Gopal Prasad
Encyclopedia
Gopal Prasad is an Indian
mathematician
. His research interests span the fields of Lie group
s, their discrete subgroups, algebraic group
s, arithmetic group
s, geometry of locally symmetric spaces, and representation theory of reductive p-adic groups.
He is the Raoul Bott
Professor of Mathematics at the University of Michigan
in Ann Arbor.
Two of Prasad's siblings are in Mathematics: Shrawan Kumar
, Professor of Mathematics at the University of North Carolina
at Chapel Hill, and Dipendra Prasad, Professor of Mathematics at the Tata Institute of Fundamental Research
, Mumbai.
In 1987, Prasad found a formula for the volume of S-arithmetic quotients of semi-simple groups. Using this formula and certain number theoretic and Galois-cohomological estimates, Armand Borel
and Gopal Prasad proved several finiteness theorems about arithmetic groups. 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. 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.
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. 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.
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 on the Bruhat-Building of the reductive p-adic group. In another joint work, Prasad and Yu determined all the quasi-reductive group schemes over a discrete valuation ring (DVR).
In collaboration with Brian Conrad
and Ofer Gabber, Prasad has classified 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. The classification of pseudo-reductive groups already has many applications. There has been a Bourbaki seminar in March 2010 on the work of Tits, Conrad-Gabber-Prasad on pseudo-reductive groups.
, 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 and in the Indian Academy of Sciences. Prasad gave an invited talk in the International Congress of Mathematicians
held in Kyoto in 1990.
Prasad is the Managing Editor of the Michigan Mathematical Journal
, Editor of the Asian Journal of Mathematics since its inception, and was formerly an Associate Editor of the Annals of Mathematics
.
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. His research interests span the fields of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, their discrete subgroups, algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s, arithmetic group
Arithmetic group
In mathematics, an arithmetic group in a linear algebraic group G defined over a number field K is a subgroup Γ of G that is commensurable with G, where O is the ring of integers of K. Here two subgroups A and B of a group are commensurable when their intersection has finite index in each of them...
s, geometry of locally symmetric spaces, and representation theory of reductive p-adic groups.
He is the Raoul Bott
Raoul Bott
Raoul Bott, FRS was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense...
Professor of Mathematics at the University of Michigan
University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...
in Ann Arbor.
Education
He earned his bachelors 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 joined TIFR for his PhD program in 1966. There Prasad 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 University of Mumbai. Prasad became an Associate Professor at TIFR in 1979, and a Professor in 1984. He left TIFR to join the faculty at the University of Michigan in Ann Arbor in 1992, where he is 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.Two of Prasad's siblings are in Mathematics: Shrawan Kumar
Shrawan Kumar
Shrawan Kumar is the John R. and Louise S. Parker distinguished professor of mathematics at the University of North Carolina at Chapel Hill. He has written two books: Kac-Moody groups, their flag varieties, and representation theory and Frobenius splitting methods in geometry and representation...
, Professor of Mathematics at the University of North Carolina
University of North Carolina
Chartered in 1789, the University of North Carolina was one of the first public universities in the United States and the only one to graduate students in the eighteenth century...
at Chapel Hill, and Dipendra Prasad, Professor of Mathematics at the Tata Institute of Fundamental Research
Tata Institute of Fundamental Research
The Tata Institute of Fundamental Research is a research institution in India dedicated to basic research in mathematics and the sciences. It is a Deemed University and works under the umbrella of the Department of Atomic Energy of the Government of India. It is located at Navy Nagar, Colaba, Mumbai...
, Mumbai.
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. 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. In collaboration with M. S. Raghunathan, Prasad determined the topological central extensions of these groups, and computed the "metaplectic kernel" for isotropic groups. Later, together with Andrei Rapinchuk, Prasad gave a precise computation of the metaplectic kernel for all simply connected semi-simple groups. Prasad and Raghunathan have also obtained results on the Kneser-Tits problem.In 1987, Prasad found a formula for the volume of S-arithmetic quotients of semi-simple groups. Using this formula and certain number theoretic and Galois-cohomological estimates, Armand Borel
Armand Borel
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993...
and Gopal Prasad proved several finiteness theorems about arithmetic groups. 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
Fake projective plane
In mathematics, a fake projective plane is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type....
s (in the theory of smooth projective complex surfaces) into 28 non-empty classes. 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
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
and harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
. 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
Admissible representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra....
and to give a classification of representations of depth zero.
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. 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.
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 on the Bruhat-Building of the reductive p-adic group. In another joint work, Prasad and Yu determined all the quasi-reductive group schemes over a discrete valuation ring (DVR).
In collaboration with Brian Conrad
Brian Conrad
Brian Conrad , is an American mathematician and number theorist, working at Stanford University. Previously he was at the University of Michigan....
and Ofer Gabber, Prasad has classified 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. The classification of pseudo-reductive groups already has many applications. There has been a Bourbaki seminar in March 2010 on the work of Tits, Conrad-Gabber-Prasad on pseudo-reductive groups.
Honors
Prasad has received the Guggenheim FellowshipGuggenheim Fellowship
Guggenheim Fellowships are American grants that have been awarded annually since 1925 by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the arts." Each year, the foundation makes...
, 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 and in the Indian Academy of Sciences. Prasad gave an invited talk in the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
held in Kyoto in 1990.
Prasad is the Managing Editor of the Michigan Mathematical Journal
Michigan Mathematical Journal
Michigan Mathematical Journal is published by the mathematics department atUniversity of Michigan.Its initial editor was George Piranian....
, Editor of the Asian Journal of Mathematics since its inception, and was formerly an Associate Editor of the Annals of Mathematics
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
.