Alek Vainshtein

Hurwitz numbers and intersections on moduli spaces of curves

This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.

Cluster Algebras and Poisson Geometry

Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping subsets (clusters) of the same cardinality (the rank of the cluster algebra) connected by exchange relations. Examples of cluster algebras include coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. The theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links to a wide range of subjects including representation theory, discrete dynamical systems, Teichmuller theory, and commutative and non-commutative algebraic geometry. This book is the first devoted to cluster algebras. After presenting the necessary introductory material about Poisson geometry and Schubert varieties in the first two chapters, the authors introduce cluster algebras and prove their main properties in Chapter 3. This chapter can be viewed as a primer on the theory of cluster algebras. In the remaining chapters, the emphasis is made on geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.|Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping subsets (clusters) of the same cardinality (the rank of the cluster algebra) connected by exchange relations. Examples of cluster algebras include coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. The theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links to a wide range of subjects including representation theory, discrete dynamical systems, Teichmuller theory, and commutative and non-commutative algebraic geometry. This book is the first devoted to cluster algebras. After presenting the necessary introductory material about Poisson geometry and Schubert varieties in the first two chapters, the authors introduce cluster algebras and prove their main properties in Chapter 3. This chapter can be viewed as a primer on the theory of cluster algebras. In the remaining chapters, the emphasis is made on geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.

Cluster algebras and Weil-Petersson forms

In our previous paper we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of a general matrix of transition exponents. Our leading idea that a relevant geometric object in this case is a certain closed 2-form compatible with the cluster algebra structure. The main example is provided by Penner coordinates on the decorated Teichmueller space, in which case the above form coincides with the classic Weil-Petersson symplectic form.

The Number of Ramified Coverings of the Sphere by the Torus and Surfaces of Higher Genera

Abstract. We obtain an explicit expression for the number of ramified coverings of the sphere by the torus with given ramification type for a small number of ramification points, and conjecture this to be true for an arbitrary number of ramification points. In addition, the conjecture is proved for simple coverings of the sphere by the torus. We obtain corresponding expressions for surfaces of higher genera for small number of ramification points, and conjecture the general form for this number in terms of a symmetric polynomial that appears to be new. The approach involves the analysis of the action of a transposition to derive a system of linear partial differential equations that give the generating series for the desired numbers.

On hurwitz numbers and Hodge integrals

Abstract In this paper we find an explicit formula for the number of topologically different ramified coverings C → CP1 (C is a compact Riemann surface of genus g) with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.

Rational functions and real Schubert calculus

We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real whenever the data are real. Our arguments explore the connection between subspaces of codimension 2 and rational functions of one variable.

Optimal Strategies for Spinning and Blocking

In parallel and distributed computing environments, threads (or processes) share access to variables and data structures. To assure consistency during updates, locks are used. When a thread attempts to acquire a lock but finds it busy, it must choose between spinning, which means repeatedly attempting to acquire the lock in the hope that it will become free, and blocking, which means suspending its execution and relinquishing its processor to some other thread. The choice between spinning and blocking involves balancing the processor time lost to spinning against the processor time required to save the context of a process when it blocks (context switch overhead). In this paper, we investigate a model that permits us to evaluate how long a process should spin before blocking. We determine conditions under which the extreme cases of immediate blocking (no spinning) and pure spinning (spin until the lock is acquired) are optimal. In other cases, we seek ways of estimating an optimal limit on spinning time before blocking. Results are obtained by a combination of analysis and simulation.

Chamber behavior of double Hurwitz numbers in genus 0

We study double Hurwitz numbers in genus zero counting the number of covers

Domination analysis of combinatorial optimization problems

\CP^1\to\CP^1

Poisson geometry of directed networks in an annulus

with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.