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This paper extends some results of Hatcher and Quinn beyond the metastable range. We give a bordism theoretic obstruction to deforming a map between manifolds simultaneously off of a finite collection of pairwise disjoint submanifolds. In a certain range of dimensions, our obstruction is the entire story.

Let M be a 1-cusped hyperbolic 3-manifold whose cusp shape is not quadratic, and M(p/q) be its p/q-Dehn filled manifold. We show that if p/q $\neq$ p'/q' for sufficiently large |p|+|q| and |p'|+|q'|, there is no orientation preserving isometry between M(p/q) and M(p'/q'). This resolves the conjecture of C. Gordon, which is so called the Cosmetic Surgery Conjecture, for hyperbolic 3-manifolds belonging to the aforementioned class except for possibly finitely many exceptions for each manifold. We also consider its generalization to more cusped manifolds. The key ingredient of the proof is the unlikely intersection theory developed by E. Bombieri, D. Masser and U. Zannier.

Let M be a 1-cusped hyperbolic 3-manifold whose cusp shape is not quadratic, and M(p/q) be its p/q-Dehn filled manifold. We show that if p/q $\neq$ p'/q' for sufficiently large |p|+|q| and |p'|+|q'|, there is no orientation preserving isometry between M(p/q) and M(p'/q'). This resolves the conjecture of C. Gordon, which is so called the Cosmetic Surgery Conjecture, for hyperbolic 3-manifolds belonging to the aforementioned class except for possibly finitely many exceptions for each manifold. We also consider its generalization to more cusped manifolds. The key ingredient of the proof is the unlikely intersection theory developed by E. Bombieri, D. Masser and U. Zannier.

Two faculty members of the Division of Education at Neumann College (Pennsylvania) have collaborated in the development of a cooperative learning based teacher preparation curriculum in early childhood education and elementary education. At every level of pre-service training, students enrolled in these programs are exposed to activities, strategies, and assignments that incorporate cooperative learning. The curriculum is grounded in the relationship that is established between cooperative learning and the three domains of learning: cognitive, affective, and psychomotor. Ultimately, students become fully engaged in recognizing when the application of cooperative learning to instruction is appropriate, why its application benefits the learner, and how to make the most effective application of it. (Contains 15 references.) (Author)

In these notes we survey the tropical intersection theory on R^n by deriving the properties for tropical cycles from the corresponding properties in Chow cohomology. For this we review the stable intersection product introduced by Mikhalkin and the push forward of tropical cycles defined by Allermann and Rau. Furthermore we define a pull back for tropical cycles based on the pull back of Minkowski weights. This pull back commutes with the tropical intersection product and satisfies the projection formula. Our main result is to deduce the latter from the corresponding projection formula in Chow cohomology.

Formulas are developed for evaluating the propagation of the various errors present in a system of photogrammetric cameras in triangulating the space position of a photographed point.

11 p. 28 cm

For an operator in a possibly infinite-dimensional Hilbert space of a certain class, we set down axioms of an abstract intersection theory, from which the Riemann hypothesis regarding the spectrum of that operator follows. In our previous paper [BU] we constructed a GNS (Gelfand-Naimark-Segal) model of abstract intersection theory. In this paper we propose another model, which we call a standard model of abstract intersection theory. We show that there is a standard model of abstract intersection theory for a given operator if and only if the Riemann hypothesis and semi-simplicity hold for that operator. (For the definition of semi-simplicity of an operator in Hilbert space, see the definition in Introduction.) We show this result under a condition for a given operator which is much weaker than the condition in the previous paper. The operator satisfying this condition can be constructed by the method of automorphic scattering in [U]. Combining this with a result from [U], we can show that an Dirichlet $L$-function, including the Riemann zeta-function, satisfies the Riemann hypothesis and its all nontrivial zeros are simple if and only if there is a corresponding standard model of abstract intersection theory. Similar results can be proven for GNS models since the same technique of proof for standard models can be applied.

For an operator of a certain class in Hilbert space, we introduce axioms of an abstract intersection theory, which we prove to be equivalent to the Riemann Hypothesis concerning the spectrum of that operator. In particular if the nontrivial zeros of the Riemann zeta-function arise from an operator of this class, the original Riemann Hypothesis is equivalent to the existence of an abstract intersection theory.

