Wall crossings for double Hurwitz numbers

Double Hurwitz numbers count covers of P¹ by genus g curves with assigned ramification profiles over 0 and ∞, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification (Goulden et al., 2005) [10], and Shadrin, Shapiro and Vainshtein have determined the chamber structure and wall crossing formulas for g=0 (Shadrin et al., 2008) [15]. This paper gives a unified approach to these results and strengthens them in several ways — the most important being the extension of the results of Shadrin et al. (2008) [15] to arbitrary genus.The main tool is the authors? previous work (Cavalieri et al., 2010) [6] expressing double Hurwitz number as a sum over certain labeled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko (1987) [17], and could have broader applications.     

Left and right centers in quasi-Poisson geometry of moduli spaces

We introduce left central and right central functions and left and right leaves in quasi-Poisson geometry, generalizing central (or Casimir) functions and symplectic leaves from Poisson geometry. They lead to a new type of (quasi-)Poisson reduction, which is both simpler and more general than known quasi-Hamiltonian reductions. We study these notions in detail for moduli spaces of flat connections on surfaces, where the quasi-Poisson structure is given by an intersection pairing on homology.     

Geometry of tropical moduli spaces and linkage of graphs

We prove the following “linkage” theorem: two p-regular graphs of the same genus can be obtained from one another by a finite alternating sequence of one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the linkage theorem to prove that various moduli spaces of tropical curves are connected through codimension one.     

Witten's formulas for intersection pairings on moduli spaces of flat G-bundles

In a 1992 paper (J. Geom. Phys. 9 (1992) 303), Witten gave a formula for the intersection pairings of the moduli space of flat G-bundles over an oriented surface, possibly with markings. In this paper, we give a general proof of Witten's formula, for arbitrary compact, simple groups, and any markings for which the moduli space has at most orbifold singularities.     

Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of Abelian threefolds

We prove that the moduli spaces of polarized Abelian threefolds with polarizations of types D=(1,1,2),(1,2,2),(1,1,3) or (1,3,3) are unirational. The result is based on the study of families of simple coverings of elliptic curves of degree 2 or 3 and on the study of the corresponding period mappings associated with holomorphic differentials with trace 0. In particular we prove the unirationality of the Hurwitz space which parametrizes simply branched triple coverings of an elliptic curve Y with determinants of the Tschirnhausen modules isomorphic to A^-1. Dedicated to the memory of Fabio BardelliMathematics Subject Classification (2000) Primary: 14K10; Secondary: 14H10, 14H30, 14D07     

Deformation theory of objects in homotopy and derived categories I: General theory

This is the first paper in a series. We develop a general deformation theory of objects in homotopy and derived categories of DG categories. Namely, for a DG module E over a DG category we define four deformation functors Defh(E), coDefh(E), Def(E), coDef(E). The first two functors describe the deformations (and co-deformations) of E in the homotopy category, and the last two - in the derived category. We study their properties and relations. These functors are defined on the category of artinian (not necessarily commutative) DG algebras.     

Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

In this paper, we define the virtual moduli cycle of moduli spaces with perfect tangent-obstruction theory. The two interesting moduli spaces of this type are moduli spaces of vector bundles over surfaces and moduli spaces of stable morphisms from curves to projective varieties. As an application, we define the Gromov-Witten invariants of smooth projective varieties and prove all its basic properties.

     

Symplectic Automorphisms and the Picard Group of a K3 Surface

We consider the symplectic action of a finite group G on a K3 surface X. The Picard group of X has a primitive sublattice determined by G. We show how to compute the rank and discriminant of this sublattice. We then investigate the classification of symplectic actions by a fixed finite group, using moduli spaces of K3 surfaces with symplectic G-action.     

Matrix models, complex geometry, and integrable systems: II

We consider certain examples of applications of the general methods based on geometry and integrability of matrix models. These methods were described in the first part of this paper. In particular, we investigate the nonlinear differential equations satisfied by semiclassical tau functions. We also discuss a similar semiclassical geometric picture arising in the context of multidimensional supersymmetric gauge theories and the AdS/CFT correspondence. [This article was written at the request of the Editorial Board. It is based on several lectures presented at schools of mathematical physics and talks at the conference “Complex Geometry and String Theory” and the Polivanov memorial seminar.] __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 399–449, June, 2006.     

Matrix models, complex geometry, and integrable systems: I

We consider the simplest gauge theories given by one-and two-matrix integrals and concentrate on their stringy and geometric properties. We recall the general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. We study the main ingredients of this geometric picture, suggesting that it can be generalized beyond one complex dimension, and formulate them in terms of semiclassical integrable systems solved by constructing tau functions or prepotentials. We discuss the complex curves and tau functions of one-and two-matrix models in detail. [This article was written at the request of the Editorial Board. It is based on several lectures presented at schools of mathematical physics and talks at the conference “Complex Geometry and String Theory” and the Polivanov memorial seminar.] __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 163–228, May, 2006.     

