Grothendiecks existence theorem in formal geometry The Picard scheme: The Picard scheme Bibliography Index.

Cohomology of the moduli of parabolic vector bundles

The purpose of this paper is to compute the Betti numbers of the moduli space ofparabolic vector bundles on a curve (see Seshadri [7], [8] and Mehta & Seshadri [4]), in the case where every semi-stable parabolic bundle is necessarily stable. We do this by generalizing the method of Atiyah and Bott [1] in the case of moduli of ordinary vector bundles. Recall that (see Seshadri [7]) the underlying topological space of the moduli of parabolic vector bundles is the space of equivalence classes of certain unitary representations of a discrete subgroup Γ which is a lattice in PSL (2,R). (The lattice Γ need not necessarily be co-compact).While the structure of the proof is essentially the same as that of Atiyah and Bott, there are some difficulties of a technical nature in the parabolic case. For instance the Harder-Narasimhan stratification has to be further refined in order to get the connected strata. These connected strata turn out to have different codimensions even when they are part of the same Harder-Narasimhan strata.If in addition to ‘stable = semistable’ the rank and degree are coprime, then the moduli space turns out to be torsion-free in its cohomology.The arrangement of the paper is as follows. In § 1 we prove the necessary basic results about algebraic families of parabolic bundles. These are generalizations of the corresponding results proved by Shatz [9]. Following this, in § 2 we generalize the analytical part of the argument of Atiyah and Bott (§ 14 of [1]). Finally in § 3 we show how to obtain an inductive formula for the Betti numbers of the moduli space. We illustrate our method by computing explicitly the Betti numbers in the special case of rank = 2, and one parabolic point.

TOPOLOGY OF QUADRIC BUNDLES

This paper begins with a description of cohomological invariants of non-degenerate quadric bundles, in terms of the cohomology rings of the classifying spaces of the general orthogonal groups. Following this, the Main Theorem of the paper determines the behavior of these invariants under the Gysin boundary map, when a quadric bundle degenerates over a divisor.

MODULI OF REGULAR HOLONOMIC D-MODULES WITH NORMAL CROSSING SINGULARITIES

This paper solves the global moduli problem for regular holonomic Dmodules with normal crossing singularities on a nonsingular complex projective variety. This is done by introducing a level structure (which gives rise to “pre-D-modules”), and then introducing a notion of (semi-)stability and applying Geometric Invariant Theory to construct a coarse moduli scheme for semistable pre-D-modules. A moduli is constructed also for the corresponding perverse sheaves, and the Riemann-Hilbert correspondence is represented by an analytic morphism between these moduli spaces.

Quasi-parabolic Siegel formula

The result of Siegel that the Tamagawa number ofSLr over a function field is 1 has an expression purely in terms of vector bundles on a curve, which is known as the Siegel formula. We prove an analogous formula for vector bundles with quasi-parabolic structures. This formula can be used to calculate the Betti numbers of the moduli of parabolic vector bundles using the Weil conjuctures

Granular self-organization by autotuning of friction

Significance Self-organization is ubiquitous in nature, although a complete understanding of the phenomena in specific cases is rare. Here we elucidate a route to self-organization in a model granular system. The local rules of motion are extracted from the experiment. When converted into an algorithm, they simulate the main aspects of the experimental results. From this, a key ingredient for achieving robustness emerges, namely, a continuously variable relative fraction of time the objects spend in two distinct motional degrees of freedom, rolling and sliding. In so doing, they access a large range of effective friction coefficients that allows self-tuning of the system to adjust its response to changing environments and guarantees a protocol-insensitive unique final state, a previously unidentified paradigm for self-organization. A monolayer of granular spheres in a cylindrical vial, driven continuously by an orbital shaker and subjected to a symmetric confining centrifugal potential, self-organizes to form a distinctively asymmetric structure which occupies only the rear half-space. It is marked by a sharp leading edge at the potential minimum and a curved rear. The area of the structure obeys a power-law scaling with the number of spheres. Imaging shows that the regulation of motion of individual spheres occurs via toggling between two types of motion, namely, rolling and sliding. A low density of weakly frictional rollers congregates near the sharp leading edge whereas a denser rear comprises highly frictional sliders. Experiments further suggest that because the rolling and sliding friction coefficients differ substantially, the spheres acquire a local time-averaged coefficient of friction within a large range of intermediate values in the system. The various sets of spatial and temporal configurations of the rollers and sliders constitute the internal states of the system. Experiments demonstrate and simulations confirm that the global features of the structure are maintained robustly by autotuning of friction through these internal states, providing a previously unidentified route to self-organization of a many-body system.

SCHEMATIC HARDER–NARASIMHAN STRATIFICATION

For any flat family of pure-dimensional coherent sheaves on a family of projective schemes, the Harder-Narasimhan type (in the sense of Gieseker semistability) of its restriction to each fiber is known to vary semicontinuously on the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder-Narasimhan stratification. In this note, we show how to endow each Harder-Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback family admits a relative Harder-Narasimhan filtration with a given Harder-Narasimhan type if and only if the base change factors through the schematic stratum corresponding to that Harder-Narasimhan type. The above schematic stratification induces a stacky stratification on the algebraic stack of pure-dimensional coherent sheaves. We deduce that coherent sheaves of a fixed Harder-Narasimhan type form an algebraic stack in the sense of Artin.

On the geometric phenomenology of static friction

In this note we introduce a hierarchy of phase spaces for static friction, which give a graphical way to systematically quantify the directional dependence in static friction via subregions of the phase spaces. We experimentally plot these subregions to obtain phenomenological descriptions for static friction in various examples where the macroscopic shape of the object affects the frictional response. The phase spaces have the universal property that for any experiment in which a given object is put on a substrate fashioned from a chosen material with a specified nature of contact, the frictional behaviour can be read off from a uniquely determined classifying map on the control space of the experiment which takes values in the appropriate phase space.

Representability ofGL E

We prove a necessary and sufficient condition for the automorphisms of a coherent sheaf to be representable by a group scheme.

A geometric framework for dynamics with unilateral constraints and friction, illustrated by an example of self-organized locomotion

We present a geometric framework to deal with mechanical systems which have unilateral constraints, and are subject to damping/friction, which cannot be treated within usual classical mechanics. In this new framework, the dynamical evolution of the system takes place on a multidimensional curvilinear polyhedron, and energetics near the corners of the polyhedron leads to qualitative behaviour such as stable entrapment and bifurcation. We illustrate this by an experiment in which dumbbells, placed inside a tilted hollow cylindrical drum that rotates slowly around its axis, climb uphill by forming dynamically stable pairs, seemingly against the pull of gravity.