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In an important paper, Zagier proved that certain half-integral weight modular forms are generating functions for traces of polynomials in the $j$-function. It turns out that Zagier's work makes it possible to algorithmically compute…

Consider the classical action of ${\rm GL}_n$ on a sum of $q$ copies of the defining representation and $p$ copies of its dual; by Howe duality, the polynomial functions on this space decompose under the joint action of ${\rm GL}_n$ and…

Let $p$ be an odd prime. Consider normalized newforms $f_1,f_2$ that both satisfy the Heegner hypothesis for an imaginary quadratic field $K$ and suppose that they induce isomorphic residual Galois representations. In the work of…

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix…

We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincar\'e series of a given level $\Gamma_0(N)$ and weight $k\ge 2$ coincides with the field generated by the single-valued periods of a certain motive…

In a recent paper, we derived the leading alpha' corrections to the Kahler potentials for moduli in (0,2) heterotic compactifications. In the same spirit as the LARGE volume stabilization scenario for type IIB orientifolds, we examine…

In this paper we give a classification of the asymptotic expansion of the $q$-expansion of reciprocals of Eisenstein series $E_k$ of weight $k$ for the modular group $\func{SL}_2(\mathbb{Z})$. For $k \geq 12$ even, this extends results of…

Poincar\'e in 1911 and Petersson in 1932 gave the now classical expression for the parabolic Fourier coefficients of holomorphic Poincar\'e series in terms of Bessel functions and Kloosterman sums. Later, in 1941, Petersson introduced…

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences…

By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory…

Extremal BPS-saturated black holes in $N=2$, $d=4$ supergravity can carry electric and magnetic charges $(q^\Lambda_{(m)},q_\Lambda^{(e)})$. It is shown that in smooth cases the moduli fields at the horizon take a fixed "rational" value…

We discuss type I -- heterotic duality in four-dimensional models obtained as a Coulomb phase of the six-dimensional U(16) orientifold model compactified on T^2 with arbitrary SU(16) Wilson lines. We show that Kahler potentials, gauge…

We describe a new class of N=2 topological amplitudes that compute a particular class of BPS terms in the low energy effective supergravity action. Specifically they compute the coupling F^2(\lambda\lambda)^{g-2}(d\phi)^2 where F, \lambda…

We investigate the structure of the moduli space of multiple BPS non-Abelian vortices in U(N) gauge theory with N fundamental Higgs fields, focusing our attention on the action of the exact global (color-flavor diagonal) SU(N) symmetry on…

We present an explicit expression for the topological invariants associated to $SU(2)$ monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding…

We study Fourier term modules on $\mathrm{SU}(2,1)$, which are the modules arising in Fourier expansions of automorphic forms. Maximal unipotent subgroups $N$ of $\mathrm{SU}(2,1)$ are non-abelian, and we consider the ``abelian'' Fourier…

The space of polynomials in two real variables with values in a 2-dimensional irreducible module of a dihedral group is studied as a standard module for Dunkl operators. The one-parameter case is considered (omitting the two-parameter case…

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical…

We give all possible holomorphic Eisenstein series on $\Gamma_0(p)$, of rational weights greater than $2$, and with multiplier systems the same as certain rational-weight eta-quotients at all cusps. We prove they are modular forms and give…

In Type II Calabi-Yau string compactifications, S-duality predicts that suitable generating series of BPS indices counting microstates of D4-D2-D0 black holes are in general mock modular forms of higher depth. The non-holomorphic…