Gourab Bhattacharya

Gourab Bhattacharya

Institut des Hautes Etudes Scientifiques
Le Bois-Marie, 35 route de Chartres
91440 Bures-sur-Yvette
France

CFM

email: bhattacharya@ihes.fr
gourabmath@gmail.com
gourabb@ihes.fr
phone: (+33) 1 60 92 66 27
office: 0S6 (26627)

I do not consider Mathematics only as a formal language, but a way of thinking, a narrative to some abstract objects that require a certain generalised but simple explanation, that can unify a vast amount of informations in a single statement, it is similar to something like, "Be Wise Generalise", I forgot who first said the quote, but whoever that person might be, it actually describes and explains Mathematics and its nature, however, a certain level of consistency and rigour is necessary for the development of Mathematical thinking, but rigour has its limitations, it may not be absolute, it is the end product, for me, Mathematics is a part of life, like breathing. At this point of time, I am working in several topics related to the celebrated Homological Mirror Symmetry of Maxim Kontsevitch, some are directly related, some are not, also the theory of the Gravitation and corresponding Gravitational Waves attract me much due to its deeply hidden Geometric structures that need to be discovered. The Thermodynamic objects in a Strong Gravitational Field attracts my attention a lot, I have lots of unanswered questions in this field of study, hopefully one day I will get my answers to all my questions.

My main interest and the object of study now is to understand the noncommutative instantons from the noncommutative Hermitian Yang-Mills equation or the infinite dimensional Nekrasov equation, actually one can show (we showed in our work with Kontsevich) that, it is a consequence of certain M-Theory (M = Matrix ) compactification of higher dimensions on noncommutative tori (Physicists call it the IKKT and the BFSS models) of the Connes-Douglas-Schwarz type (more Mathematically), also I am interested in Fukaya Category via the Kontsevich-Soibelman Wall-Crossing structures, specifically those wall-crosiing structures coming from the Blackhole attractors in the Superstrings and Supergravity theories.

Besides that, we are hoping our future paper on the infinite diemensional King's equation will shed considerable amout of light on the q-Difference equations and the Discrete Integrable Systems coming from Geometry, Physics and, Arithmetic. I am also interested in the theories of Gravitation of various kinds (not necessarily of Einsteinian one's) , Quantum Field Theories, Conformal Field Theories, the Foundational problems in Quantum Mechanics, and, different Noncommutative Geometries.

  • A Generalization of King's Equation via Noncommutative Geometry.
    Part of this work is a generalisation of the talk I gave in the conference, "New Trends in Mathematical and Theoretical Physics", at the Steklov Institute, Moscow, on 3rd October, 2016, and, a talk given in the conference, "101e rencontre entre mathématiciens et physiciens théoriciens" at University of Strasbourg, 5th April 2018, the proof and the generalisation is documented in page 21 of the article. Below in the section of "talks", one can find a presentation of our recent work, there Professor Kontsevich ( in an online seminar or webinar, virtually hosted by the University of Geneva) explained our work where we tried to have a language that formally unifies different Matrix/Differential equations of Physics in a single context under the name of King's equation of Representation theory, namely, the ADHM-construction (due to Donaldson for two-dimensional Complex Projective plane), the Deformed ADHM-construction due to Nekrasov et al, the Hermitain Yang-Mills in 4-dimensions, the Vafa-Witten equations for the smooth Complex curves and the smooth Kaehler Manifolds of arbitrary finite complex dimension, the Bogomolny's Monopole equation for the smooth Real 3-Manifolds and, the Nahm's one-parameter Monopole Equations, and many other equations in both quiver and smooth categories are shown to be a particular case of King's equation, the constructions done in the paper are partly motivated by "M-theory" based on a Noncommutative Geometry, where M, without any ambiguities, stands for Matrix.
    Gourab Bhattacharya and Maxim Kontsevich.
    Published at, Geometry and Topology, A Collection of Essays Dedicated to Vladimir G. Turaev (ed. Athanase Papadopoulos), European Mathematical Society Publishing House, Berlin, 2021 (available in print, 2021).
  • On the dual version of Seiberg-Witten Gauge Theory and a proof of Narasimhan-Seshadri theorem.
  • Vafa-Witten equation as Hitchin's equation over compact complex curves.
  • On Deformed Hermitian Yang-Mills and Bridgeland Stability Structure (talk at the University of Strasbourg, 5th April 2018)
  • BPS States and invariants of 3 and 4- Manifolds (talk at the conference "encounter between Mathematicians and theoretical physicists")
  • A Generalization of King's Equation via Noncommutative Geometry Part II
  • Short bio

