Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle product rule allows the possibility of a combinatorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with certain repeated arguments. We also prove a similar cyclic sum identity. Finally, we present extensive computational evidence supporting an infinite family of conjectured MZV identities that simultaneously generalize the Zagier identity.

J. M. Borwein D. M. Bradley D. J. Broadhurst P. Lisonek

In this paper we construct the modular Cauchy kernel on the Hilbert modular surface $\Xi_{\mathrm{Hil},m}(z)(z_2-\bar{z_2})$, i.e. the function of two variables, $(z_1, z_2) \in \mathbb{H} \times \mathbb{H}$, which is invariant under the action of the Hilbert modular group, with the first order pole on the Hirzebruch-Zagier divisors. The derivative of this function with respect to $\bar{z_2}$ is the function $\omega_m (z_1, z_2)$ introduced by Don Zagier in \cite{Za1}. We consider the question of the convergence and the Fourier expansion of the kernel function. The paper generalizes the first part of the results obtained in the preprint \cite{Sa}

Nina Sakharova

We give a formula for the radial asymptotics to all orders of the special $q$-hypergeometric series known as Nahm sums at complex roots of unity. This result is used in~\cite{CGZ} to prove one direction of Nahm's conjecture relating the modularity of Nahm sums to the vanishing of a certain invariant in $K$-theory. The power series occurring in our asymptotic formula are identical to the conjectured asymptotics of the Kashaev invariant of a knot once we convert Neumann-Zagier data into Nahm data, suggesting a deep connection between asymptotics of quantum knot invariants and asymptotics of Nahm sums that will be discussed further in a subsequent publication.

Stavros Garoufalidis Don Zagier

We show that certain hypergeometric series used to formulate mirror symmetry for Calabi-Yau hypersurfaces, in string theory and algebraic geometry, satisfy a number of interesting properties. Many of these properties are used in separate papers to verify the BCOV prediction for the genus one Gromov-Witten invariants of a quintic threefold and more generally to compute the genus one Gromov-Witten invariants of any Calabi-Yau projective hypersurface.

Don Zagier Aleksey Zinger

We give a short proof of the trace formula for Hecke operators on modular forms for the modular group, using the action of Hecke operators on the space of period polynomials.

Alexandru A. Popa Don Zagier

We prove that a curious generating series identity implies Faber's intersection number conjecture (by showing that it implies a combinatorial identity already given in arXiv:1902.02742) and give a new proof of Faber's conjecture by directly proving this identity.

Elba Garcia-Failde Don Zagier

In this note we study the eigenvalue problem for a quadratic form associated with Strichartz estimates for the Schr\"{o}dinger equation, proving in particular a sharp Strichartz inequality for the case of odd initial data. We also describe an alternative method that is applicable to a wider class of matrix problems.

Felipe Gonçalves Don Zagier

This is an article about the work of Walter Neumann on hyperbolic geometry, ideal triangulations of 3-manifolds, the volume and Chern-Simons invariants of 3-manifolds and their elements of the the Bloch group. The article focuses on the relations of these classical invariants and the impact of Neumann's work to quantum topology and to mathematical physics.

Stavros Garoufalidis Don Zagier

In this paper we give a trace formula for Hecke operators acting on the cohomology of a Fuchsian group of finite covolume, with coefficients in a module $V$. The proof is based on constructing an operator whose trace on $V$ equals the Lefschetz number of the Hecke correspondence on cohomology, generalizing the operator introduced together with Don Zagier for the modular group.

Alexandru A. Popa

We offer an alternative viewpoint on Dyson's original paper regarding the application of Brownian motion to random matrix theory (RMT). In particular we show how one may use the same approach in order to study the stochastic motion in the space of matrix traces $t_n = \sum_{\nu=1}^{N} \lambda_\nu^n$, rather than the eigenvalues $\lambda_\nu$. In complete analogy with Dyson we obtain a Fokker-Planck equation that exhibits a stationary solution corresponding to the joint probability density function in the space $t = (t_1,\ldots,t_n)$, which can in turn be related to the eigenvalues $\lambda = (\lambda_1,\ldots,\lambda_N)$. As a consequence two interesting combinatorial identities emerge, which are proved algebraically in the appendix. We also offer a number of comments on this version of Dyson's theory and discuss its potential advantages.

Christopher H. Joyner Uzy Smilansky