# From de Jonquières Counts to Cohomological Field Theories

Size: px
Start display at page:

Transcription

1 From de Jonquières Counts to Cohomological Field Theories Mara Ungureanu Women at the Intersection of Mathematics and High Energy Physics 9 March 2017

2 What is Enumerative Geometry? How many geometric structures of a given type satisfy a given collection of geometric conditions?

3 What is Enumerative Geometry? Appolonius problem (approx. 200 BC) 8 circles tangent to 3 other circles

4

5 What is Enumerative Geometry? 3264 conics tangent to 5 given conics (1864 Chasles)

6 What is Enumerative Geometry? 3264 conics tangent to 5 given conics (1864 Chasles) 27 lines on cubic surface (1849 Cayley-Salmon)

7 What is Enumerative Geometry? 3264 conics tangent to 5 given conics (1864 Chasles) 27 lines on cubic surface (1849 Cayley-Salmon) 2875 lines on a quintic threefold (1886 Schubert)

8 What is Enumerative Geometry? 3264 conics tangent to 5 given conics (1864 Chasles) 27 lines on cubic surface (1849 Cayley-Salmon) 2875 lines on a quintic threefold (1886 Schubert) conics on a quintic threefold (1985 Katz)

9 What is Enumerative Geometry? 3264 conics tangent to 5 given conics (1864 Chasles) 27 lines on cubic surface (1849 Cayley-Salmon) 2875 lines on a quintic threefold (1886 Schubert) conics on a quintic threefold (1985 Katz) cubics on a quintic threefold (1991 Ellingsrud-Strømme)

10 What is Enumerative Geometry? Question Number of rational curves of any degree on quintic threefold?

11 What is Enumerative Geometry? Question Number of rational curves of any degree on quintic threefold? Answer Mirror symmetry! 1991 Candelas, de la Ossa, Green, Parkes

12 What is Enumerative Geometry? Question Number of rational curves of any degree on quintic threefold? Answer Mirror symmetry! 1991 Candelas, de la Ossa, Green, Parkes 1995 Kontsevich 1996 Givental Clemens conjecture?

13 What is Enumerative Geometry? Question How many points in the plane lie at the intersection of two given lines?

14 What is Enumerative Geometry? Question How many points in the plane lie at the intersection of two given lines? Answer It depends! Lines in general position exactly one Parallel lines none Lines coincide an infinite number of points

15 What is Enumerative Geometry? Parameter (moduli) space Compactify! Do excess intersection theory

16 Ernest de Jonquières

17 Disclaimer

18 de Jonquières multitangency conditions C smooth, genus g f : C P r non-degenerate Degree of f = #{f(c) H} =: d d = 3

19 de Jonquières multitangency conditions C smooth, genus g f : C P r non-degenerate Degree of f = #{f(c) H} =: d d = 3

20 de Jonquières multitangency conditions C smooth, genus g f : C P r non-degenerate Degree of f = #{f(c) H} =: d d = 3

21 de Jonquières multitangency conditions C smooth, genus g f : C P r non-degenerate Degree of f = #{f(c) H} =: d f 1 {f(c) H} = p 1 + 2p 2

22 de Jonquières multitangency conditions de Jonquières counts the number of pairs (p 1, p 2 ) such that there exists a hyperplane H P r with f 1 {f(c) H} = p 1 + 2p 2

23 de Jonquières multitangency conditions de Jonquières (and Mattuck, Macdonald) count the n-tuples (p 1,..., p n ) such that there exists a hyperplane H P r with f 1 {f(c) H} = a 1 p a n p n where a a n = d

24 de Jonquières multitangency conditions The (virtual) de Jonquières numbers are the coefficients of t 1... t n in (1 + a 2 1t a 2 nt n ) g (1 + a 1 t a n t n ) d r g

25 Constructing the moduli space An embedding f : C P r of degree d is given by A pair (L, V ) a line bundle L of degree d on C an (r + 1)-dimensional vector space V of sections of L

26 Constructing the moduli space An embedding f : C P r of degree d is given by A pair (L, V ) a line bundle L of degree d on C an (r + 1)-dimensional vector space V of sections of L Choose (σ 0,..., σ r ) basis of V f : C P r p [σ 0 (p) :... : σ r (p)]

