We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight $2$. Moreover, we determine the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and we study Faltings heights of arithmetic Hirzebruch-Zagier divisors

We introduce the inertial cohomology ring ${\mathrm{NH}}_T^{*,◊}(Y)$ of a stably almost complex manifold carrying an action of a torus $T$. We show that in the case where $Y$ has a locally free action by $T$, the inertial cohomology ring is isomorphic to the Chen-Ruan orbifold cohomology ring $H_{\mathrm{CR}}^{*}(Y/T)$ (as defined in [CR]) of the quotient orbifold $Y/T$.

For $Y$ a compact Hamiltonian $T$-space, we extend to orbifold cohomology two techniques that are standard in ordinary cohomology. We show that ${\mathrm{NH}}_T^{*,◊}(Y)$ has a natural ring surjection onto $H_{\mathrm{CR}}^{*}(Y//T)$, where $Y//T$ is the symplectic reduction of $Y$ by $T$ at a regular value of the moment map. We extend to ${\mathrm{NH}}_T^{*,◊}(Y)$ the graphical Goresky-Kottwitz-MacPherson (GKM) calculus (as detailed in, e.g., [HHH]) and the kernel computations of [TW] and [G1], [G2].

We detail this technology in two examples: toric orbifolds and weight varieties, which are symplectic reductions of flag manifolds. The Chen-Ruan ring has been computed for toric orbifolds, with $\mathbb{Q}$-coefficients, in [BCS]; we reproduce their results over $\mathbb{Q}$ for all symplectic toric orbifolds obtained by reduction by a connected torus (though with different computational methods) and extend them to $\mathbb{Z}$-coefficients in certain cases, including weighted projective spaces

Using the main properties of the skew polynomial rings ${\mathbb{F}}_{q^d}\{\tau \}$ and some related rings, we describe the explicit construction of Ramanujan hypergraphs, which are certain simplicial complexes introduced in the author's thesis [29] (see also [30]) as generalizations of Ramanujan graphs. Such hypergraphs are described in terms of Cayley graphs of various groups. We give an explicit description of our hypergraph as the Cayley graph of the groups ${\mathrm{PSL}}_d({\mathbb{F}}_r)$ and ${\mathrm{PGL}}_d({\mathbb{F}}_r)$ with respect to a certain set of generators, over a finite field ${\mathbb{F}}_r$ with $r$ elements

Let $X$ be a Fano manifold of pseudo-index at least $3$ such that $c_1(X{)}^2-2c_2(X)$ is nef. Irreducibility of some spaces of rational curves on $X$ implies that a general point of $X$ is contained in a rational surface

Assuming the continuum hypothesis, we show that the Calkin algebra has $2^{{\aleph }_1}$ outer automorphisms