Gopal Prasad
Gopal Prasad  

Born 
Ghazipur, British India 
31 July 1945
Alma mater 
Patna University IITK TIFR Institute for Advanced Study 
Awards 
Indian Academy of Sciences (1984), 
Scientific career  
Fields  Mathematics 
Institutions  University of Michigan 
Doctoral advisor  M. S. Raghunathan 
Gopal Prasad (born 31 July 1945 in Ghazipur, India) is an IndianAmerican mathematician. His research interests span the fields of Lie groups, their discrete subgroups, algebraic groups, arithmetic groups, geometry of locally symmetric spaces, and representation theory of reductive padic groups.
He is the Raoul Bott Professor of Mathematics^{[1]} at the University of Michigan in Ann Arbor.
Contents
Education
Prasad earned his bachelor's degree with honors in Mathematics from Magadh University in 1963. Two years later, in 1965, he received his masters in Mathematics from Patna University. After a brief stay at the Indian Institute of Technology Kanpur in their Ph.D. program for Mathematics, Prasad entered the Ph.D. program at the Tata Institute of Fundamental Research (TIFR) in 1966. There he began a long and extensive collaboration with his advisor M. S. Raghunathan on several topics including the study of lattices in semisimple Lie groups. In 1976, Prasad received his Ph.D. from the University of Mumbai. Prasad became an Associate Professor at TIFR in 1979, and a Professor in 1984. In 1992 he left TIFR to join the faculty at the University of Michigan in Ann Arbor, where he is currently the Raoul Bott Professor of Mathematics.
Family
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In 1969, he married Indu Devi of Deoria. Gopal Prasad and Indu Devi have a son and a daughter and five grandchildren. Shrawan Kumar, a professor of mathematics at the University of North Carolina at Chapel Hill, and Dipendra Prasad, a professor of mathematics at the Tata Institute of Fundamental Research, are his younger brothers.
Some contributions to mathematics
Prasad's early work was on discrete subgroups of real and padic semisimple groups. He proved the "strong rigidity" of lattices in real semisimple groups of rank 1 and also of lattices in padic groups, see [1] and [2]. He then tackled grouptheoretic and arithmetic questions on semisimple algebraic groups. He proved the "strong approximation" property for simply connected semisimple groups over global function fields [3]. In collaboration with M. S. Raghunathan, Prasad determined the topological central extensions of these groups, and computed the "metaplectic kernel" for isotropic groups, see [11], [12] and [10]. Later, together with Andrei Rapinchuk, Prasad gave a precise computation of the metaplectic kernel for all simply connected semisimple groups, see [14]. Prasad and Raghunathan have also obtained results on the KneserTits problem, [13].
In 1987, Prasad found a formula for the volume of Sarithmetic quotients of semisimple groups, [4]. Using this formula and certain number theoretic and Galoiscohomological estimates, Armand Borel and Gopal Prasad proved several finiteness theorems about arithmetic groups, [6]. The volume formula, together with numbertheoretic and BruhatTits theoretic considerations led to a classification, by Gopal Prasad and SaiKee Yeung, of fake projective planes (in the theory of smooth projective complex surfaces) into 28 nonempty 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 padic groups with Allen Moy. The filtrations of parahoric subgroups, referred to as the "MoyPrasad filtration", is widely used in representation theory and harmonic analysis. Moy and Prasad used these filtrations and BruhatTits theory to prove the existence of "unrefined minimal Ktypes", 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 Zariskidense subgroups of semisimple groups and proved the existence in such a subgroup of regular semisimple 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 "weakcommensurability" of arithmetic subgroups and determined "weak commensurability classes" of arithmetic groups in a given semisimple group. They used their results on weakcommensurability to obtain results on lengthcommensurable and isospectral arithmetic locally symmetric spaces, see [17], [18] and [19].
Together with JiuKang Yu, Prasad has studied the fixed point set under the action of a finite group of automorphisms of a reductive padic group G on the BruhatBuilding of G, [24]. In another joint work, Prasad and Yu determined all the quasireductive group schemes over a discrete valuation ring (DVR), [25].
In collaboration with Brian Conrad and Ofer Gabber, Prasad has studied the structure of pseudoreductive 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 pseudoreductive groups, including a Titsstyle classification and also many interesting examples. The classification of pseudoreductive groups already has many applications. There was a Bourbaki seminar in March 2010 on the work of Tits, ConradGabberPrasad on pseudoreductive groups.
Honors
Prasad has received the Guggenheim Fellowship, the Humboldt Senior Research Award, and the Raoul Bott Professorship at the University of Michigan. He was awarded the Shanti Swarup Bhatnagar prize (by the Council of Scientific and Industrial Research of the Government of India). He has received Fellowships in the Indian National Science Academy, the Indian Academy of Sciences and the American Mathematical Society. Prasad gave an invited talk in the International Congress of Mathematicians held in Kyoto in 1990. In 2012 he became a fellow of the American Mathematical Society.^{[2]}
Prasad was the Managing Editor of the Michigan Mathematical Journal for over a decade, an Associate Editor of the Annals of Mathematics for six years, and is an editor of the Asian Journal of Mathematics since its inception.