We define a bordism invariant for the fiberwise intersection theory. Under some certain conditions, this invariant is an obstruction for the theory.

We show how to equip the cone complexes of toroidal embeddings with additional structure that allows to define a balancing condition for weighted subcomplexes. We then proceed to develop the foundations of an intersection theory on cone complexes including push-forwards, intersections with tropical divisors, and rational equivalence. These constructions are shown to have an algebraic interpretation: Ulirsch's tropicalizations of subvarieties of toroidal embeddings carry natural multiplicities making them tropical cycles, and the induced tropicalization map for cycles respects push-forwards, intersections with boundary divisors, and rational equivalence. As an application we prove a correspondence between the genus 0 tropical descendant Gromov-Witten invariants introduced by Markwig and Rau and the genus 0 logarithmic descendant Gromov-Witten invariants of toric varieties.

In this paper, an intersection theory for generic differential polynomials is presented. The intersection of an irreducible differential variety of dimension $d$ and order $h$ with a generic differential hypersurface of order $s$ is shown to be an irreducible variety of dimension $d-1$ and order $h+s$. As a consequence, the dimension conjecture for generic differential polynomials is proved. Based on the intersection theory, the Chow form for an irreducible differential variety is defined and most of the properties of the Chow form in the algebraic case are established for its differential counterpart. Furthermore, the generalized differential Chow form is defined and its properties are proved. As an application of the generalized differential Chow form, the differential resultant of $n+1$ generic differential polynomials in $n$ variables is defined and properties similar to that of the Macaulay resultant for multivariate polynomials are proved.

This is the third of a series of papers relating intersections of special cycles on the integral model of a Shimura surface to Fourier coefficients of Hilbert modular forms. More precisely, we embed the Shimura curve over Q associated to a rational quaternion algebra into the Shimura surface associated to the base change of the quaternion algebra to a real quadratic field. After extending the associated moduli problems over Z we obtain an arithmetic threefold with a embedded arithmetic surface, which we view as a cycle of codimension one. We then construct a family, indexed by totally positive algebraic integers in the real quadratic field, of codimension two cycles (complex multiplication points) on the arithmetic threefold. The intersection multiplicities of the codimension two cycles with the fixed codimension one cycle are shown to agree with the Fourier coefficients of a (very particular) Hilbert modular form of weight 3/2. The results are higher dimensional variants of results of Kudla-Rapoport-Yang, which relate intersection multiplicities of special cycles on the integral model of a Shimura curve to Fourier coefficients of a modular form in two variables.

We introduce a notion of super-potential for positive closed currents of bidegree (p,p) on projective spaces. This gives a calculus on positive closed currents of arbitrary bidegree. We define in particular the intersection of such currents and the pull-back operator by meromorphic maps. One of the main tools is the introduction of structural discs in the space of positive closed currents which gives a "geometry" on that space. We apply the theory of super-potentials to construct Green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms.

The operational Chow cohomology classes of a complete toric variety are identified with certain functions, called Minkowski weights, on the corresponding fan. The natural product of Chow cohomology classes makes the Minkowski weights into a commutative ring; the product is computed by a displacement in the lattice, which corresponds to a deformation in the toric variety. We show that, with rational coefficients, this ring embeds in McMullen's polytope algebra, and that the polytope algebra is the direct limit of these Chow rings, over all compactifications of a given torus. In the nonsingular case, the Minkowski weight corresponding to the Todd class is related to a certain Ehrhart polynomial.

The operational Chow cohomology classes of a complete toric variety are identified with certain functions, called Minkowski weights, on the corresponding fan. The natural product of Chow cohomology classes makes the Minkowski weights into a commutative ring; the product is computed by a displacement in the lattice, which corresponds to a deformation in the toric variety. We show that, with rational coefficients, this ring embeds in McMullen's polytope algebra, and that the polytope algebra is the direct limit of these Chow rings, over all compactifications of a given torus. In the nonsingular case, the Minkowski weight corresponding to the Todd class is related to a certain Ehrhart polynomial.

The operational Chow cohomology classes of a complete toric variety are identified with certain functions, called Minkowski weights, on the corresponding fan. The natural product of Chow cohomology classes makes the Minkowski weights into a commutative ring; the product is computed by a displacement in the lattice, which corresponds to a deformation in the toric variety. We show that, with rational coefficients, this ring embeds in McMullen's polytope algebra, and that the polytope algebra is the direct limit of these Chow rings, over all compactifications of a given torus. In the nonsingular case, the Minkowski weight corresponding to the Todd class is related to a certain Ehrhart polynomial.

