Colored graphs and matrix integrals

We discuss applications of generating functions for colored graphs to asymptotic expansions of matrix integrals. The described technique provides an asymptotic expansion of the Kontsevich integral. We prove that this expansion is a refinement of the Kontsevich expansion, which is the sum over the set of classes of isomorphic ribbon graphs. This yields a proof of Kontsevich’s results that is independent of the Feynman graph technique. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 8–24.     

Wheels, wheeling, and the Kontsevich integral of the Unknot

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of uni-trivalent diagrams. The two formulas use the related notions of “Wheels” and “Wheeing”. We prove these formulas ‘on the level of Lie algebras’ using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras. In a brief epilogue we report on recent proofs of our full conjectures, by Kontsevich [Ko2] and by DBN, DPT, and T. Q. T. Le, [BLT]. This paper is available electronically     

M-theory of matrix models

Small M-theories incorporate various models representing a unified family in the same way that the M-theory incorporates a variety of superstring models. We consider this idea applied to the family of eigenvalue matrix models: their M-theory unifies various branches of the Hermitian matrix model (including the Dijkgraaf-Vafa partition functions) with the Kontsevich τ-function. Moreover, the corresponding duality relations are reminiscent of instanton and meron decompositions, familiar from the Yang-Mills theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 179–192, February, 2007.     

On denominators of the Kontsevich integral and the universal perturbative invariant of 3-manifolds

 The integrality of the Kontsevich integral and perturbative invariants is discussed. It is shown that the denominator of the degree n part of the Kontsevich integral of any knot or link is a divisor of (2!3!…n!)⁴(n+1)!. We prove this by establishing the existence of a Drinfeld's associator in the space of chord diagrams with special denominators. We also show that the denominator of the degree n part of the universal perturbative invariant of homology 3-spheres is not divisible by any prime greater than 2n+1. Oblatum 20-VI-1997 & 28-IV-1998 / Published online: 12 November 1998     

The loop expansion of the Kontsevich integral, the null-move and S-equivalence

The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Euler degree, and a geometric null-move on the set of knots. We explain the relation of the null-move to S-equivalence, and the relation to the Euler grading of the Kontsevich integral. The null-move leads in a natural way to the introduction of trivalent graphs with beads, and to a conjecture on a rational version of the Kontsevich integral, formulated by the second author and proven in Geom. Top 8 (2004) 115 (see also Kricker, preprint 2000, math/GT.0005284).     

A Remark on the sl<Subscript>2</Subscript> Approximation of the Kontsevich Integral of the Unknot

The Kontsevich integral of a knot K is a sum over all chord diagrams with suitable coefficients. Here A_n is the space of chord diagrams with n chords. A simple explicit formula for the coefficients a_D is not known even for the unknot. Let E₁, E₂,... be elements of A = ⊕_n A_n. Say that the sum is an sl₂ approximation of the Kontsevich integral if the values of the sl₂ weight system W_sl2 on both sums are equal: W_sl2 (I(K)) = W_sl2 (I′(K)). For any natural n fix points a₁,..., a_2n on a circle. For any permutation σ ∈ S_2n of 2n elements, one defines the chord diagram D(σ) with n chords as the diagram with chords formed by pairs a_{σ (2-1)} and a_σ(2i), i = 1,...,n. It is shown that

On Kontsevich’s characteristic classes for higher dimensional sphere bundles I: the simplest class

This paper studies the simplest one of the sequence of characteristic classes of framed smooth fiber bundles constructed by M. Kontsevich. By introducing a correction term to the characteristic number of the Kontsevich class, we obtain an invariant of unframed sphere bundles over a sphere. The correction term is given by a multiple of Hirzebruch’s signature defect. We observe that a reduction of our invariant modulo a certain integer agrees with a multiple of Milnor’s λ′-invariant of exotic spheres. Furthermore, our invariant is non-trivial for many fiber dimensions. Hence we can detect some ‘exotic’ non-trivial subspace of π _i (Diff(S ^d )) ⊗ for some pairs (i, d) which are not in Igusa’s stable range. Dedicated to Professor Akio Kawauchi for his 60th birthday.     

On rational Drinfeld associators

We prove an estimate on denominators of rational Drinfeld associators. To obtain this result, we prove the corresponding estimate for the p-adic associators stable under the action of suitable elements of Gal([`(\mathbbQ)]/\mathbbQ){{\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q})} . As an application, we settle in the positive Duflo’s question on the Kashiwara–Vergne factorizations of the Jacobson element J _p (x, y) = (x + y) ^p − x ^p − y ^p in the free Lie algebra over a field of characteristic p. Another application is a new estimate on denominators of the Kontsevich knot invariant.     

The Global Mckay-Ruan Correspondence Via Motivic Integration

The purpose of this paper is to show how the methods of motivicintegration of Kontsevich, Denef–Loeser (Invent. Math.135 (1999) 201–232 and Compositio Math. 131 (2002) 267–290)and Looijenga (Astérisque 276 (2002) 267–297) canbe adapted to prove the McKay–Ruan correspondence, a generalizationof the McKay–Reid correspondence to orbifolds that arenot necessarily global quotients. 2000 Mathematics Subject Classification14A20, 14E15, 14F43.     

Matrix Models: Geometry of Moduli Spaces and Exact Solutions

We study the connection between characteristics of moduli spaces of Riemann surfaces with marked points and matrix models. The Kontsevich matrix model describes intersection indices on continuous moduli spaces, and the Kontsevich–Penner matrix model describes intersection indices on discretized moduli spaces. Analyzing the constraint algebras satisfied by various generalized Kontsevich matrix models, we derive time transformations that establish exact relations between different models appearing in mathematical physics. We solve the Hermitian one-matrix model using the moment technique in the genus expansion and construct a recursive procedure for solving this model in the double scaling limit.     