References
 ^ G. Prasad Archived 23 March 2010 at the Wayback Machine.
 ^ List of Fellows of the American Mathematical Society Archived 26 June 2015 at the Wayback Machine., retrieved 20130526.
[1]. Strong rigidity of Qrank 1 lattices, Inventiones Math. 21(1973), 255286.
[2]. Lattices in semisimple groups over local fields, Adv.in Math. Studies in Algebra and Number Theory, 1979, 285356.
[3]. Strong approximation for semisimple groups over function fields, Annals of Math. 105(1977), 553572.
[4]. Volumes of Sarithmetic quotients of semisimple groups, Publ.Math.IHES 69(1989), 91117.
[5]. Semisimple groups and arithmetic subgroups, Proc.Int.Congress of Math., Kyoto, 1990, Vol. II, 821832.
[6]. Finiteness theorems for discrete subgroups of bounded covolume in semisimple groups, Publ.Math.IHES 69(1989), 119171; Addendum: ibid, 71(1990); with A.Borel.
[7]. Values of isotropic quadratic forms at Sintegral points, Compositio Mathematica, 83 (1992), 347372; with A.Borel.
[8]. Unrefined minimal Ktypes for padic groups, Inventiones Math. 116(1994), 393408; with Allen Moy.
[9]. Jacquet functors and unrefined minimal Ktypes, Commentarii Math.Helv. 71(1996), 98121; with Allen Moy.
[10]. On the congruence subgroup problem: Determination of the "Metaplectic Kernel", Inventiones Math. 71(1983), 2142; with M.S.Raghunathan.
[11]. Topological central extensions of semisimple groups over local fields, Annals of Math. 119(1984), 143268; with M.S.Raghunathan.
[12]. Topological central extensions of SL_1(D), Inventiones Math. 92(1988), 645689; with M.S.Raghunathan.
[13]. On the KneserTits problem, Commentarii Math.Helv. 60(1985), 107121; with M.S.Raghunathan.
[14]. Computation of the metaplectic kernel, Publ.Math.IHES 84(1996), 91187; with A.S.Rapinchuk.
[15]. Existence of irreducible Rregular elements in Zariskidense subgroups, Math.Res.Letters 10(2003), 2132; with A.S.Rapinchuk.
[16]. Zariskidense subgroups and transcendental number theory, Math.Res.Letters 12(2005), 239249; with A.S.Rapinchuk.
[17]. Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ.Math.IHES 109(2009), 113184; with A.S.Rapinchuk.
[18]. Localglobal principles for embedding of fields with involution into simple algebras with involution, Commentarii Math.Helv. 85(2010), 583645; with A.S.Rapinchuk.
[19]. On the fields generated by the lengths of closed geodesics in locally symmetric spaces, preprint; with A.S.Rapinchuk.
[20]. Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre, In "Collected papers of John Milnor", vol.V, AMS (2010), 307325; with A.S.Rapinchuk.
[21]. Fake projective planes, Inventiones Math. 168(2007), 321370, "Addendum", ibid, 182(2010), 213227; with SaiKee Yeung.
[22]. Arithmetic fake projective spaces and arithmetic fake Grassmannians, Amer.J.Math. 131(2009), 379407; with SaiKee Yeung.
[23]. Nonexistence of arithmetic fake compact hermitian symmetric spaces of type other than A_n, n<5, J.Math.Soc.Japan; with SaiKee Yeung.
[24]. On finite group actions on reductive groups and buildings, Inventiones Math. 147(2002), 545560; with JiuKang Yu.
[25]. On quasireductive group schemes, J.Alg.Geom. 15(2006), 507549; with JiuKang Yu.
[26]. Pseudoreductive groups, second edition, New Mathematical Monographs #26, xxiv+665 pages, Cambridge University Press, 2015; with Brian Conrad and Ofer Gabber.
[27]. Classification of Pseudoreductive groups, Annals of Mathematics Studies #191, 245 pages, Princeton University Press, 2015; with Brian Conrad.
External links
 Gopal Prasad at the Mathematics Genealogy Project
 Recipients of the Shanti Swarup Bhatnagar Award in Mathematical Science
 1945 births
 Living people
 20thcentury Indian mathematicians
 Magadh University alumni
 Patna University alumni
 University of Mumbai alumni
 University of Michigan faculty
 People from Ghazipur
 Tata Institute of Fundamental Research alumni
 Fellows of the American Mathematical Society
 Guggenheim Fellows
 Scientists from Uttar Pradesh