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Mon, 25 Jan 2010 18:12:06 PST

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Mon, 25 Jan 2010 18:12:06 PST

Sigma Function as A Tau Function

Nakayashiki, A. — Mon, 25 Jan 2010 18:12:07 PST

The tau function corresponding to the affine ring of a certain plane algebraic curve, called (n, s)-curve, embedded in the universal Grassmann manifold is studied. It is neatly expressed by the multivariate sigma function. This expression is in turn used to prove fundamental properties on the series expansion of the sigma function established in a previous article in a different method.

The Infinitesimal Hopf Algebra and the Operads of Planar Forests

Foissy, L. — Mon, 25 Jan 2010 18:12:07 PST

We introduce two operads based on the set of planar forests. With its usual product and two other products defined by different types of graftings, the algebra of planar rooted trees becomes an algebra over these operads. The compatibility with the infinitesimal coproduct of and these structures is studied. As an application, an inductive way of computing the dual basis of for its infinitesimal pairing is given. Moreover, three Cartier–Quillen–Milnor–Moore theorems are given for the operads of planar forests and a rigidity theorem for one of them.

Wegner Estimate and Level Repulsion for Wigner Random Matrices

Erdos, L., Schlein, B., Yau, H.-T. — Mon, 25 Jan 2010 18:12:07 PST

We consider N x N Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order 1/ N. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales >> N^–1. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result [6]. We then show a Wegner estimate, i.e., that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.

Elementary Combinatorics of the HOMFLYPT Polynomial

Chmutov, S., Polyak, M. — Mon, 25 Jan 2010 18:12:07 PST

We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.

Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian

Ferapontov, E. V., Hadjikos, L., Khusnutdinova, K. R. — Mon, 25 Jan 2010 18:12:07 PST

We investigate integrable second-order equations of the form which typically arise as the Hirota-type relations for various (2 + 1)-dimensional dispersionless hierarchies. Familiar examples include the Boyer–Finley equation , the potential form of the dispersionless Kadomtsev–Petviashvili (dKP) equation , the dispersionless Hirota equation , etc. The integrability is understood as the existence of an infinity of hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit.

Isotropic Curvature and the Ricci Flow

Nguyen, H. T. — Mon, 25 Jan 2010 18:12:07 PST

In this paper, we study the Ricci flow on higher dimensional compact manifolds. We prove that nonnegative isotropic curvature is preserved by the Ricci flow in dimensions greater than or equal to four. In order to do so, we introduce a new technique to prove that curvature functions defined on the orthonormal frame bundle are preserved by the Ricci flow. At a minimum of such a function, we compute the first and second derivatives in the frame bundle. Using an algebraic construction, we can use these expressions to show that the nonlinearity is positive at a minimum. Finally, using the maximum principle, we can show that the Ricci flow preserves the cone of curvature operators with nonnegative isotropic curvature.

Combinatorial R-Matrices for Kirillov-Reshetikhin Crystals of Type D(1)n, B(1)n, A(2)2n-1

Okado, M., Sakamoto, R. — Mon, 25 Jan 2010 18:12:07 PST

We calculate the image of the combinatorial R-matrix for any classical highest weight element in the tensor product of Kirillov–Reshetikhin crystals B^r,k B^1,l of type D⁽¹⁾_n, B⁽¹⁾_n, A⁽²⁾_2n–1. The notion of ±-diagrams is effectively used for the identification of classical highest weight elements in B^1,l B^r,k.