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Heat Kernels and Dirac Operators
The past few years have seen the emergence of new insights into the Atiyah-Singer Index Theorem for Dirac operators. In this book, elementary proofs of this theorem, and some of its more recent
On the Segal-Shale-Weil representations and harmonic polynomials
In this paper, we give the answer to the following two intimately related problems. (a) To decompose the tensor products of the harmonic representations into irreducible components to get a series of
Lattice points in simple polytopes
in terms of fP(h) q(x)dx where the polytope P(h) is obtained from P by independent parallel motions of all facets. This extends to simple lattice polytopes the EulerMaclaurin summation formula of
Residue formulae, vector partition functions and lattice points in rational polytopes
We obtain residue formulae for certain functions of several vari- ables. As an application, we obtain closed formulae for vector partition func- tions and for their continuous analogs. They imply an
Kostant Partitions Functions and Flow Polytopes
This paper discusses volumes and Ehrhart polynomials in the context of flow polytopes. The general approach that studies these functions via rational functions with poles on arrangement of
Modular forms of weight 1/2
We will construct θ-series of weight 1/2 and 3/2 for some congruence subgroups of SL(2,ℤ) by taking appropriate coefficients of the representation \( \mathop{R}\limits^{\sim } \) of SL(2;ℝ),
Residues formulae for volumes and Ehrhart polynomials of convex polytopes.
In these notes, we explain residue formulae for volumes of convex polytopes, and for Ehrahrt polynomials based on the notion of total residue. We apply this method to the computation of the volume of
How to integrate a polynomial over a simplex
It is proved that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus, and if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, it is proven that integration can be done inPolynomial time.
Toric reduction and a conjecture of Batyrev and Materov
We present a new integration formula for intersection numbers on toric quotients, extending the results of Witten, Jeffrey and Kirwan on localization. Our work was motivated by the Toric Residue