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{{Short description|Indian-American mathematician}}
{{BLP sources|date=April 2021}}
{{Use Indian English|date=August 2015}}
{{Use dmy dates|date=August 2015}}
{{Infobox scientist
| name = Gopal Prasad
| image = GPrasad.jpg
| image_size =
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| birth_date = {{Birth date and age|df=yes|1945|07|31}}
| birth_place = [[Ghazipur]], [[British India]]
| death_date = 
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| nationality =
| fields = [[Mathematics]]
| workplaces = [[University of Michigan]]
| alma_mater = [[Patna University]] <br> [[IITK]] <br> [[TIFR]] <br> [[Institute for Advanced Study]]
| doctoral_advisor = [[M. S. Raghunathan]]
| academic_advisors =
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| notable_students =
| known_for =
| influences =
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| awards = {{plainlist | style = white-space: nowrap; |
* [[Shanti Swarup Bhatnagar Award]] (1989)
* [[Guggenheim Fellowship]] (1998)
* [[Humboldt Research Award]] (2006)
* Fellow of:
*[[American Mathematical Society]] (2012),
*[[Indian Academy of Sciences]] (1984),
*[[Indian National Science Academy]] (1982)
}}
| signature = 
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}}

'''Gopal Prasad''' (born 31 July 1945 in [[Ghazipur]], [[India]]) is an Indian-American [[mathematician]]. His research interests span the fields of [[Lie group]]s, their [[discrete subgroup]]s, [[algebraic group]]s, [[arithmetic group]]s, geometry of [[symmetric space|locally symmetric spaces]], and representation theory of reductive [[P-adic Lie group|p-adic groups]].

He is the [[Raoul Bott]] Professor of Mathematics<ref>[http://www.math.lsa.umich.edu/~gprasad/ G. Prasad] {{webarchive|url=https://web.archive.org/web/20100323061532/http://www.math.lsa.umich.edu/~gprasad/ |date=23 March 2010 }}</ref> at the [[University of Michigan]] in [[Ann Arbor, Michigan|Ann Arbor]].

==Education==
{{BLP unsourced section|date=August 2021}}
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 semi-simple Lie groups and the congruence subgroup problem. 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==
{{BLP unsourced section|date=March 2018}}
Gopal Prasad's parents were Ram Krishna Prasad and Lakshmi Devi. Ram Krishna Prasad was a social worker, philantropist, and was jailed by the British for his participation in the Indian freedom struggle against British rule. The family was involved in reinsurance, retail, and wholesale businesses, which Gopal Prasad helped with in his childhood. In 1969, he married Indu Devi (nee Poddar) of [[Deoria, Uttar Pradesh|Deoria]]. Gopal Prasad and Indu Devi have a son, Anoop Prasad, who is Managing Director at D.E. Shaw & Co, and a daughter, [[Ila Fiete]], who is Professor of Neuroscience at MIT, and five grand-children. [[Shrawan Kumar]], Professor of Mathematics at the [[University of North Carolina at Chapel Hill]], [[Pawan Kumar (astrophysicist)|Pawan Kumar]], Professor of Astrophysics at the [[University of Texas, Austin]] and [[Dipendra Prasad]], 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 p-adic semi-simple groups. He proved the "[[strong rigidity]]" of lattices in real semi-simple groups of rank 1 and also of lattices in p-adic groups, see [1] and [2]. He then tackled group-theoretic and arithmetic questions on semi-simple algebraic groups. He proved the "[[strong approximation]]" property for simply connected semi-simple groups over global function fields [3]. Prasad determined the topological central extensions of these groups and computed the "metaplectic kernel" for isotropic groups in collaboration with [[M. S. Raghunathan]], see [11], [12] and [10]. Prasad and Raghunathan have also obtained results on the Kneser-Tits problem, [13]. Later, together with Andrei Rapinchuk, Prasad gave a precise computation of the metaplectic kernel for all simply connected semi-simple groups, see [14].

In 1987, 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 plane]]s (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 building|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]. The results and techniques introduced in these two papers [8],[9] enabled a series of important developments in the field.

In collaboration with Andrei Rapinchuk, Prasad has studied [[Zariski-dense| 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, that has been used in the geometric Langlands program, 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. A second monograph [27] contains a complete classification of pseudo-reductive groups, including a Tits-style classification and also many interesting examples. The classification of pseudo-reductive groups already has many applications. There was a Bourbaki seminar in March 2010 on the work of Tits, Conrad-Gabber-Prasad on pseudo-reductive groups. Prasad has developed new methods for unramified and tamely descents in Bruhat-Tits theory [28][29]. Together with Tasho Kaletha, he has recently written a book on Bruhat-Tits theory which contains new proofs of several results [30].

==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. 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]].<ref>[https://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society] {{webarchive|url=https://web.archive.org/web/20150626101726/http://www.ams.org/profession/fellows-list |date=26 June 2015 }}, retrieved 2013-05-26.</ref>
He has served on the Mathematical Sciences jury of the [[Infosys Prize]] from 2011 to 2018.

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==
{{reflist}}
[1]. ''Strong rigidity of '''Q'''-rank 1 lattices'', Inventiones Math. '''21'''(1973), 255-286.