We study intersection theory on the relative Hilbert scheme of a family of nodal-or-smooth curves, over a base of arbitrary dimension. We introduce an additive group called 'discriminant module', generated by diagonal loci, node scrolls, and twists thereof, and determine the action of the discriminant or big diagonal divisor on this group by intersection. We show that this suffices to determine arbitrary polynomials in Chern classes, in particular Chern numbers, for the tautological vector bundles on the Hilbert schemes, which are closely related to enumerative geometry. The latest version includes some new explicit computations and reference to a computer program due to G. Liu implementing our intersection calculus.

We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on the relation between tropical and complex intersection theories which is also established here. We give two applications of the methods developed in this paper. First we classify all locally irreducible approximable 3-valent fan tropical curves in a non-singular fan tropical plane. Secondly, we prove that a generic non-singular tropical surface in tropical projective 3-space contains finitely many approximable tropical lines if it is of degree 3, and contains no approximable tropical lines if it is of degree 4 or more.

Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate hyperplanes, these formulae are also the tightest possible upper bounds on the number of isolated roots. We also characterize, in terms of sparse resultants, precisely when these upper bounds are attained. Finally, we reformulate and extend some of the prior combinatorial results of the author on which subsets of coefficients must be chosen generically for our formulae to be exact. Our underlying framework provides a new toric variety setting for computational intersection theory in affine space minus an arbitrary union of coordinate hyperplanes. We thus show that, at least for root counting, it is better to work in a naturally associated toric compactification instead of always resorting to products of projective spaces.

For an acyclic quiver with three vertices, we consider the canonical decomposition of a non-Schurian root and associate certain representations of a generalized Kronecker quiver. These representations correspond to points contained in the intersection of two subvarieties of a Grassmannian and give rise to representations of the original quiver, preserving indecomposability. We show that these subvarieties intersect using Schubert calculus. Provided that the intersection contains a Schurian representation, it already contains an open subset of Schurian representations whose dimension is what we expect by Kac's Theorem.

A number of current research directions in the fields of digital signal processing and modern control and estimation theory were studied. Topics such as stability theory, linear prediction and parameter identification, system analysis and implementation, two-dimensional filtering, decentralized control and estimation, image processing, and nonlinear system theory were examined in order to uncover some of the basic similarities and differences in the goals, techniques, and philosophy of the two disciplines. An extensive bibliography is included.

We define a theory of descendent integration on the moduli spaces of stable pointed disks. The descendent integrals are proved to be coefficients of the $\tau$-function of an open KdV heirarchy. A relation between the integrals and a representation of half the Virasoro algebra is also proved. The construction of the theory requires an in depth study of homotopy classes of multivalued boundary conditions. Geometric recursions based on the combined structure of the boundary conditions and the moduli space are used to compute the integrals. We also provide a detailed analysis of orientations. Our open KdV and Virasoro constraints uniquely specify a theory of higher genus open descendent integrals. As a result, we obtain an open analog (governing all genera) of Witten's conjectures concerning descendent integrals on the Deligne-Mumford space of stable curves.

The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko-Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of CP^1 by CP^1 with prescribed ramification points and ramification types is related to the degree of the Lyashko--Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko-Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations between the homology classes represented by various strata.

In this paper, we extend the GKZ-system method to the more general case: compact Complete Intersection Calabi-Yau manifolds (CICY). For several one-deformation modulus compact CICYs with D-branes, the on-shell superpotentials in this paper from the extended GKZ-system method are exactly consistent with published results obtained from other methods. We further compute the off-shell superpotentials of these models. Then we obtain both the on-shell and off-shell superpotentials for sev- eral two-deformation moduli compact CICYs with D-branes by using the extended GKZ-system method. The discrete symmetrical groups, Z2, Z3 and Z4, of the holo- morphic curves wrapped by D-branes play the important roles in computing the superpotentials, in some sense, they are the quantum symmetries of these models. Furthermore, through the mirror symmetry, the Ooguri-Vafa invariants are extracted from the A-model instanton expansion.

