Narrow Results

We review some recent results on the problem of the choice of polarization in geometric quantization. Specifically, we describe the general philosophy, developed by the author together with his collaborators, of treating real polarizations as limits of degenerating families of holomorphic polarizations. We first review briefly the general framework of geometric quantization, with a particular focus on the problem of the dependence of quantization on the choice of polarization. The problem of quantization in real polarizations is emphasized. We then describe the relation between quantization in real and Kähler polarizations in some families of symplectic manifolds, that can be explicitly quantized and that constitute an important class of examples: cotangent bundles of Lie groups, abelian varieties and toric varieties. Applications to theta functions and moduli spaces of vector bundles on curves are also reviewed.

In this paper we extend the result on base point freeness of the powers of the determinant bundle on the moduli space of vector bundles on a curve. We describe the parabolic analogues of parabolic theta functions, then we determine a uniform bound depending only on the rank of the parabolic bundles. In order to get this bound, we construct a parabolic analogue of Grothendieck's scheme of quotients, which parametrizes quotient bundles of a parabolic bundle, of fixed parabolic Hilbert polynomial. We prove an estimate for its dimension, which extends the result of Popa and Roth on the dimension of the Quot scheme. As an application of the theorem on base point freeness, we characterize parabolic semistability on the algebraic stack of quasi-parabolic bundles as the base locus of the linear system of the parabolic determinant bundle.

We introduce various Ruelle type zeta functions ζ_L(s) according to a choice of homogeneous "length functions" for a lattice L in via Euler products. The logarithm of each ζ_L(s) yields naturally a certain arithmetic function. We study the asymptotic distribution of averages of such arithmetic functions. Asymptotic behavior of the zeta functions at the origin s=0 are also investigated.

It is known that holomorphic sections of an ample line bundle L (and its tensor power L^k) over an Abelian variety A are given by theta functions. Moreover, a natural basis of the space of holomorphic sections of L or L^k is related to a certain Lagrangian fibration of A. In our previous paper, we studied projective embeddings of A defined by these basis for L^k. For a natural torus action on the ambient projective space, it is proved that its moment map, restricted to A, approximates the Lagrangian fibration of A for large k, with respect to the "Gromov–Hausdorff topology". In this paper, we prove that the same is true for the Kummer variety associated to A.

We propose a potential function $W$ for the cohomology ring of partial flag manifolds. We prove a formula expressing integrals over partial flag manifolds by residues, which generalizes [E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, in Geometry, Topology, Physics (International Press, 1995), pp. 357–422]. Using this formula, we prove a Landweber–Stong type vanishing theorem for generalized ${\mathrm{\text{string}}}^c$ complete intersections in flag manifolds, which serves as evidence for the ${\mathrm{\text{string}}}^c$ version of Stolz conjecture [Q. Chen, F. Han and W. Zhang, Generalized Witten genus and vanishing theorems, J. Differential Geom.88(1) (2011) 1–39].

The goal of this paper is to investigate the optimality of the $d$-dimensional rock-salt structure, i.e. the cubic lattice $V^{1/d}{\mathbb{Z}}^d$ of volume $V$ with an alternation of charges $\pm 1$ at lattice points, among periodic distributions of charges and lattice structures. We assume that the charges are interacting through two types of radially symmetric interaction potentials, according to their signs. We first restrict our study to the class of orthorhombic lattices. We prove that, for our energy model, the $d$-dimensional rock-salt structure is always a critical point among periodic structures of fixed density. This holds for a large class of potentials. We then investigate the minimization problem among orthorhombic lattices with an alternation of charges for inverse power laws and Gaussian interaction potentials. High density minimality results and low-density non-optimality results are derived for both types of potentials.

Numerically, we investigate several particular cases in dimensions $2$, $3$ and $8$. The numerics support the conjecture that the rock-salt structure is the global optimum among all lattices and periodic charges, satisfying some natural constraints. For $d=2$, we observe a phase transition of the type “triangular-rhombic-square-rectangular” for the minimizer’s shape as the density decreases.

