Some combinatorial identities related to commuting varieties and Hilbert schemes

In this article we explore some of the combinatorial consequences of recent results relating the isospectral commuting variety and the Hilbert scheme of points in the plane

Quantization on Curves

Deformation quantization on varieties with singularities offers perspectives that are not found on manifolds. Essential deformations are classified by the Harrison component of Hochschild cohomology, that vanishes on smooth manifolds and reflects information about singularities. The Harrison 2-cochains are symmetric and are interpreted in terms of abelian $*$-products. This paper begins a study of abelian quantization on plane curves over \Crm, being algebraic varieties of the form R2/I where I is a polynomial in two variables; that is, abelian deformations of the coordinate algebra C[x,y]/(I). To understand the connection between the singularities of a variety and cohomology we determine the algebraic Hochschild (co-)homology and its Barr-Gerstenhaber-Schack decomposition. Homology is the same for all plane curves C[x,y]/(I), but the cohomology depends on the local algebra of the singularity of I at the origin.Comment: 21 pages, LaTex format. To appear in Letters Mathematical Physic

Constrained Willmore Surfaces

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy $W=\int H^2$ under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the critical points of arbitrary geometric functionals on the space of immersions under the constraint that the admissible variations infinitesimally preserve the conformal structure. Besides constrained Willmore surfaces we discuss in some detail examples of constrained minimal and volume critical surfaces, the critical points of the area and enclosed volume functional under the conformal constraint.Comment: 17 pages, 8 figures; v2: Hopf tori added as an example, minor changes in presentation, numbering changed; v3: new abstract and appendix, several changes in presentatio

Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams

We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries these numbers of matrices are not polynomials in q (Stembridge 98); however, when the set of entries is a Young diagram, the numbers, up to a power of q-1, are polynomials with nonnegative coefficients (Haglund 98). In this paper, we give a number of conditions under which these numbers are polynomials in q, or even polynomials with nonnegative integer coefficients. We extend Haglund's result to complements of skew Young diagrams, and we apply this result to the case when the set of entries is the Rothe diagram of a permutation. In particular, we give a necessary and sufficient condition on the permutation for its Rothe diagram to be the complement of a skew Young diagram up to rearrangement of rows and columns. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros

of cyclotomic polynomials with interlacing zero

Debye screening

The existence and exponential clustering of correlation functions for a classical coulomb system at low density or high temperature are proven using methods from constructive quantum field theory, the sine gordon transformation and the Glimm, Jaffe, Spencer expansion about mean field theory. This is a vindication of a belief of long standing among physicists, known as Debye screening. That is, because of special properties of the coulomb potential, the configurations of significant probability are those in which the long range parts of r −1 are mostly cancelled, leaving an effective exponentially decaying potential acting between charge clouds. This paper generalizes a previous paper of one of the authors in which these results were obtained for a special lattice system. The present treatment covers the continuous mechanics situation, with essentially arbitrary short range forces and charge species. Charge symmetry is not assumed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46519/1/220_2005_Article_BF01197700.pd

Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.