We define a new family of open Gromov-Witten type invariants based on intersection theory on the moduli space of pseudoholomorphic curves of arbitrary genus with boundary in a Lagrangian submanifold. We assume the Lagrangian submanifold arises as the fixed points of an anti-symplectic involution and has dimension 2 or 3. In the strongly semi-positive genus 0 case, the new invariants coincide with Welschinger's invariant counts of real pseudoholomorphic curves. Furthermore, we calculate the new invariant for the real quintic threefold in genus 0 and degree 1 to be 30.

We define a new family of open Gromov-Witten type invariants based on intersection theory on the moduli space of pseudoholomorphic curves of arbitrary genus with boundary in a Lagrangian submanifold. We assume the Lagrangian submanifold arises as the fixed points of an anti-symplectic involution and has dimension 2 or 3. In the strongly semi-positive genus 0 case, the new invariants coincide with Welschinger's invariant counts of real pseudoholomorphic curves. Furthermore, we calculate the new invariant for the real quintic threefold in genus 0 and degree 1 to be 30.

xxxiii, 395 p. : 24 cm

Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology $3$-spheres. As an application, we give explicit constrains on the intersection forms of spin $4$-manifolds bounded by Brieskorn spheres $\pm\Sigma(2,3,6k\pm1)$. Along the way, we also give an alternative proof of Furuta-Kametanni's improvement of 10/8-theorem for closed spin-4 manifolds.

We study the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension, via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We work out the action of the blowup or 'discriminant' polarization on some natural cycles in the Hilbert scheme, including generalized diagonals and cycles, called 'node scrolls', parametrizing schemes supported on singular points. We derive an intersection calculus for Chern classes of tautological vector bundles, which are closely related to enumerative geometry.

We study the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension, via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We work out the action of the blowup or 'discriminant' polarization on some natural cycles in the Hilbert scheme, including generalized diagonals and cycles, called 'node scrolls', parametrizing schemes supported on singular points. We derive an intersection calculus for Chern classes of tautological vector bundles, which are closely related to enumerative geometry.

The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these numbers belong to a very particular subalgebra of the algebra of power series.

In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological quantum field theories. We explain in particular why matrix integrals of the type considered by Kontsevich naturally appear as tau-functions associated to minimal models. Our starting point is the extremely simple form of the string equation for the topological (p,1) models, where the so-called Baker-Akhiezer function is given by a (generalized) Airy function.

This historical introduction is in two parts. The first is reprinted with permission from ``A century of mathematics in America, Part II,'' Hist. Math., 2, Amer. Math. Soc., 1989, pp.543-585. Virtually no change has been made to the original text. In particular, Section 8 is followed by the original list of references. However, the text has been supplemented by a series of endnotes, collected in the new Section 9 and followed by a second list of references. If a citation is made to the first list, then its reference number is simply enclosed in brackets -- for example, [36]. However, if a citation is made to the second list, then its number is followed by an `S' -- for example, [36S]. Further, if a subject in the reprint is elaborated on in an endnote, then the subject is flagged in the margin by the number of the corresponding endnote, and the endnote includes in its heading, between parentheses, the page number or numbers on which the subject appears in the reprint below. Finally, all cross-references appear as hypertext links in the dvi and pdf copies.

The rational Chow ring A?(S[n],Q) of the Hilbert scheme S[n] parametrising the length n zero-dimensional subschemes of a toric surface S can be described with the help of equivariant techniques. In this paper, we explain the general method and we illustrate it through many examples. In the last section, we present results on the intersection theory of graded Hilbert schemes.

We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in R^n and then for "abstract" cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in R^n.

Speech communication instructors must think through the consequences of adopting a pedagogy of empowerment before they leap into the classroom with a handy packet of reading and pre-packaged activities. Schools are contested spheres and the struggle over what forms of authority and types of knowledge should be legitimated and transmitted to students can be seen in the demands of right-wing religious groups, feminists, ecologists, minorities, and other interest groups. Teaching with empowering students as a goal can be particularly challenging for speech communication instructors because so much of what happens in their classrooms is public. Instructors should keep student developmental levels in mind prior to trying out new critical classroom approaches. According to the theories of William Perry, students walk into class at many different places on the hierarchy of development, and instructors cannot reasonably expect all of their students to end up at the same place by the end of the semester. Incorporating sensitivity to multiculturalism into a public speaking class can be an extremely challenging endeavor. By exploring tenets of critical pedagogy and looking for opportunities to incorporate them into public speaking classrooms, instructors will be doing a service to their students. By using developmental theory as an "overlay" for critical activities, instructors can better understand their students' successes and failures. (Contains nine references.) (RS)