[2]. ''Lattices in semi-simple groups over local fields'', Adv.in Math. Studies in Algebra and Number Theory, 1979, 285-356.

[3]. ''Strong approximation for semi-simple groups over function fields'', Annals of Mathematics '''105'''(1977), 553-572.

[4]. ''Volumes of S-arithmetic quotients of semi-simple groups'', Publ.Math.IHES '''69'''(1989), 91-117.

[5]. ''Semi-simple groups and arithmetic subgroups'', Proc.Int.Congress of Math., Kyoto, 1990, Vol. II, 821-832.

[6]. ''Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups'', Publ.Math.IHES '''69'''(1989), 119-171; Addendum: ibid, '''71'''(1990); with A.Borel.

[7]. ''Values of isotropic quadratic forms at S-integral points'', Compositio Mathematica, 83 (1992), 347-372; with A.Borel.

[8]. ''Unrefined minimal K-types for p-adic groups'', Inventiones Math. '''116'''(1994), 393-408; with Allen Moy.

[9]. ''Jacquet functors and unrefined minimal K-types'', Commentarii Math.Helv. '''71'''(1996), 98-121; with Allen Moy.

[10]. ''On the congruence subgroup problem: Determination of the "Metaplectic Kernel"'', Inventiones Math. '''71'''(1983), 21-42; with M.S.Raghunathan.

[11]. ''Topological central extensions of semi-simple groups over local fields'', Annals of Mathematics '''119'''(1984), 143-268; with M.S.Raghunathan.

[12]. ''Topological central extensions of SL_1(D)'', Inventiones Math. '''92'''(1988), 645-689; with M.S.Raghunathan.

[13]. ''On the Kneser-Tits problem'', Commentarii Math.Helv. '''60'''(1985), 107-121; with M.S.Raghunathan.

[14]. ''Computation of the metaplectic kernel'', Publ.Math.IHES '''84'''(1996), 91-187; with A.S.Rapinchuk.

[15]. ''Existence of irreducible '''R'''-regular elements in Zariski-dense subgroups'',
Math.Res.Letters '''10'''(2003), 21-32; with A.S.Rapinchuk.

[16]. ''Zariski-dense subgroups and transcendental number theory'', Math.Res.Letters '''12'''(2005), 239-249; with A.S.Rapinchuk.

[17]. ''Weakly commensurable arithmetic groups and isospectral locally symmetric spaces'', Publ.Math.IHES '''109'''(2009), 113-184; with A.S.Rapinchuk.

[18]. ''Local-global principles for embedding of fields with involution into simple algebras with involution'', Commentarii Math.Helv. '''85'''(2010), 583-645; 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), 307-325; with A.S.Rapinchuk.

[21]. ''Fake projective planes'', Inventiones Math. '''168'''(2007), 321-370, "Addendum", ibid, '''182'''(2010), 213-227; with Sai-Kee Yeung.

[22]. ''Arithmetic fake projective spaces and arithmetic fake Grassmannians'', Amer.J.Math. '''131'''(2009), 379-407; with Sai-Kee Yeung.

[23]. ''Nonexistence of arithmetic fake compact hermitian symmetric spaces of type other than A_n, n<5'', J.Math.Soc.Japan; with Sai-Kee Yeung.

[24]. ''On finite group actions on reductive groups and buildings'', Inventiones Math. '''147'''(2002), 545-560; with Jiu-Kang Yu.

[25]. ''On quasi-reductive group schemes'', J.Alg.Geom. '''15'''(2006), 507-549; with Jiu-Kang Yu.

[26]. ''Pseudo-reductive groups'', second edition, New Mathematical Monographs #'''26''', xxiv+665 pages, Cambridge University Press, 2015; with Brian Conrad and Ofer Gabber.

[27]. ''Classification of Pseudo-reductive groups'', Annals of Mathematics Studies #'''191''', 245 pages, Princeton University Press, 2015; with Brian Conrad.

[28]. ''A new approach to unramified descent in Bruhat-Tits theory'', Amer. J. Math. vol. 142 #'''1''' (2020), 215--253.

[29]. ''Finite group actions on reductive groups and buildings and tamely-ramified descent in Bruhat-Tits theory'', Amer. J. Math. vol. 142 #'''4''' (2020), 1239--1267.

[30]. ''Bruhat--Tits theory: a new approach'', Cambridge University Press, UK, 2021; with Tasho Kaletha.

==External links==
* {{MathGenealogy|id=7920}}
{{SSBPST recipients in Mathematical Science}}

{{Authority control}}

{{DEFAULTSORT:Prasad, Gopal}}
[[Category:1945 births]]
[[Category:Living people]]
[[Category:20th-century Indian mathematicians]]
[[Category:Magadh University alumni]]
[[Category:Patna University alumni]]
[[Category:University of Mumbai alumni]]
[[Category:University of Michigan faculty]]
[[Category:People from Ghazipur]]
[[Category:Tata Institute of Fundamental Research alumni]]
[[Category:Fellows of the American Mathematical Society]]
[[Category:Scientists from Uttar Pradesh]]
[[Category:Recipients of the Shanti Swarup Bhatnagar Award in Mathematical Science]]

Gopal Prasad - Revision history

imported>Eevee01 at 06:02, 1 September 2021