Mathai Varghese
Encyclopedia
Mathai Varghese is a mathematician and an Australian Research Council
(ARC) Australian Professorial Fellow at the University of Adelaide
. His most influential contribution to date is the Mathai-Quillen formalism, which he formulated together with Daniel Quillen, and which has since found applications in index theory and topological quantum field theory
. He became a full professor in 2006. He became Director of the Institute for Geometry and its Applications in 2009. In 2011, he was elected to the Australian Academy of Science.
at the Illinois Institute of Technology
. He then proceeded to the Massachusetts Institute of Technology
, where he was awarded a doctorate
under the supervision of Daniel Quillen, a Fields medallist
.
Varghese's work may be considered to fall under the ambit of geometric analysis
. His research interests are in
analysis, index theory and noncommutative geometry
. He currently works on mathematical problems that have their roots in physics, for example, topological field theories, fractional quantum Hall effect
, and D-branes in the presence of B-fields. The main focus of his research is on the application of noncommutative geometry and index theory to mathematical physics, with particular emphasis on string theory
. His current work on index theory is ongoing joint work with Richard Melrose and Isadore Singer
, on the fractional analytic index and on the index theorem for projective families of elliptic operators. His current work on string theory is ongoing joint work with Peter Bouwknegt, Jarah Evslin, Keith Hannabuss and Jonathan Rosenberg, on T-duality in the presence of background flux.
The Mathai–Quillen formalism appeared in Topology
shortly after Varghese completed his PhD. Using the superconnection formalism of Quillen, they obtained a refinement of the Riemann–Roch formula, which links together the Thom classes in K-theory
and cohomology
, as an equality on the level of differential forms. This has an interpretation in physics as the computation of the classical and quantum (super) partition functions
for the fermionic analogue of a harmonic oscillator
with source term. In particular, they obtained a nice Gaussian shaped representative of the Thom class in cohomology, which has a peak along the zero section. Its universal representative is obtained using the machinery of equivariant differential forms.
Varghese was awarded the Australian Mathematical Society
Medal in 2000. From August 2000 to August 2001, he was also a Clay Mathematics Institute
Research Scholar and Visiting Scientist at the Massachusetts Institute of Technology
. From March to June 2006, he was a Senior Research Fellow at the Erwin Schrödinger Institute in Vienna.
Australian Research Council
The Australian Research Council is the Australian Government’s main agency for allocating research funding to academics and researchers in Australian universities. Its mission is to advance Australia’s capacity to undertake research that brings economic, social and cultural benefit to the...
(ARC) Australian Professorial Fellow at the University of Adelaide
University of Adelaide
The University of Adelaide is a public university located in Adelaide, South Australia. Established in 1874, it is the third oldest university in Australia...
. His most influential contribution to date is the Mathai-Quillen formalism, which he formulated together with Daniel Quillen, and which has since found applications in index theory and topological quantum field theory
Topological quantum field theory
A topological quantum field theory is a quantum field theory which computes topological invariants....
. He became a full professor in 2006. He became Director of the Institute for Geometry and its Applications in 2009. In 2011, he was elected to the Australian Academy of Science.
Biography
Varghese received his BABachelor of Arts
A Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...
at the Illinois Institute of Technology
Illinois Institute of Technology
Illinois Institute of Technology, commonly called Illinois Tech or IIT, is a private Ph.D.-granting university located in Chicago, Illinois, with programs in engineering, science, psychology, architecture, business, communications, industrial technology, information technology, design, and law...
. He then proceeded to the Massachusetts Institute of Technology
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
, where he was awarded a doctorate
Doctorate
A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder to teach in a specific field, A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder...
under the supervision of Daniel Quillen, a Fields medallist
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
.
Varghese's work may be considered to fall under the ambit of geometric analysis
Geometric analysis
Geometric analysis is a mathematical discipline at the interface of differential geometry and differential equations. It includes both the use of geometrical methods in the study of partial differential equations , and the application of the theory of partial differential equations to geometry...
. His research interests are in
analysis, index theory and noncommutative geometryNoncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
. He currently works on mathematical problems that have their roots in physics, for example, topological field theories, fractional quantum Hall effect
Fractional quantum Hall effect
The fractional quantum Hall effect is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values of e^2/h. It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations...
, and D-branes in the presence of B-fields. The main focus of his research is on the application of noncommutative geometry and index theory to mathematical physics, with particular emphasis on string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
. His current work on index theory is ongoing joint work with Richard Melrose and Isadore Singer
Isadore Singer
Isadore Manuel Singer is an Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology...
, on the fractional analytic index and on the index theorem for projective families of elliptic operators. His current work on string theory is ongoing joint work with Peter Bouwknegt, Jarah Evslin, Keith Hannabuss and Jonathan Rosenberg, on T-duality in the presence of background flux.
The Mathai–Quillen formalism appeared in Topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
shortly after Varghese completed his PhD. Using the superconnection formalism of Quillen, they obtained a refinement of the Riemann–Roch formula, which links together the Thom classes in K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
and cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
, as an equality on the level of differential forms. This has an interpretation in physics as the computation of the classical and quantum (super) partition functions
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
for the fermionic analogue of a harmonic oscillator
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
with source term. In particular, they obtained a nice Gaussian shaped representative of the Thom class in cohomology, which has a peak along the zero section. Its universal representative is obtained using the machinery of equivariant differential forms.
Varghese was awarded the Australian Mathematical Society
Australian Mathematical Society
The Australian Mathematical Society was founded in 1956 and is the national society of the mathematics profession in Australia. One of the Society's listed purposes is to promote the cause of mathematics in the community by representing the interests of the profession to government. The Society...
Medal in 2000. From August 2000 to August 2001, he was also a Clay Mathematics Institute
Clay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...
Research Scholar and Visiting Scientist at the Massachusetts Institute of Technology
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
. From March to June 2006, he was a Senior Research Fellow at the Erwin Schrödinger Institute in Vienna.
External links
- An abbreviated curriculum vitae (c. 2001) at the Massachusetts Institute of TechnologyMassachusetts Institute of TechnologyThe Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
. - Mathai Varghese's homepage at the University of Adelaide.
- Research citation for the award of the Australian Mathematical Society Medal in 2000.