On the compatibility between cup products,the Alekseev–Torossian connection and the Kashiwara–Vergne conjecture,II

We give a different proof of the famous result on compatibility between cup product (Kontsevich, 2003, [3, Section 8]) in cohomology of degree 0, for a finite-dimensional Lie algebra, from which we deduce an alternative way of re-writing Kontsevich?s star product by means of the Alekseev–Torossian connection (Alekseev and Torossian, 2010, [1]).     

G ∞-Formality Theorem in Terms of Graphs and Associated Chevalley–Eilenberg–Harrison Cohomology #

In 1997, M. Kontsevich proved the L _∞-formality conjecture (which implies the existence of star-products for any Poisson manifold) using graphs. A year later, D. Tamarkin gave another proof of a more general conjecture (for G _∞-structures) using operads and cohomological methods. In this article, we show how Tamarkin's construction can be written using graphs. For that, we introduce a generalization of Kontsevich graphs on which we define a “Chevalley–Eilenberg–Harrison” complex. We show that this complex on graphs is related to the “Chevalley–Eilenberg–Harrison” complex for maps on polyvector fields, which is trivial and give Tamarkin's formality theorem as a consequence. This formality reduces to an L _∞-formality.     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998     

Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration

A graph is called a semi-regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficient condition for an automorphism of the graph F to be an automorphism of a map with the underlying graph F is obtained. Using this result, all orientation-preserving automorphisms of maps on surfaces (orientable and non-orientable) or just orientable surfaces with a given underlying semi-regular graph F are determined. Formulas for the numbers of non-equivalent embeddings of this kind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, the non-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable and general surfaces are enumerated.     

The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient C ₀(B, P _1,T ) is the regular value at s = 0 of the zeta function , where B = P ₊ + G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus)—for example the solution operator of a classical elliptic problem—and P _1,T is a realization of an elliptic differential operator P ₁, having a ray free of eigenvalues. Relative formulas (e.g., for the difference between the constants with two different choices of P _1,T ) have been known for some time and are local. We here determine C ₀(B, P _1,T ) itself (with even-order P ₁), showing how it is put together of local residue-type integrals (generalizing the noncommutative residues of Wodzicki, Guillemin, Fedosov–Golse–Leichtnam–Schrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich and Vishik, formed of Hadamard finite parts). Our formula generalizes a formula shown recently by Paycha and Scott for manifolds without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving log P _1,T . Since the complex powers of P _1,T lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here; instead we have developed a resolvent parametric method, where results from our calculus of parameter-dependent boundary operators can be used.     

The mathematics of eigenvalue optimization

 Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briefly on semidefinite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread. Received: December 4, 2002 / Accepted: April 22, 2003 Published online: May 28, 2003 Key Words.  eigenvalue optimization – convexity – nonsmooth analysis – duality – semidefinite program – subdifferential – Clarke regular – chain rule – sensitivity – eigenvalue perturbation – partly smooth – spectral function – unitarily invariant norm – hyperbolic polynomial – stability – robust control – pseudospectrum – H _∞ norm Mathematics Subject Classification (2000): 90C30, 15A42, 65F15, 49K40     

Strong Formulations of Robust Mixed 0–1 Programming

We introduce strong formulations for robust mixed 0–1 programming with uncertain objective coefficients. We focus on a polytopic uncertainty set described by a ``budget constraint' for allowed uncertainty in the objective coefficients. We show that for a robust 0–1 problem, there is an α–tight linear programming formulation with size polynomial in the size of an α–tight linear programming formulation for the nominal 0–1 problem. We give extensions to robust mixed 0–1 programming and present computational experiments with the proposed formulations.     

Volume-preserving Fields and Reeb Fields on 3-manifolds

Volume-preserving field X on a 3-manifold is the one that satisfies LxΩ = 0 for some volume Ω. The Reeb vector field of a contact form is of volume-preserving, but not conversely. On the basis of Geiges-Gonzalo＇s parallelization results, we obtain a volume-preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields. From many aspects, we discuss the distinction between volume-preserving fields and Reeb-like fields. We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction. We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction.     

One-regular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer

A graph G is one-regular if its automorphism group Aut（G） acts transitively and semiregularly on the arc set. A Cayley graph Cay（Г, S） is normal if Г is a normal subgroup of the full automorphism group of Cay（Г, S）. Xu, M. Y., Xu, J. （Southeast Asian Bulletin of Math., 25, 355-363 （2001）） classified one-regular Cayley graphs of valency at most 4 on finite abelian groups. Marusic, D., Pisanski, T. （Croat. Chemica Acta, 73, 969-981 （2000）） classified cubic one-regular Cayley graphs on a dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4-valent one-regular normal Cayley graphs G on a dihedral group whose vertex stabilizers in Aut（G） are cyclic. A classification of the same kind of graphs of valency 6 is also discussed.     

Numerical solution to the optimal feedback control of continuous casting process

Using a semi-discrete model that describes the heat transfer of a continuous casting process of steel, this paper is addressed to an optimal control problem of the continuous casting process in the secondary cooling zone with water spray control. The approach is based on the Hamilton–Jacobi–Bellman equation satisfied by the value function. It is shown that the value function is the viscosity solution of the Hamilton–Jacobi–Bellman equation. The optimal feedback control is found numerically by solving the associated Hamilton–Jacobi–Bellman equation through a designed finite difference scheme. The validity of the optimality of the obtained control is experimented numerically through comparisons with different admissible controls. Detailed study of a low-carbon billet caster is presented.