Toric gerbes are étale gerbes over toric Deligne-Mumford stacks which are constructed out of suitably chosen toric data. In this paper we study the genus 0 Gromov-Witten theory of toric gerbes. Our main result equates the genus 0 Gromov-Witten theory of a toric gerbe with a suitable twist of the genus 0 Gromov-Witten theory of a disjoint union of several copies of the base. Our result can be interpreted in the context of the decomposition conjecture in physics. The main tool used in this paper is the calculation of Gromov-Witten theory of toric Deligne-Mumford stacks by Coates-Corti-Iritani-Tseng.

We derive closed formula for the heat kernel $K_{\mathbb H, k}$ associated to the Maass-Laplacian operator $D_k$ for any real weight $k$ and prove that the heat kernel $K_{\mathbb H, k}$ is strictly monotone decreasing function of the hyperbolic distance. We derive small time and large time asymptotic formulae for the heat kernel $K_{\mathbb H, k}$ and describe its behavior as a function of the real weight $k$.

Recently, Otake and Shaska have given a formula for the discriminant of quadrinomials of the form $f(x) = x^n + t(x^2 + ax + b)$. In their concluding remarks, they ask if a formula can be found for the discriminant of $f(x) = x^n + tg(x)$ when $n > \deg(g) = 3$. Assuming that $f(x) = x^n + tg(x)$ is irreducible, and under certain restrictions on a polynomial related to $g(x)$, in this article we give a formula for the discriminant of $f(x)$, regardless of $\deg(g) \ge 1$. We then use our discriminant formula to generate some new infinite families of monogenic polynomials $f(x) = x^n + tg(x)$ with $n > \deg(g)$, when $g(x)$ is monic and $\deg(g)\in 2 \{2, 3, 4\}$.

$\tau$-Li coefficients describe if a function satisfies the Generalized Riemann Hypothesis or not. In this paper we prove that certain values of the $\tau$-Li coefficients lead to existence or non-existence of certain zeros. The first main result gives explicit numbers $N_1$ and $N_2$ such that if all real parts of the $\tau$-Li coefficients are non-negative for all indices between $N_1$ and $N_2$, then the function has non zeros outside a certain region. According to the second result, if some of the real parts of the $\tau$-Li coefficients are negative for some index n between numbers $n_1$ and $n_2$, then there is at least one zero outside a certain region.

Let $W \subset \mathbb P^n, n \ge 3$, be a degree k hypersurface. Consider a "general" nodal union of $d$ lines $L_1, \dots L_d$ with $L_i \cap L_j \neq \emptyset$ if and only if $|i - j| \le 1$ (here called a degree $d$ bamboo). We study the Hilbert function of the set $Y \cap W$ with cardinality $k \deg(Y)$ and prove that it is the expected one (with a few classified exceptions $(n,k,d)$) when $W$ is either a quadric hypersurface of rank at least $2$ or $n = 3$ and $W$ is an integral cubic surface.