Mathai Varghese is a mathematician at the

University of Adelaide The University of Adelaide (informally Adelaide University) is a public research university located in Adelaide, South Australia. Established in 1874, it is the third-oldest university in Australia. The university's main campus is located on ...

. His first most influential contribution is the

Mathai–Quillen formalism In mathematics, the Mathai–Quillen formalism is an approach to topological quantum field theory introduced by , based on the Mathai–Quillen form constructed in . In more detail, using the superconnection formalism of Quillen, they obtained a r ...

, which he formulated together with

Daniel Quillen Daniel Gray "Dan" 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 19 ...

, and which has since found applications in

index theory Index (or its plural form 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 a Halo megastru ...

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 which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathe ...

. He was appointed a full professor in 2006. He was appointed Director of the

Institute for Geometry and its Applications An institute is an organisational body created for a certain purpose. They are often research organisations ( research institutes) created to do research on specific topics, or can also be a professional body. In some countries, institutes c ...

in 2009. In 2011, he was elected a

Fellow of the Australian Academy of Science The Fellowship of the Australian Academy of Science is made up of about 500 Australian scientists. Scientists judged by their peers to have made an exceptional contribution to knowledge in their field may be elected to Fellowship of the Academy ...

. In 2013, he was appointed the (Sir Thomas)

Elder Professor of Mathematics Sir Thomas Elder, (5 August 1818 – 6 March 1897), was a Scottish-Australian pastoralist, highly successful businessman, philanthropist, politician, race-horse owner and breeder, and public figure. Amongst many other things, he is notable fo ...

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 The Australian Laureate Fellowship is an Australian professorial research fellowship awarded by the Australian Research Council. Up to 17 fellows are chosen each year for five-year awards. Kathleen Fitzpatrick and Georgina Sweet fellowships In 20 ...

. 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 be ...

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 Illinois Institute of Technology (IIT) is a private research university in Chicago, Illinois. Tracing its history to 1890, the present name was adopted upon the merger of the Armour Institute and Lewis Institute in 1940. The university has pro ...

. He then proceeded to the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private Land-grant university, land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern t ...

, where he was awarded a

doctorate A doctorate (from Latin ''docere'', "to teach"), doctor's degree (from Latin ''doctor'', "teacher"), or doctoral degree is an academic degree awarded by universities and some other educational institutions, derived from the ancient formalism '' ...

under the supervision of

, a Fields Medallist. 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 In mathem ...

. 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 (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of e^2/h. It is a property of a collective state in which elec ...

, and

D-branes In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...

in the presence of

B-fields A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...

. 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 Richard Burt Melrose is an Australian mathematician and professor at the Massachusetts Institute of Technology who works on geometric analysis, partial differential equations, and differential geometry. Education Melrose received in 1974 his P ...

and

Isadore Singer Isadore Manuel Singer (May 3, 1924 – February 11, 2021) was an American mathematician. He was an Emeritus Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology and a Professor Emeritus of Mathemat ...

, 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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

'', 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 geom ...

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 viewe ...

, 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: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...

for the fermionic analogue of a

harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...

with source term. In particular, they obtained a nice Gaussian shape representative of the

Thom class In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact s ...

in cohomology, which has a peak along the zero section. Its universal representative is obtained using the machinery of

equivariant differential forms In mathematics, equivariance is a form of symmetry for function (mathematics), functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are Group action ( ...

. 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 The Clay Mathematics Institute (CMI) is a private, non-profit foundation dedicated to increasing and disseminating mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address is now in Denver, Colorado. CMI's sc ...

Research Fellow and visiting scientist at the

. From March to June 2006, he was a senior research fellow at the

Erwin Schrödinger Institute Erwin may refer to: People Given name * Erwin Chargaff (1905–2002), Austrian biochemist * Erwin Dold (1919–2012), German concentration camp commandant in World War 2 * Erwin Hauer (1926–2017), Austrian-born American sculptor * Egon Erwin ...

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, st ...

'' 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 University of Adelaide faculty Year of birth missing (living people) Living people Fellows of the Australian Academy of Science