Pseudomoments of the Riemann zeta-function and pseudomagic squares

We compute integral moments of partial sums of the Riemann zeta function on the critical line and obtain an expression for the leading coefficient as a product of the standard arithmetic factor and a geometric factor. The geometric factor is equal to the volume of the convex polytope of substochastic matrices and is equal to the leading coefficient in the expression for moments of truncated characteristic polynomial of a random unitary matrix

Quantum curves and topological recursion

This is a survey article describing the relationship between quantum curves and topological recursion. A quantum curve is a Schr\"odinger operator-like noncommutative analogue of a plane curve which encodes (quantum) enumerative invariants in a new and interesting way. The Schr\"odinger operator annihilates a wave function which can be constructed using the WKB method, and conjecturally constructed in a rather different way via topological recursion.Comment: This article arose out of the Banff workshop Quantum Curves and Quantum Knot Invariants. Comments welcome. 20 pages, 1 figur

Supersymmetric U(N)U(N) Chern-Simons-matter theory and phase transitions

We study ${\mathcal{N}}=2$ supersymmetric $U(N)$ Chern-Simons with $N_{f}$ fundamental and $N_{f}$ antifundamental chiral multiplets of mass $m$ in the complete parameter space spanned by $(g,\,m,\,N,\,N_{f})$, where $g$ denotes the coupling constant. In particular, we analyze the matrix model description of its partition function, both at finite $N$ using the method of orthogonal polynomials together with Mordell integrals and, at large $N$ with fixed $g$, using the theory of Toeplitz determinants. We show for the massless case that there is an explicit realization of the Giveon-Kutasov duality. For finite $N$, with $N>N_{f}$, three regimes that exactly correspond to the known three large $N$ phases of theory are identified and characterized.Comment: 28 pages, v3: Minor modification to match published versio