27 Constructing the moduli space Space of all divisors of degree d on C For example C d = C... C /S }{{} d d times p 1 + 2p 2 C 3

28 Constructing the moduli space Space of all divisors of degree d on C For example We define de Jonquières divisors C d = C... C /S }{{} d d times p 1 + 2p 2 C 3 p p n C n such that f 1 {f(c) H} = a 1 p a n p n

29 Constructing the moduli space D = p p n is de Jonquières divisor there exists a section σ whose zeros are a 1 p a n p n

30 Constructing the moduli space D = p p n is de Jonquières divisor there exists a section σ whose zeros are a 1 p a n p n the map has nonzero kernel β D : V V a1 p a np n σ σ a1 p a np n

31 Constructing the moduli space the map β D : V V a1 p a np n σ σ a1 p a np n has nonzero kernel rank(β D ) dim V 1 = r

32 Constructing the moduli space V β D V a1 p a np n D = p p n C n

33 Constructing the moduli space V O Cn β F C n De Jonquières divisors: DJ n = {D C n rank(β D ) r} determinantal variety

34 To summarise... Fix curve C of genus g Fix embedding given by (L, V ) C n := space of divisors of degree n Defined de Jonquières divisors via multitangency conditions Described space DJ n of de Jonquières divisors as determinantal variety over C n

35 Analysing the moduli space dim DJ n n d + r

36 Analysing the moduli space dim DJ n n d + r Relevant questions n d + r < 0 non-existence of de Jonquières divisors n d + r 0 existence of de Jonquières divisors n d + r = 0 finite number of de Jonquières divisors dim DJ n = n d + r

37 Taking a variational perspective Allow C to vary in M g,n Vary the de Jonquières structure with it

38 Taking a variational perspective M g,n = moduli space of smooth curves of genus g with n marked points (C; p 1,..., p n ) M g,n dim M g,n = 3g 3 + n compactification M g,n

39 Taking a variational perspective M g,n = moduli space of smooth curves of genus g with n marked points (C; p 1,..., p n ) M g,n dim M g,n = 3g 3 + n compactification M g,n Question What is the cohomology of M g,n?

40 Taking a variational perspective L = K C = bundle of differential forms on C (L, V ) = (K C, Γ(C, K C )) Now d = 2g 2 and r = g 1

41 Taking a variational perspective L = K C = bundle of differential forms on C (L, V ) = (K C, Γ(C, K C )) Now d = 2g 2 and r = g 1 Fix partition µ = (a 1,..., a n ) of 2g 2 H g (µ) = {(C; p 1,..., p n ) such that K C admits the de Jonquières divisor a 1 p a n p n }

42 Taking a variational perspective H g (µ) M g,n determinantal subvariety flat surfaces, dynamical systems, Teichmüller theory: Masur, Eskin, Zorich, Kontsevich,... Bainbridge-Chen-Gendron-Grushevsky-Möller ( 16),... algebraic geometry: Diaz ( 84), Polishchuk ( 03), Farkas-Pandharipande ( 15)

43 Taking a variational perspective H g (µ) M g,n Take closure: H g (µ) M g,n Question What is the fundamental class [H g (µ)]? Answer (potentially) Cohomological field theory!

44 M g,n and 2-dimensional CFT CFT 2-dimensional QFT invariant under conformal transformations defined over compact Riemann surfaces Stick to holomorphic side CFT = 2-dimensional QFT covariant w.r.t. holomorphic coordinate changes

45 M g,n and 2-dimensional CFT Infinitesimal change of holomorphic coordinate Local holomorphic vector field z z + ɛf(z) f(z) d dz

46 M g,n and 2-dimensional CFT Infinitesimal change of holomorphic coordinate Local meromorphic vector field z z + ɛf(z) f(z) d dz

47 M g,n and 2-dimensional CFT Infinitesimal change of holomorphic coordinate Local meromorphic vector field z z + ɛf(z) f(z) d dz Virasoro algebra: L n = z n+1 d dz [L n, L m ] = (m n)l m+n, n Z etc...

