Mathai Varghese is a mathematician at the

University of Adelaide The University of Adelaide is a public university, public research university based in Adelaide, South Australia. Established in 1874, it is the third-oldest university in Australia. Its main campus in the Adelaide city centre includes many Sa ...

. His first most influential contribution is the Mathai–Quillen formalism, which he formulated together with

Daniel Quillen Daniel Gray Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978. Fr ...

, and which has since found applications in

index theory Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...

and

topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While TQFTs were invented by physicists, they are also of mathemati ...

. He was appointed a full professor in 2006. He was appointed Director of the Institute for Geometry and its Applications in 2009. In 2011, he was elected a Fellow of the Australian Academy of Science. In 2013, he was appointed the (Sir Thomas) Elder Professor of Mathematics at the University of Adelaide, and was elected a Fellow of the Royal Society of South Australia. In 2017, he was awarded an ARC Australian Laureate Fellowship. In 2021, he was awarded the prestigious Hannan Medal and Lecture from the Australian Academy of Science, recognizing an outstanding career in Mathematics. In 2021, he was also awarded the prestigious

George Szekeres Medal The George Szekeres Medal is awarded by the Australian Mathematical Society for outstanding research contributions over a fifteen-year period. This award, established in 2001, was given biennially in even-numbered years until 2021 and has since bee ...

which is the Australian Mathematical Society’s most prestigious medal, recognising research achievement and an outstanding record of promoting and supporting the discipline.

Biography

Mathai studied at Bishop Cotton Boys' School, Bangalore. Mathai received a BA at the

Illinois Institute of Technology The Illinois Institute of Technology, commonly referred to as Illinois Tech and IIT, is a Private university, private research university in Chicago, Illinois, United States. Tracing its history to 1890, the present name was adopted upon the m ...

. He then proceeded to the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...

, where he was awarded a

doctorate A doctorate (from Latin ''doctor'', meaning "teacher") or doctoral degree is a postgraduate academic degree awarded by universities and some other educational institutions, derived from the ancient formalism '' licentia docendi'' ("licence to teach ...

under the supervision of

, a

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...

list. Mathai's work is in the area of

geometric analysis Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...

. His research interests are in

L^2

analysis, index theory, and

noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions, possibly in some g ...

. He currently works on mathematical problems that have their roots in physics, for example, topological field theories,

fractional quantum Hall effect The fractional quantum Hall effect (fractional QHE or FQHE) is the observation of precisely quantized plateaus in the Hall conductance of 2-dimensional (2D) electrons at fractional values of e^2/h, where ''e'' is the electron charge and ''h'' i ...

, and

D-branes In string theory, D-branes, short for Dirichlet membrane, are a class of extended objects upon which open string (physics), strings can end with Dirichlet boundary conditions, after which they are named. D-branes are typically classified by their ...

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 In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

. 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 Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

'', shortly after Mathai completed his Ph.D. Using the superconnection formalism of Quillen, they obtained a refinement of the Riemann–Roch formula, which links together the Thom classes in

K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...

and

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

, 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 may refer to: Arts and entertainment Film and television * Partition (1987 film), ''Partition'' (1987 film), directed by Ken McMullen * Partition (2007 film), ''Partition'' (2007 film), directed by Vic Sarin * ''Partition: 1947'', or '' ...

for the fermionic analogue of a

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 const ...

with source term. In particular, they obtained a nice Gaussian shape 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. Mathai was awarded the

Australian Mathematical Society The Australian Mathematical Society (AustMS) 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 ...

Medal in 2000. From August 2000 to August 2001, he was also a Clay Mathematics Institute Research Fellow and visiting scientist at the

. From March to June 2006, he was a senior research fellow at the Erwin Schrödinger Institute in Vienna.

Selected publications

* * Bouwknegt, Peter, Evslin, Jarah and Mathai, Varghese. (2004) "T-duality: Topology Change from H-flux". ''

Communications in Mathematical Physics ''Communications in Mathematical Physics'' is a peer-reviewed academic journal published by Springer. The journal publishes papers in all fields of mathematical physics, but focuses particularly in analysis related to condensed matter physics, sta ...

'' 249 (2), 383–415. *

Notes

References

* Blau, Matthia
"The Mathai-Quillen Formalism and Topological Field Theory"
Infinite-dimensional geometry in physics (Karpacz, 1992). J. Geom. Phys. 11 (1993), no. 1-4, 95–127 * Wu, Siy
"Mathai-Quillen Formalism"
J. Geom. Phys. 17 (1995), no. 4, 299–309

External links

Mathai Varghese's research page
at the University of Adelaide. * {{DEFAULTSORT:Mathai, Varghese 20th-century Indian mathematicians 21st-century Indian mathematicians Australian mathematicians Australian people of Malayali descent Massachusetts Institute of Technology alumni Illinois Institute of Technology alumni Scientists from Kerala Academic staff of the University of Adelaide Year of birth missing (living people) Living people Fellows of the Australian Academy of Science