Deformation theory of objects in homotopy and derived categories II: Pro-representability of the deformation functor

This is the second paper in a series. In part I we developed deformation theory of objects in homotopy and derived categories of DG categories. Here we extend these (derived) deformation functors to an appropriate bicategory of artinian DG algebras and prove that these extended functors are pro-representable in a strong sense.     

Phase Spaces and Deformation Theory

In the papers (Laudal in Contemporary Mathematics, vol. 391, [2005]; Geometry of time-spaces, Report No. 03, [2006/2007]), we introduced the notion of (non-commutative) phase algebras (spaces) Ph ⁿ (A), n=0,1,…,∞ associated to any associative algebra A (space), defined over a field k. The purpose of this paper is to study this construction in some more detail. This seems to give us a possible framework for the study of non-commutative partial differential equations. We refer to the paper (Laudal in Phase spaces and deformation theory, Report No. 09, [2006/2007]), for the applications to non-commutative deformation theory, Massey products and for the construction of the versal family of families of modules. See also (Laudal in Homology, Homotopy, Appl. 4:357–396, [2002]; Proceedings of NATO Advanced Research Workshop, Computational Commutative and Non-Commutative Algebraic Geometry, [2004]).      

Jet schemes and minimal toric embedded resolutions of rational double point singularities

Using the structure of the jet schemes of rational double point singularities, we construct “minimal embedded toric resolutions” of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every “minimal embedded toric resolution”. We prove that this correspondence is bijective except for the E₈ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.     

An efficient algorithm for accelerating the convergence of oscillatory series,useful for computing the polylogarithm and Hurwitz zeta functions

This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function” by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating Li _s (z) for general values of complex s and a kidney-shaped region of complex z values given by ∣z ²/(z–1)∣<4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler–Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler–Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor’s series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.      

Analysis and geometry of Floer theory of Landau-Ginzburg model on ℂn

This article studies the Floer theory of Landau-Ginzburg (LG) model on ℂⁿ: We perturb the Kähler form within a xed Kähler class to guarantee the transversal intersection of Lefschetz thimbles. The C⁰ estimate for solutions of the LG Floer equation can be derived then by our analysis tools. The Fredholm property is guaranteed by all these results.     

Effect of inquiry-based instruction enriched with origami activities on achievement,and self-efficacy in geometry

The purpose of the study was to investigate the effect of inquiry-based instruction enriched with origami activities on 7th grade students’ achievement in reflection symmetry and self-efficacy in geometry. Two classes, instructed by the first author of the paper, were randomly assigned as experimental and control groups. In order to gather data, participants were administered Reflection Symmetry Achievement Test, and Geometry Self-Efficacy Scale as pre-test and post-test. The Analysis of Covariance was performed in order to answer the research questions. Moreover, five participants were interviewed to examine self-efficacy sources which are determinant of the change in self-efficacy levels. Findings revealed that the inquiry-based instruction enriched with origami activities had a significantly positive effect on students’ achievement in reflection symmetry and self-efficacy in geometry. Interviews showed that all four sources were influenced by the intervention.     

An explanatory,transformation geometry proof of a classic treasure-hunt problem and its generalization

This paper discusses an interesting, classic problem that provides a nice classroom investigation for dynamic geometry, and which can easily be explained (proved) with transformation geometry. The deductive explanation (proof) provides insight into why it is true, leading to an immediate generalization, thus illustrating the discovery function of logical reasoning (proof).     

Partial geometries pg (s, t, 2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry

Let be a proper partial geometry pg(s,t,2), and let G be an abelian group of automorphisms of acting regularly on the points of . Then either t≡2±od s+1 or is a pg(5,5,2) isomorphic to the partial geometry of van Lint and Schrijver (Combinatorica 1 (1981), 63–73). This result is a new step towards the classification of partial geometries with an abelian Singer group and further provides an interesting characterization of the geometry of van Lint and Schrijver.The author is Postdoctoral Fellow of the Fund for Scientific Research Flanders (FWO-Vlaanderen).     

Simplicial collapsibility,discrete Morse theory,and the geometry of nonpositively curved simplicial complexes

Understanding the conditions under which a simplicial complex collapses is a central issue in many problems in topology and combinatorics. Let K be a finite simplicial complex of dimension three or less endowed with the piecewise Euclidean geometry given by declaring edges to have unit length, and satisfying the property that every 2-simplex is a face of at most two 3-simplices in K. Our main result is that if |K| is nonpositively curved [in the sense of CAT(0)] then K simplicially collapses to a point. The main tool used in the proof is Forman’s discrete Morse theory, a combinatorial analog of the classical smooth theory developed in the 1920s. A key ingredient in our proof is a combinatorial analog of the fact that a minimal surface in has nonpositive Gauss curvature.