Last Update 2018/09/05
My research interests have always rotated around string theory and quantum field theory, spanning over various fields of mathematics. The main result of my Ph.D. thesis established the super Mumford isomorphism, confirming mathematically the dimension of space-time in superstring theory to be 10. Later, I developed semi-infinite homological algebra, providing foundations for BRST cohomology. Takashi Kimura, Jim Stasheff, and I explained homotopy Lie-algebraic structures appearing in string-field theory, using operadic properties of moduli spaces and constructing a real version of the Deligne-Mumford compactification. Afterwards, I introduced the Swiss-cheese operad and the cactus operad to handle algebraic structures arising in open-closed string theory and string topology, respectively. I gave historically the second (after Dmitry Tamarkin, among quite a few) proof of Deligne's Hochschild cohomology conjecture. Kontsevich used the Swiss-cheese operad to formulate his generalization of Deligne's conjecture, which we subsequently proved with Po Hu and Igor Kriz. I also worked on higher-categorical foundations of topological quantum field theory with Mark Feshbach. More recently, my former Ph.D. student Denis Bashkirov and I have been working on homotopical algebra related to the BV formalism and to Lie algebroids and bialgebroids. With my other former Ph.D. student Amit Sharma, we developed a categorification of Dijkgraaf-Witten theory. I have also been working on quantum deformation theory, which is based on the quantum master equation instead of the Maurer-Cartan equation. I have founded the “Tea Break" rubric in Kavli IPMU News magazine and been a frequent contributor to that section.
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