    I consider myself mainly as a "Physical Geometer". It is hard to define what a Geometry is, and, it is more difficult to define what a Physical Geometer might mean, however, I love Physics, and I extract various Geometries from Physics. What I sense and realise that our understanding of Science is very limited, for say, it is hard to say what is the general definition of Mass of an object in Classical Gravitational Theories. No one knows in what context one can understand the principle of conservation of the Mass or the Energy separately, together they form an Energy-Momentum Tensor but individually they constitute what is called a pseudo-tensor, so the energy or the mass becomes background dependent. The case for the Angular-Momentum becomes more complicated, and it is impossibly hard to have a general definition. It seems all these concepts to have any meaning in the presence of Gravity, requires a new Breakthrough.

  • Curriculum Vitae
  • The following is a talk at Steklov Institute, Moscow, Russia, where I explained how to prove the Narasimhan-Seshadri Theorem for Compact Riemann Surfaces using Vafa-Witten Equations. These are the equations coming from String theory or more precisely via the use of the S-Duality in non-perturbative String theory, inveneted by Physicists, while describing the proof, I showed how one can get a slightly different stability structure compared to the existing Mumford' stability condition, namely the slope stability. If the link is not working, then one can copy paste the following link, http://www.mathnet.ru/eng/present14889 into one's browser.
  • A Dual Version of Seiberg-Witten Theory
  • Quantum Minimal Surface and Noncommutative Kaehler Geometry. The talk below is given by Professor Maxim Kontsevich on our joint work (one can consult, https://arxiv.org/pdf/2003.03171.pdf). A version of Noncommutative Kaehler Geometry is presented. One can find Open String Theory as a motivation to some of the constructions made above. Open strings end on Branes, and, they are partly Mathematically determined by some higher dimensional variants of Hermitian Yang-Mills equations (or Deformed Hermitian Yang-Mills Equations). To describe these Branes, one needs certain machinery of Noncommutative Geometry, or more specifically Noncommutative Kaehler Geometry in a sense described in the talk.
  • Research statement I am currently working on the Infinite dimensional King's Equation that Generalises Nekrasov's construction and, trying to understand the Geometry that lives in the Kernel of the King's equation! Also, I am thinking on Noncommutative Kaehler Geometry as initiated in our joint work with Professor Maxim Kontsevich. This notion of Noncommutative Kaehler Geometry generalises certain aspects of the Open String Theory. Since, the Open Strings end on the Branes and, the Branes are in certain sense determined by the Deformed Hermitian Yang-Mills Equations, one can exepct, our systematic study of a Mathematical notion of Noncommutative Kaehler Geometry might shed some light on the solutions of Deformed Hermitian Yang-Mills Equations. One of the main problems is to study the Donaldson-Uhlenbeck-Yau (DUY) type theorems in our context. On a different note, I am interested in the Geometries coming from Gravitation. As shown by Robinson, Einstein-Maxwell's vaccuum solutions describe high-frequency massless gravitons, one gets Robinson's Congruence and, that somehow predicts the existence of flat space Twistors. I am curious to understand what more one can get if one takes some other Gravitational Waves in account, namely, as I understand Twistors are roughly the consequence of the pp-wave type solutions for Einstein's Vacuum Equations, but to understand the curved Twistors, should we take the full Gravitational Waves into account? I do not know the answer, but wish to understand it in full details.