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The Burgers--KdV hierarchy was introduced towards understanding intersection numbers on the moduli space of Riemann surfaces with boundaries. The goal of this paper is to derive the Dubrovin--Zhang type loop equation for the topological…
By elaborating on the recent progress made in the area of Feynman integrals, we apply the intersection theory for twisted de Rham cohomologies to simple integrals involving orthogonal polynomials, matrix elements of operators in Quantum…
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting…
In two-dimensional statistical physics, correlation functions of the O(N) and Potts models may be written as sums over configurations of non-intersecting loops. We define sums associated to a large class of combinatorial maps (also known as…
The models of triangulated random surfaces embedded in (extended) Dynkin diagrams are formulated as a gauge-invariant matrix model of Weingarten type. The double scaling limit of this model is described by a collective field theory with…
The Matrix String Theory, i.e. the two dimensional U(N) SYM with N=(8,8) supersymmetry, has classical BPS solutions that interpolate between an initial and a final string configuration via a bordered Riemann surface. The Matrix String…
We consider the random matrix ensemble with an external source \[ \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM \] defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with only two eigenvalues $\pm a$ of equal…
The paper contains some new results and a review of recent achievements, concerning the multisupport solutions to matrix models. In the leading order of the 't Hooft expansion for matrix integral, these solutions are described by…
Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller…
Let A be a commutative noetherian local DG-ring with bounded cohomology. The Intersection Theorem for DG-modules is examined and some of its applications are provided. The first is to prove the DG-setting of the amplitude inequality, New…
A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the…
Among solutions of n-Gelfand-Dikii's hierarchy there exists a remarkable solution W, which satisfies the string equation. We call it Witten's solution because according to the Witten conjecture the function F(x_1, x_2, x_3,...) =…
We investigate the geometry of the matrix model associated with an N=1 super Yang-Mills theory with three adjoint fields, which is a massive deformation of N=4. We study in particular the Riemann surface underlying solutions with arbitrary…
Gaussian and Chiral Beta-Ensembles, which generalise well known orthogonal (Beta=1), unitary (Beta=2), and symplectic (Beta=4) ensembles of random Hermitian matrices, are considered. Averages are shown to satisfy duality relations like…
We study intersection theory and Chern classes of reflexive sheaves on normal varieties. In particular, we define generalization of Mumford's intersection theory on normal surfaces to higher dimensions. We also define and study the second…
We calculate the genus zero cobordism-valued Gromov-Witten invariants of a point by refining the string equation on $\overline{\mathcal{M}}_{0,n}$ from the Chow ring to algebraic cobordism. This gives inductive formulas for cobordism-valued…
In an earlier work we had considered a Gaussian ensemble of random matrices in the presence of a given external matrix source. The measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials, or on the…
We associate a noncommutative curve to a periodic, bipartite, planar dimer model with polygonal boundary. It determines the inverse Kasteleyn matrix and hence all correlations. It may be seen as a quantization of the limit shape…
We derive an explicit generating function of correlations functions of an arbitrary tau-function of the KdV hierarchy. In particular applications, our formulation gives closed formul\ae\ of a new type for the generating series of…
Several intersection matrices of $s$-subsets vs. $k$-subsets of a $v$-set are introduced in the literature. We study these matrices systematically through counting arguments and generating function techniques. A number of new or known…