We prove in this paper that the minimizer of Lennard–Jones energy per particle among Bravais lattices is a triangular lattice, i.e. composed of equilateral triangles, in ℝ² for large density of points, while it is false for sufficiently small density. We show some characterization results for the global minimizer of this energy and finally we also prove that the minimizer of the Thomas–Fermi energy per particle in ℝ² among Bravais lattices with fixed density is triangular.

We prove a conjecture by Brown, Dilcher and Manna on the asymptotic behavior of sparse binomial-type polynomials arising naturally in a graph-theoretical context in connection with the expected number of independent sets of a graph.

Two fundamental theta identities, a three-term identity due to Weierstrass and a five-term identity due to Jacobi, both with products of four theta functions as terms, are shown to be equivalent. One half of the equivalence was already proved by R. J. Chapman in 1996. The history and usage of the two identities, and some generalizations are also discussed.

We study results related to a conjecture formulated by Strohmer and Beaver about optimal Gaussian Gabor frame set-ups. Our attention will be restricted to the case of Gabor systems with standard Gaussian window and rectangular lattices of density 2. Although this case has been fully treated by Faulhuber and Steinerberger, the results in this work are new and quite curious. Indeed, the optimality of the square lattice for the Tolimieri and Orr bound already implies the optimality of the square lattice for the sharp lower frame bound. Our main tools include determinants of Laplace–Beltrami operators on tori as well as special functions from analytic number theory, in particular Eisenstein series, zeta functions, theta functions and Kronecker’s limit formula. We note that our results also carry over to energy minimization problems over lattices and a heat distribution problem over flat tori.

We present a new class of elliptic-like strings on two-dimensional manifolds of constant curvature. Our solutions are related to a class of identities between Jacobi theta functions and to the geometry of the light-cone in one dimension more. We show in particular that two well-known fundamental identities among theta functions have a natural interpretation as expressing the Virasoro constraints of dS or AdS strings.

Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form

Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form

In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four-parameter theta-function identity due to Jacobi. New shifted and shiftless partition identities are proved.

Some new theta function identities are proved and used to determine the number of representations of a positive integer n by certain quaternary quadratic forms.

We describe an mth order generalization of Jacobi's theta functions and use these functions to construct classes of theta function identities in multiple variables. These identities are equivalent to the Macdonald identities for the seven infinite families of irreducible affine root systems. They are also equivalent to some elliptic determinant evaluations proven recently by Rosengren and Schlosser.

Runge gave the ring of genus three Siegel modular forms as a quotient ring, R₃/〈J⁽³⁾〉 where R₃ is the genus three ring of code polynomials and J⁽³⁾ is the difference of the weight enumerators for the e₈ ⊕ e₈ and codes. Freitag and Oura gave a degree 24 relation, , of the corresponding ideal in genus four; where is also a linear combination of weight enumerators. We take another step towards the ring of Siegel modular forms in genus four. We explain new techniques for computing with Siegel modular forms and actually compute six new relations, classifying all relations through degree 32. We show that the local codimension of any irreducible component defined by these known relations is at least 3 and that the true ideal of relations in genus four is not a complete intersection. Also, we explain how to generate an infinite set of relations by symmetrizing first order theta identities and give one example in degree 32. We give the generating function of R₅ and use it to reprove results of Nebe [25] and Salvati Manni [37].

We derive relations between theta functions evaluated at period matrices of cyclic covers of order 3 ramified above 3k points.

Some theta function identities are proved and used to give formulae for the number of representations of a positive integer by certain quaternary forms x² + ey² + fz² + gt² with e, f, g ∈ {1, 2, 4, 8}.

Formulae, originally conjectured by Liouville, are proved for the number of representations of a positive integer n by each of the eight sextenary quadratic forms , , , , , , , .

We give a bijective proof of the quintuple product identity using bijective proofs of Jacobi's triple product identity and Euler's recurrence relation.

In this paper, using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we establish some theta function identities. From these identities, we obtain some formulas for the number of representations of a natural number as a sum of quadratic polynomials involving generalized pentagonal numbers. In particular, we derive a formula for the number of representations of a natural number as a sum of twelve generalized pentagonal numbers.