Destination page number search scope search text search scope search text. Enumerative geometry beyond numbers introductory workshop msri, january 22 to 26, 2018 1 a. They provide a rich class of examples of smooth holomor. Pdf towards an enumerative geometry of the moduli space. Ruan, orbifolds and stringy topology, cambridge tracts in mathe. That is, counting the solutions to a geometrically meaningful system of polynomial equations.
Enumerative geometry from string theory sciencedirect. This workshop will consist of expository minicourses and lectures introducing various aspects of modern enumerative geometry, among which. In the case of degreed instantons, for example, there is the conjecture of clemens that for the quintics in cp4, 53 nd for all d. Topological string theory and enumerative geometry a dissertation submitted to the department of physics and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy yun s. Ktheoretic enumerative geometry and the hilbert scheme of. Enumerative algebraic geometry of conics mathematical.
Katz, mirror symmetry and algebraic geometry, mathematical surveys and mono. Given five conics in the plane, are there any conics that. In mathematics, enumerative geometry is the branch of algebraic geometry concerned with. The enumerative geometry of the hilbert schemes of points of. My research interests lie broadly in algebraic geometry, representation theory and mathematical physics, motivated by the beautiful and rich principle of symmetry and duality. In this thesis,1 i concern with several problems that appear in two di. Enumerative algebraic geometry of conics andrew bashelor, amy ksir, and will traves 1. Enumerative geometry and classical algebraic geometry lebarz. The methods used may be of independent interest, especially i the surprisingly intricate geometry of maps of pointed curves to p1, and ii the study of the space of curves in pn via a smooth bration from an open set to the space of maps of curves to p1. Enumerative problems come up in various areas of mathematical research.
One of its most spectacular applications is that of the complex and real enumerative geometry. Topics in enumerative algebraic geometry these are lecture notes on gromovwitten theory based on the topics cource i taught at uc berkeley in 1997. From enumerative geometry to solving systems of polynomial. Insection 4we discuss stable maps with lagrangian boundary conditions and the idea of a virtual fundamental class on moduli spaces of stable maps. What links here related changes upload file special pages permanent link page. The notes were typeset by the students and generously shared with the others. Perhaps the most famous example of how ideas from modern physics have revolutionized mathematics is the way string theory has led to an overhaul of enumerative geometry, an area of mathematics that started in the eighteen hundreds. Enumerative geometry, intersection theory and moduli spaces. Eva silverstein abstract in this thesis we investigate several problems which have their roots in both topolog. In this paper, we study the geometry of moduli spaces of genus 0 and 1 curves in pn with specified contact with a hyperplane h. Enumerative geometry of double spin curves dissertation zur erlangung des akademischen grades doctor rerum naturalium dr. Topological string theory and enumerative geometry yun s. In 1848 jakob steiner, professor of geometry at the university of berlin, posed the following problem 19.
The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genus g. For any given enumerative question three problems naturally arise. Quantum field theory, supersymmetry, and enumerative. The enumerative geometry of the hilbert schemes of points of a k3 surface a dissertation submitted to attain the degree of doctor of sciences of eth zuric h dr. In this talk i will discuss a probabilistic approach schubert calculus the computation of the cohomology ring of the grassmannian. This problem asks for the number and construction of circles that are tangent to three given circles, points or lines. The enumerative geometry of rational and elliptic curves in projective space ravi vakil abstract. Enumerative geometry on pa in this section, we will consider only irreducible plane curves of degree d, with equation fu,v,w0, 2. Buy quantum field theory, supersymmetry, and enumerative geometry iaspark city mathematics series on free shipping on qualified orders. Quantum field theory, supersymmetry, and enumerative geometry. Oct 27, 2016 probabilistic enumerative geometry abstract. Enumerative geometry and geometric representation theory. Traditional enumerative geometry asks certain questions to which the expected answer is a number. The enumerative geometry of rational and elliptic tropical.
For us, enumerative geometry is concerned with enumerating geometric gures of some kind having speci ed positions with respect to general xed gures. Integrals of characteristic classes of tautological sheaves on the hilbert scheme of points on a surface frequently arise in enumerative problems. We study the geometry of varieties parametrizing degree drational and elliptic curves in pn intersecting. Here is our book, computations in algebraic geometry with macaulay 2, edited by david eisenbud, daniel r. By this we mean setting up a chow ring for the moduli.
Algebraic curves over finite fields wei ho this course will study the arithmetic and geometry of algebraic curves, especially over finite fields. By this we mean setting up a chow ring for the moduli space m g of curves of genus g and its compactification m g, defining what seem to be the most important classes in this ring and calculating the class of some geometrically important loci in. Pdf towards an enumerative geometry of the moduli space of. The di culties lie in solving the rst two problems. Translate the enumerative problem into an intersection problem.
It has however been recognized for some time that the numerics is often just. A surface is the set of points in 3dimensional space whose coordinates satisfy a single polynomial equation. Yet the notes are unedited so use at your own risk. Enumerative geometry and string theory pdf free download. Rearrange individual pages or entire files in the desired order.
Kevin mcgerty 1 since their introduction in the seminal paper n1, nakajimas quiver varieties have played a central role in geometric representations theory. One takes a ver sion of string theory or mtheory, or ftheory and compactifies the theory on a calabiyau manifold, obtaining an effective physical theory. Enumerative geometry of stable maps 3 riemann surfaces and doubling constructions that we will use. Ktheoretic enumerative geometry and the hilbert scheme of points on a surface. Towards an enumerative geometry of the moduli space of curves. Eth zuric h presented by georg oberdieck accepted on the recommendation of prof. The right answer is that the enumerative constraints and deformations of curves in x put a certain sheaf o on the set of solutions, and we should take the euler characteristic. Eugenii shustin, tropical enumerative geometry the tropical geometry is a rapidly developing area which brings together algebraic geometry, symplectic geometry, geometry and combinatorics of polyhedral complexes, various algebraic structures. Enumerative geometry and classical algebraic geometry. The unifying theme has been the application of representation theoretic and harmonic analytic techniques to geometric, enumerative and sometimes, arithmetic problems. In this talk i will discuss a probabilistic approach schubert calculus the computation of the cohomology ring of. In 1848 jakob steiner, professor of geometry at the univer sity of berlin, posed the following problem 19.
Enumerative geometry beyond numbers connections for women workshop msri, january 18 to 19, 2018 1 d. The remarkable properties of the zeta function for a curve over a finite field, which gives a close link between the arithmetic of the curve such as the number. Broadly speaking, enumerative geometry is about geometric questions the answer for which is a number. Enumerative geometry of stable maps with lagrangian boundary. We use macaulay 2 to investigate some enumerative geometric problems from this point of view. Enumerative geometry and classical algebraic geometry progress in mathematics softcover reprint of the original 1st ed. Some of them can be formulated in purely combinatorial terms, while for others even such a formulation can be the sole purpose of a highly nontrivial investigation. The problem we will consider about matroid varieties is an enumerative geometry problem, a generalization of the linear gromovwitten invariants of projective spaces. Enumerative geometry is also currently one of the most active areas of research in algebraic geometry, mainly due to a recent in. The problem of apollonius is one of the earliest examples of enumerative geometry.
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