48 M g,n and 2-dimensional CFT Local meromorphic vector field f(z) d dz Infinitesimal deformation of complex structure Infinitesimal deformation of an algebraic curve

49 M g,n and 2-dimensional CFT (C; p 1,..., p n ) M g,n λ = (λ1,..., λ n ) representation labels V λ (C; p 1,..., p n ) space of conformal blocks

50 M g,n and 2-dimensional CFT (C; p 1,..., p n ) M g,n λ = (λ1,..., λ n ) representation labels V λ (C; p 1,..., p n ) space of conformal blocks

51 M g,n and 2-dimensional CFT ( C; p 1,..., p n, q +, q ) M g,n+2 λ = (λ1,..., λ n, λ, λ ) representation labels V λ ( C; p 1,..., p n, q +, q ) space of conformal blocks

52 M g,n and 2-dimensional CFT Verlinde bundle V λ M g,n Each fibre is given by space of conformal blocks V λ (C; p 1,..., p n ) (C; p 1,..., p n )

53 M g,n and CohFT The characters ch(v λ ) define a CohFT on M g,n!

54 M g,n and CohFT The characters ch(v λ ) define a CohFT on M g,n! A CohFT a vector space of fields U a non-degenerate pairing η a distinguished vector 1 U a family of correlators Ω g,n H (M g,n, Q) (U ) n satisfying gluing...

55 M g,n and CohFT Quantum multiplication on U η(v 1 v 2, v 3 ) = Ω 0,3 (v 1 v 2 v 3 ) Q (U, ) Frobenius algebra of the CohFT Teleman: classification of all CohFT with semisimple Frobenius algebra

56 M g,n and CohFT H g (µ) = {(C; p 1,..., p n ) such that K C admits the de Jonquières divisor a 1 p a n p n } [H g (µ)] =? Maybe [H g (µ)] is one of the Ω g,n

57 H g (µ) and CohFT [H g (µ)] is not a CohFT class! Conjecture (Pandharipande, Pixton, Zvonkine): it is related to one Witten R-spin class Wg,µ R H 2g 2 (M g,n, Q)

58 To summarise... Tour of enumerative geometry

59 To summarise... Tour of enumerative geometry Described de Jonquières divisors on fixed curve C with fixed embedding (L, V )

60 To summarise... Tour of enumerative geometry Described de Jonquières divisors on fixed curve C with fixed embedding (L, V ) Allowed C to vary in moduli

61 To summarise... Tour of enumerative geometry Described de Jonquières divisors on fixed curve C with fixed embedding (L, V ) Allowed C to vary in moduli Obtained subspace of M g,n for particular case L = K C

62 To summarise... Tour of enumerative geometry Described de Jonquières divisors on fixed curve C with fixed embedding (L, V ) Allowed C to vary in moduli Obtained subspace of M g,n for particular case L = K C What if L K C?

### Enumerative Geometry: from Classical to Modern

: from Classical to Modern February 28, 2008 Summary Classical enumerative geometry: examples Modern tools: Gromov-Witten invariants counts of holomorphic maps Insights from string theory: quantum cohomology:

### Counting curves on a surface

Counting curves on a surface Ragni Piene Centre of Mathematics for Applications and Department of Mathematics, University of Oslo University of Pennsylvania, May 6, 2005 Enumerative geometry Specialization

### Topics in Geometry: Mirror Symmetry

MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:

### arxiv: v4 [math.ag] 3 Apr 2016

THE MODULI SPACE OF TWISTED CANONICAL DIVISORS GAVRIL FARKAS AND RAHUL PANDHARIPANDE arxiv:1508.07940v4 [math.ag] 3 Apr 2016 ABSTRACT. The moduli space of canonical divisors (with prescribed zeros and

### THE MODULI SPACE OF TWISTED CANONICAL DIVISORS

THE MODULI SPACE OF TWISTED CANONICAL DIVISORS GAVRIL FARKAS AND RAHUL PANDHARIPANDE ABSTRACT. The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact

### Dynamics and Geometry of Flat Surfaces

IMPA - Rio de Janeiro Outline Translation surfaces 1 Translation surfaces 2 3 4 5 Abelian differentials Abelian differential = holomorphic 1-form ω z = ϕ(z)dz on a (compact) Riemann surface. Adapted local

