# Gopal Prasad

**Gopal Prasad** (born 31 July 1945 in Ghazipur, India) is an Indian-American 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 p-adic groups.

Gopal Prasad | |
---|---|

Born | |

Alma mater | Patna University IITK TIFR Institute for Advanced Study |

Awards | - Shanti Swarup Bhatnagar Award (1989)
- Guggenheim Fellowship (1998)
- Humboldt Research Award (2006)
- Fellow of American Mathematical Society (2012),
Indian Academy of Sciences (1984), |

Scientific career | |

Fields | Mathematics |

Institutions | University of Michigan |

Doctoral advisor | M. S. Raghunathan |

He is the Raoul Bott Professor of Mathematics[1] at the University of Michigan in Ann Arbor.

## 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 semi-simple 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

In 1969, he married Indu Devi of Deoria. Gopal Prasad and Indu Devi have a son and a daughter and five grand-children. 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 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]. 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 semi-simple groups, see [14]. Prasad and Raghunathan have also obtained results on the Kneser-Tits problem, [13].

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 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. 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.

## 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 2013-05-26.

[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.