This is the first part of an investigation concerning the formulation of 2D gravity in the framework of the uniformization theory of Riemann surfaces. As a first step in this direction we show that the classical Liouville action appears in the expression of the correlators of topological gravity. Next we derive an inequality involving the cutoff of 2D gravity and the background geometry. Another result, always related to uniformization theory, concerns a relation between the higher genus normal ordering and the Liouville action. Furthermore, we show that the chirally split anomaly of CFT is equivalent to the Krichever-Novikov cocycle. By means of the inverse map of uniformization we give a realization of the Virasoro algebra on arbitrary Riemann surfaces and find the eigenfunctions for {\it holomorphic} covariant operators defining higher order cocycles and anomalies which are related to $W$-algebras. Finally we attack the problem of considering the positivity of $e^\sigma$, with $\sigma$ the Liouville field, by proposing an explicit construction for the Fourier modes on compact Riemann surfaces.

This is the first part of an investigation concerning the formulation of 2D gravity in the framework of the uniformization theory of Riemann surfaces. As a first step in this direction we show that the classical Liouville action appears in the expression of the correlators of topological gravity. Next we derive an inequality involving the cutoff of 2D gravity and the background geometry. Another result, always related to uniformization theory, concerns a relation between the higher genus normal ordering and the Liouville action. Furthermore, we show that the chirally split anomaly of CFT is equivalent to the Krichever-Novikov cocycle. By means of the inverse map of uniformization we give a realization of the Virasoro algebra on arbitrary Riemann surfaces and find the eigenfunctions for {\it holomorphic} covariant operators defining higher order cocycles and anomalies which are related to $W$-algebras. Finally we attack the problem of considering the positivity of $e^\sigma$, with $\sigma$ the Liouville field, by proposing an explicit construction for the Fourier modes on compact Riemann surfaces.

In this paper we develop an equivariant intersection theory for actions of algebraic groups on algebraic schemes. The theory is based on our construction of equivariant Chow groups. They are algebraic analogues of equivariant cohomology groups which satsify the formal properties of ordinary Chow groups. In addition, they enjoy many of the properties of equivariant cohomology. The principal results are: (1) We prove the existence of canonical intersection products on the Chow groups of geometric quotients of smooth varieties- even when the stabilizers of geometric points are non-reduced. (2) We construct a Todd class map from equivariant $K$-theory of coherent sheaves to a completion of equivariant Chow groups, and prove that a completion of equivariant $K$-theory is isomorphic to the completion of equivariant Chow groups. (3) We prove a localization theorem for torus actions and use it to give a characteristic free proof of the Bott residue formula for actions of tori on complete smooth varieties.

We consider two mixed curve $C,C'\subset {\Bbb C}^2$ which are defined by mixed functions of two variables $\bf z=(z_1,z_2)$. We have shown in \cite{MC}, that they have canonical orientations. If $C$ and $C'$ are smooth and intersect transversely at $P$, the intersection number $I_{top}(C,C';P)$ is topologically defined. We will generalize this definition to the case when the intersection is not necessarily transversal or either $C$ or $C'$ may be singular at $P$ using the defining mixed polynomials.

Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to Fourier coefficients of Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of a Shimura curve, has been completed by Kudla-Rapoport-Yang. In the present paper we prove results in a higher dimensional setting. On the integral model of a Shimura surface we consider the intersection of a Shimura curve with a codimension two cycle of complex multiplication points, and relate the intersection to certain cycles classes constructed by Kudla-Rapoport-Yang. As a corollary we deduce that our intersection multiplicities appear as Fourier coefficients of a Hilbert modular form of half-integral weight.

We consider a class of tautological top intersection products on the moduli space of stable pairs consisting of semistable vector bundles together with N sections on a smooth complex projective curve C. We show that when N is large, these intersection numbers can equally be computed on the Grothendieck Quot scheme of coherent sheaf quotients of the rank N trivial sheaf on C. The result has applications to the calculation of the intersection theory of the moduli space of semistable bundles on C.

Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K).T_m, where CM(K) is the zero-cycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K, and T_m is the Hirzebruch-Zagier divisors parameterizing products of elliptic curves with an m-isogeny between them. In this paper, we examine fields not covered by Yang's proof of the conjecture. We give numerical evidence to support the conjecture and point to some interesting anomalies. We compare the conjecture to both the denominators of Igusa class polynomials and the number of solutions to the embedding problem stated by Goren and Lauter.

This paper addresses the problem of constructing a cycle-level intersection theory for toric varieties. We show that by making one global choice, we can determine a cycle representative for the intersection of an equivariant Cartier divisor with an invariant cycle on a toric variety. For a toric variety defined by a fan in N, the choice consists of giving an inner product or a complete flag for M_Q=Hom(N,Q), or more generally giving for each cone sigma in the fan a linear subspace of M_Q complementary to the subspace of M_Q perpendicular to sigma, satisfying certain compatibility conditions. We show that these intersection cycles have properties analogous to the usual intersections modulo rational equivalence. If X is simplicial (for instance, if X is non-singular), we obtain a commutative ring structure on the invariant cycles of X with rational coefficients. This ring structure determines cycles representing certain characteristic classes of the toric variety. We also discuss how to define intersection cycles that require no choices, at the expense of increasing the size of the coefficient field.

To any spectral curve S, we associate a topological class {\Lambda}(S) in a moduli space M^b_{g,n} of "b-colored" stable Riemann surfaces of given topology (genus g, n boundaries), whose integral coincides with the topological recursion invariants W_{g,n}(S) of the spectral curve S. This formula can be viewed as a generalization of the ELSV formula (whose spectral curve is the Lambert function and the associated class is the Hodge class), or Marino-Vafa formula (whose spectral curve is the mirror curve of the framed vertex, and the associated class is the product of 3 Hodge classes), but for an arbitrary spectral curve. In other words, to a B-model (i.e. a spectral curve) we systematically associate a mirror A-model (integral in a moduli space of "colored" Riemann surfaces). We find that the mirror map, i.e. the relationship between the A-model moduli and B-model moduli, is realized by the Laplace transform.

To any spectral curve S, we associate a topological class {\Lambda}(S) in a moduli space M^b_{g,n} of "b-colored" stable Riemann surfaces of given topology (genus g, n boundaries), whose integral coincides with the topological recursion invariants W_{g,n}(S) of the spectral curve S. This formula can be viewed as a generalization of the ELSV formula (whose spectral curve is the Lambert function and the associated class is the Hodge class), or Marino-Vafa formula (whose spectral curve is the mirror curve of the framed vertex, and the associated class is the product of 3 Hodge classes), but for an arbitrary spectral curve. In other words, to a B-model (i.e. a spectral curve) we systematically associate a mirror A-model (integral in a moduli space of "colored" Riemann surfaces). We find that the mirror map, i.e. the relationship between the A-model moduli and B-model moduli, is realized by the Laplace transform.

Complex Healthcare systems require Leaders with the requisite level of adult developmental complexity to cope with the leadership challenges of their role. This study will explore the extent and nature of the evidence for the use of constructive development theory, within the context of healthcare leadership.

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two. Moreover, we determine the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and study Faltings heights of arithmetic Hirzebruch-Zagier divisors.

The goal of the paper is two-fold. At first, we attempt to give a survey of some recent applications of symmetric polynomials and divided differences to intersection theory. We discuss: polynomials universally supported on degeneracy loci; some explicit formulas for the Chern and Segre classes of Schur bundles with applications to enumerative geometry; flag degeneracy loci; fundamental classes, diagonals and Gysin maps; intersection rings of G/P and formulas for isotropic degeneracy loci; numerically positive polynomials for ample vector bundles. Apart of surveyed results, the paper contains also some new results as well as some new proofs of earlier ones: how to compute the fundamental class of a subvariety from the class of the diagonal of the ambient space; how to compute the class of the relative diagonal using Gysin maps; a new formula for pushing forward Schur's Q- polynomials in Grassmannian bundles; a new formula for the total Chern class of a Schur bundle; another proof of Schubert's and Giambelli's enumeration of complete quadrics; an operator proof of the Jacobi-Trudi formula; a Schur complex proof of the Giambelli-Thom-Porteous formula.