Talk

In this talk, we consider exact identities for infinite sums of even divisor 
functions weighted by Laurent polynomials with logarithms for which the sum is 
absolutely convergent. Such identities are motivated by computations in string theory 
related to the scattering of 4-gravitons. We show that in general they give Fourier 
coefficients of holomorphic cusp forms using the method of holomorphic projection 
and discuss an application of spectral techniques to such sums. The talk is based on 

We extend the work of Ledrappier-Wang and Besson-Courtois-Gallot 's barycenter technique to apply it to RCD measure metric spaces.
We prove minimal and maximal volume entropy rigidity results and express them with a new notion of homotopic degree suitable for RCD spaces.
For example, we show that an RCD(-(N-1), N) space homotopic to a hyperbolic manifold M has total measure bounded below by M's hyperbolic
volume, and equality occurs if and only if the space is isometric to M.
Joint with Dai, Connell, Perales, Núñez Zimbrón, and Wei.

Using finite dimensional Nichols algebras with automorphisms, we construct R-matrices with entries polynomials in the structure constants of the Nichols algebras. The R-matrices satisfy the Yang-Baxter equation and can be used to define multivariable polynomials of knots that generalize the colored Jones polynomial. For a Nichols algebra of rank 2, we define a sequence of polynomials $V_n(t,q)$ in two variables, whose first knot invariant is the Links-Gould polynomial and the second one $V_2$ is a new polynomial invariant that gives bounds for the Seifert-genus of a knot. Joint work with Rinat Kashaev.