### COMPACTIFICATION OF STRATA OF ABELIAN DIFFERENTIALS

COMPACTIFICATION OF STRATA OF ABELIAN DIFFERENTIALS MATT BAINBRIDGE, DAWEI CHEN, QUENTIN GENDRON, SAMUEL GRUSHEVSKY, AND MARTIN MÖLLER Abstract. We describe the closure of the strata of abelian differentials

### COMPACTIFICATION OF STRATA OF ABELIAN DIFFERENTIALS

COMPACTIFICATION OF STRATA OF ABELIAN DIFFERENTIALS MATT BAINBRIDGE, DAWEI CHEN, QUENTIN GENDRON, SAMUEL GRUSHEVSKY, AND MARTIN MÖLLER Abstract. We describe the closure of the strata of abelian differentials

### Mirror symmetry. Mark Gross. July 24, University of Cambridge

University of Cambridge July 24, 2015 : A very brief and biased history. A search for examples of compact Calabi-Yau three-folds by Candelas, Lynker and Schimmrigk (1990) as crepant resolutions of hypersurfaces

### The Chern character of the Verlinde bundle over M g,n

The Chern character of the Verlinde bundle over M g,n A. Marian D. Oprea R. Pandharipande A. Pixton D. Zvonkine November 2013 Abstract We prove an explicit formula for the total Chern character of the

### THE QUANTUM CONNECTION

THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,

### Enumerative Geometry Steven Finch. February 24, 2014

Enumerative Geometry Steven Finch February 24, 204 Given a complex projective variety V (as defined in []), we wish to count the curves in V that satisfy certain prescribed conditions. Let C f denote complex

### AFFINE GEOMETRY OF STRATA OF DIFFERENTIALS

AFFINE GEOMETRY OF STRATA OF DIFFERENTIALS DAWEI CHEN Abstract. Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic

### The geometry of Landau-Ginzburg models

Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror

### Hodge structures from differential equations

Hodge structures from differential equations Andrew Harder January 4, 2017 These are notes on a talk on the paper Hodge structures from differential equations. The goal is to discuss the method of computation

### Gromov-Witten invariants of hypersurfaces

Gromov-Witten invariants of hypersurfaces Dem Fachbereich Mathematik der Technischen Universität Kaiserslautern zur Erlangung der venia legendi vorgelegte Habilitationsschrift von Andreas Gathmann geboren

### POSITIVITY OF DIVISOR CLASSES ON THE STRATA OF DIFFERENTIALS

POSITIVITY OF DIVISOR CLASSES ON THE STRATA OF DIFFERENTIALS DAWEI CHEN Abstract. Three decades ago Cornalba-Harris proved a fundamental positivity result for divisor classes associated to families of

### The tangent space to an enumerative problem

The tangent space to an enumerative problem Prakash Belkale Department of Mathematics University of North Carolina at Chapel Hill North Carolina, USA belkale@email.unc.edu ICM, Hyderabad 2010. Enumerative

### Mini-Course on Moduli Spaces

Mini-Course on Moduli Spaces Emily Clader June 2011 1 What is a Moduli Space? 1.1 What should a moduli space do? Suppose that we want to classify some kind of object, for example: Curves of genus g, One-dimensional

### Invariance of tautological equations

Invariance of tautological equations Y.-P. Lee 28 June 2004, NCTS An observation: Tautological equations hold for any geometric Gromov Witten theory. Question 1. How about non-geometric GW theory? e.g.

### On the Virtual Fundamental Class

On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14, 2014 http://www.math.ubc.ca/~behrend/talks/seoul14.pdf Overview Donaldson-Thomas theory: counting invariants

### Geometry of Conformal Field Theory

Geometry of Conformal Field Theory Yoshitake HASHIMOTO (Tokyo City University) 2010/07/10 (Sat.) AKB Differential Geometry Seminar Based on a joint work with A. Tsuchiya (IPMU) Contents 0. Introduction

### Introduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway

Curves and surfaces Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway What is algebraic geometry? IMA, April 13, 2007 Outline Introduction Curves Surfaces Curves on surfaces

### String-Theory: Open-closed String Moduli Spaces

String-Theory: Open-closed String Moduli Spaces Heidelberg, 13.10.2014 History of the Universe particular: Epoch of cosmic inflation in the early Universe Inflation and Inflaton φ, potential V (φ) Possible

### Curve counting and generating functions

Curve counting and generating functions Ragni Piene Università di Roma Tor Vergata February 26, 2010 Count partitions Let n be an integer. How many ways can you write n as a sum of two positive integers?

### STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES

STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES DAWEI CHEN AND IZZET COSKUN Contents 1. Introduction 1 2. Preliminary definitions and background 3 3. Degree two maps to Grassmannians 4 4.

### Periods and generating functions in Algebraic Geometry

Periods and generating functions in Algebraic Geometry Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil www.impa.br/ hossein/ Abstract In 1991 Candelas-de la Ossa-Green-Parkes predicted

### SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e on a general complete intersection of multidegree

### Two simple ideas from calculus applied to Riemannian geometry

Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University

### On the WDVV-equation in quantum K-theory

On the WDVV-equation in quantum K-theory Alexander Givental UC Berkeley and Caltech 0. Introduction. Quantum cohomology theory can be described in general words as intersection theory in spaces of holomorphic

### I. Why Quantum K-theory?

Quantum groups and Quantum K-theory Andrei Okounkov in collaboration with M. Aganagic, D. Maulik, N. Nekrasov, A. Smirnov,... I. Why Quantum K-theory? mathematical physics mathematics algebraic geometry

### ON THE USE OF PARAMETER AND MODULI SPACES IN CURVE COUNTING

ON THE USE OF PARAMETER AND MODULI SPACES IN CURVE COUNTING RAGNI PIENE In order to solve problems in enumerative algebraic geometry, one works with various kinds of parameter or moduli spaces:chow varieties,

### Geometry of moduli spaces

Geometry of moduli spaces 20. November 2009 1 / 45 (1) Examples: C: compact Riemann surface C = P 1 (C) = C { } (Riemann sphere) E = C / Z + Zτ (torus, elliptic curve) 2 / 45 (2) Theorem (Riemann existence

### Knot Homology from Refined Chern-Simons Theory

Knot Homology from Refined Chern-Simons Theory Mina Aganagic UC Berkeley Based on work with Shamil Shakirov arxiv: 1105.5117 1 the knot invariant Witten explained in 88 that J(K, q) constructed by Jones

### Intersections in genus 3 and the Boussinesq hierarchy

ISSN: 1401-5617 Intersections in genus 3 and the Boussinesq hierarchy S. V. Shadrin Research Reports in Mathematics Number 11, 2003 Department of Mathematics Stockholm University Electronic versions of

### PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013

PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013 1. Problems on moduli spaces The main text for this material is Harris & Morrison Moduli of curves. (There are djvu files

### Gauged Linear Sigma Model in the Geometric Phase

Gauged Linear Sigma Model in the Geometric Phase Guangbo Xu joint work with Gang Tian Princeton University International Conference on Differential Geometry An Event In Honour of Professor Gang Tian s

### PRINCIPAL BOUNDARY OF MODULI SPACES OF ABELIAN AND QUADRATIC DIFFERENTIALS

PRINCIPAL BOUNDARY OF MODULI SPACES OF ABELIAN AND QUADRATIC DIFFERENTIALS DAWEI CHEN AND QILE CHEN Abstract. The seminal work of Eskin-Masur-Zorich described the principal boundary of moduli spaces of

### A tourist s guide to intersection theory on moduli spaces of curves

A tourist s guide to intersection theory on moduli spaces of curves The University of Melbourne In the past few decades, moduli spaces of curves have attained notoriety amongst mathematicians for their

### arxiv:alg-geom/ v2 26 Oct 1996

arxiv:alg-geom/9609021v2 26 Oct 1996 Introduction Mathematical Aspects of Mirror Symmetry David R. Morrison Mirror symmetry is the remarkable discovery in string theory that certain mirror pairs of Calabi

### SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree

### CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat

### DEGENERATIONS OF ABELIAN DIFFERENTIALS. Dawei Chen. Abstract

DEGENERATIONS OF ABELIAN DIFFERENTIALS Dawei Chen Abstract Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear

### Heterotic Mirror Symmetry

Heterotic Mirror Symmetry Eric Sharpe Physics Dep t, Virginia Tech Drexel University Workshop on Topology and Physics September 8-9, 2008 This will be a talk about string theory, so lemme motivate it...

