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Later, together with Andrei Rapinchuk, Prasad gave a precise computation of the metaplectic kernel for all simply connected semi-simple groups, see [14]. Prasad and Raghunathan have also obtained results on the Kneser-Tits problem, [13]. In , Prasad found a formula for the volume of S-arithmetic quotients of semi-simple groups, [4]. Using this formula and certain number theoretic and Galois-cohomological estimates, Armand Borel and Gopal Prasad proved several finiteness theorems about arithmetic groups, [6].

The volume formula, together with number-theoretic and Bruhat-Tits theoretic considerations led to a classification, by Gopal Prasad and Sai-Kee Yeung, of fake projective planes in the theory of smooth projective complex surfaces into 28 non-empty classes [21] see also [22] and [23]. This classification, together with computations by Donald Cartwright and Tim Steger, has led to a complete list of fake projective planes.

This list consists of exactly 50 fake projective planes, up to isometry distributed among the 28 classes. This work was the subject of a talk in the Bourbaki seminar. Prasad has worked on the representation theory of reductive p-adic groups with Allen Moy. The filtrations of parahoric subgroups, referred to as the "Moy-Prasad filtration", is widely used in representation theory and harmonic analysis.

Moy and Prasad used these filtrations and Bruhat-Tits theory to prove the existence of "unrefined minimal K-types", to define the notion of "depth" of an irreducible admissible representation and to give a classification of representations of depth zero, see [8] and [9]. In collaboration with Andrei Rapinchuk, Prasad has studied Zariski-dense subgroups of semi-simple groups and proved the existence in such a subgroup of regular semi-simple elements with many desirable properties, [15], [16].

These elements have been used in the investigation of geometric and ergodic theoretic questions. Prasad and Rapinchuk introduced a new notion of "weak-commensurability" of arithmetic subgroups and determined "weak- commensurability classes" of arithmetic groups in a given semi-simple group. They used their results on weak-commensurability to obtain results on length-commensurable and isospectral arithmetic locally symmetric spaces, see [17], [18] and [19].

Together with Jiu-Kang Yu, Prasad has studied the fixed point set under the action of a finite group of automorphisms of a reductive p-adic group G on the Bruhat-Building of G, [24]. In another joint work, Prasad and Yu determined all the quasi-reductive group schemes over a discrete valuation ring DVR , [25]. In collaboration with Brian Conrad and Ofer Gabber, Prasad has studied the structure of pseudo-reductive groups, and also provided proofs of the conjugacy theorems for general smooth connected linear algebraic groups, announced without detailed proofs by Armand Borel and Jacques Tits; their research monograph [26] contains all this.

The monograph [27] contains a complete classification of pseudo-reductive groups, including a Tits-style classification and also many interesting examples.

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The classification of pseudo-reductive groups already has many applications. In this second edition there is new material on relative root systems and Tits systems for general smooth affine groups, including the extension to quasi-reductive groups of famous simplicity results of Tits in the semisimple case. Chapter 9 has been completely rewritten to describe and classify pseudo-split absolutely pseudo-simple groups with a non-reduced root system over arbitrary fields of characteristic 2 via the useful new notion of 'minimal type' for pseudo-reductive groups.

Researchers and graduate students working in related areas, such as algebraic geometry, algebraic group theory, or number theory will value this book, as it develops tools likely to be used in tackling other problems. Root groups and root systems. Basic structure theory.

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- Pseudo-Reductive Groups (New Mathematical Monographs);
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- Classification of Pseudo-reductive Groups (AM-191).

Variation of G kkT C. Ubiquity of the standard construction.

Classification results. The exotic constructions. A Primer for Undergraduate Research. Aaron Wootton. Catalan's Conjecture. Sheaves and Functions Modulo p. Lenny Taelman. Topological Galois Theory.

## REDUCTIVE SUBGROUP SCHEMES OF A PARAHORIC GROUP SCHEME

Askold Khovanskii. John Coates. Jim Cogdell. Algebra 1.

## Pseudo-reductive Groups (New Mathematical Monographs) - [PDF Document]

David A. New Directions in Locally Compact Groups. Pierre-Emmanuel Caprace. Tullio Ceccherini-Silberstein. Automorphic Forms. Bernhard Heim. Ian Chiswell. Drinfeld Moduli Schemes and Automorphic Forms. Yuval Z Flicker. Yuval Z. John N. Bangming Deng. Multiplicative Ideal Theory and Factorization Theory. Scott Chapman. Basic Modern Algebra with Applications. Mahima Ranjan Adhikari. Teo Mora. Jay Jorgenson. Iwasawa Theory Otmar Venjakob. Hyperbolic Manifolds. Albert Marden.

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The Theory of Nilpotent Groups. Marcos Zyman. Appalachian Set Theory. Professor James Cummings. Introduction to Random Graphs.