Recording: https://disk.pku.edu.cn:443/link/421E9A256FF207AEF1CD23364820C76B
Valid Until: 2027-01-31 23:59
Abstract: The Fredholm index of an elliptic differential operator on a smooth manifold is a global homotopy invariant and it is equal to the topological index defined by Atiyah and Singer. Later Atiyah, Bott and Patodi obtained a local index theorem for twisted Dirac operator which refines the Atiyah--Singer index formula to a pointwise equality, which after integration over the manifold recovers the Atiyah--Singer index formula. Noncommutative geometry is a natural framework for index theories. Connes and Moscovici obtained a general local index formula in noncommutative geometry in terms of coefficients of heat trace expansions. We compute these coefficients and hence obtain an explicit local index formula in the following situation.
On Rn, we consider the Euler operator (an elliptic differential operator of order one and index one, written in terms of creation and annihilation operators) acting on differential forms and show that this operator is local with respect to the action of the affine unitary group by metaplectic operators. Our main result then is an explicit algebraic expression for the Connes-Moscovici coefficients. As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds. The results were obtained in a joint work with Elmar Schrohe (Hannover) and published in: Savin A., Schrohe E. Local index formulae on noncommutative orbifolds and equivariant zeta functions for the affine metaplectic group. Adv. Math. 409 (2022), part A, Paper No. 108624, 37 pp.
Bio: Anton Savin is a Professor at the Peoples' Friendship University of Russia (RUDN University). He obtained his PhD from Moscow State University in 2000 and a Doctor of Sciences degree from RUDN University in 2012. He worked at Moscow State University, the University of Potsdam, the Independent University of Moscow and the University of Hannover before joining RUDN University. He was a winner of the Pierre Deligne Contest for young Russian mathematicians. His research interests include the index theory of elliptic operators and noncommutative geometry.
