This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups It aims to provide the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields Involutions are viewed as twisted forms of similarity classes of hermitian or bilinear forms, leading to new developments on the model of the algebraic theory of quadratic forms Besides classical groups, phenomena related to triality are also discussed, as well as groups of type F_4 or G_2 arising from exceptional Jordan or composition algebras Several results and notions appear here for the first time, notably the discriminant algebra of an algebra with unitary involution and the algebra-theoretic counterpart to linear groups of type D_4 For research mathematicians and graduate students working in central simple algebras, algebraic groups, nonabelian Galois cohomology or Jordan algebras

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The Book of Involutions

A groupoid should be thought of as a group with many objects, or with many identities. A precise definition is given below. A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, flexibility and range of applications, so that even for purely group-theoretical work, it can be useful to take a path through the world of groupoids. A succinct definition is that a groupoid G is a small category in which every morphism is an isomorphism. Thus G has a set of morphisms, which we shall call just elements of G, a set Ob(G) of objects or vertices, together with functions s, t : G→ Ob(G), i : Ob(G) → G such that si = ti = 1. The functions s, t are sometimes called the source and target maps respectively. If a,b ∈ G and ta = sb, then a product or composite ab exists such that s(ab) = sa, t(ab) = tb. Further, this product is associative; the elements ix, x ∈ Ob(J), act as identities; and each element a has an inverse a with s(a) = ta, t(a) = sa,aa = isa,aaa = ita. An element a is often written as an arrow a : sa→ ta.

http://groupoids.org.uk/pdffiles/groupoidsurvey.pdf

From Groups to Groupoids: a Brief Survey

We prove that the general quartic double solid with $k\le 7$ nodes does not admit a Chow theoretic decomposition of the diagonal, (or equivalently has a nontrivial universal $\mathrm{CH}_0$ group,) and the same holds if we replace in this statement “Chow theoretic” by “cohomological”. In particular, it is not stably rational. We also deduce that the general quartic double solid with seven nodes does not admit a universal codimension $2$ cycle parameterized by its intermediate Jacobian, and even does not admit a parametrization with rationally connected fibers of its Jacobian by a family of $1$-cycles. This finally implies that its third unramified cohomology group is not universally trivial.

/pdf/unirational-threefolds-with-no-universal-codimension-2-cycle-3ei6jm6l9m.pdf

Unirational threefolds with no universal codimension $2$ cycle

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Revetements Etales Et Groupe Fondamental Sga 1

We introduce a ``geometric'' method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the level aspect over number fields, which implies subconvexity for certain standard and Rankin-Selberg $L$-functions; and to bounding Fourier coefficients of automorphic forms.

Sparse equidistribution problems, period bounds, and subconvexity

Galois cohomology of the classical groups over fields of cohomological dimension≦2

https://www.cmi.ac.in/~ypandey/R.Sridharan/01.pdf

Hermitian Analogue of a Theorem of Springer

Real components of algebraic varieties and étale cohomology

Classical groups and the Hasse principle

The u-invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field. It is an open question whether the u-invariant of function fields of p-aidc curves is 8. In this paper, we answer this question in the affirmative for function fields of nondyadic p-adic curves.

/pdf/the-u-invariant-of-the-function-fields-of-p-adic-curves-1ycjhce4ga.pdf

Raman Parimala

Papers

Galois cohomology of the classical groups over fields of cohomological dimension≦2

Hermitian Analogue of a Theorem of Springer

Real components of algebraic varieties and étale cohomology

Classical groups and the Hasse principle

The u-invariant of the function fields of p-adic curves