### 1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3

Compact course notes Riemann surfaces Fall 2011 Professor: S. Lvovski transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Structures 2 2 Framework of Riemann surfaces

### Mathematical Research Letters 1, (1994) A MATHEMATICAL THEORY OF QUANTUM COHOMOLOGY. Yongbin Ruan and Gang Tian

Mathematical Research Letters 1, 269 278 (1994) A MATHEMATICAL THEORY OF QUANTUM COHOMOLOGY Yongbin Ruan and Gang Tian 1. Introduction The notion of quantum cohomology was first proposed by the physicist

### K-stability and Kähler metrics, I

K-stability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates

### Flat Surfaces. Marcelo Viana. CIM - Coimbra

CIM - Coimbra Main Reference Interval Exchange Transformations and Teichmüller Flows www.impa.br/ viana/out/ietf.pdf Rauzy, Keane, Masur, Veech, Hubbard, Kerckhoff, Smillie, Kontsevich, Zorich, Eskin,

### BRILL-NOETHER THEORY. This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve.

BRILL-NOETHER THEORY TONY FENG This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve." 1. INTRODUCTION Brill-Noether theory is concerned

### Instantons in string theory via F-theory

Instantons in string theory via F-theory Andrés Collinucci ASC, LMU, Munich Padova, May 12, 2010 arxiv:1002.1894 in collaboration with R. Blumenhagen and B. Jurke Outline 1. Intro: From string theory to

### Rational curves on general type hypersurfaces

Rational curves on general type hypersurfaces Eric Riedl and David Yang May 5, 01 Abstract We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an

### Gromov-Witten invariants and Algebraic Geometry (II) Jun Li

Gromov-Witten invariants and Algebraic Geometry (II) Shanghai Center for Mathematical Sciences and Stanford University GW invariants of quintic Calabi-Yau threefolds Quintic Calabi-Yau threefolds: X =

### COUNTING ELLIPTIC PLANE CURVES

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 12, December 1997, Pages 3471 3479 S 0002-9939(97)04136-1 COUNTING ELLIPTIC PLANE CURVES WITH FIXED j-invariant RAHUL PANDHARIPANDE (Communicated

### STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES

STABLE BASE LOCUS DECOMPOSITIONS OF KONTSEVICH MODULI SPACES DAWEI CHEN AND IZZET COSKUN Abstract. In this paper, we determine the stable base locus decomposition of the Kontsevich moduli spaces of degree

### Overview of classical mirror symmetry

Overview of classical mirror symmetry David Cox (notes by Paul Hacking) 9/8/09 () Physics (2) Quintic 3-fold (3) Math String theory is a N = 2 superconformal field theory (SCFT) which models elementary

### Contributors. Preface

Contents Contributors Preface v xv 1 Kähler Manifolds by E. Cattani 1 1.1 Complex Manifolds........................... 2 1.1.1 Definition and Examples.................... 2 1.1.2 Holomorphic Vector Bundles..................

### Part II. Riemann Surfaces. Year

Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 96 Paper 2, Section II 23F State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised

### Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex

### Current algebras and higher genus CFT partition functions

Current algebras and higher genus CFT partition functions Roberto Volpato Institute for Theoretical Physics ETH Zurich ZURICH, RTN Network 2009 Based on: M. Gaberdiel and R.V., arxiv: 0903.4107 [hep-th]

### Moduli spaces of graphs and homology operations on loop spaces of manifolds

Moduli spaces of graphs and homology operations on loop spaces of manifolds Ralph L. Cohen Stanford University July 2, 2005 String topology:= intersection theory in loop spaces (and spaces of paths) of

### Intersection theory on moduli spaces of curves via hyperbolic geometry

Intersection theory on moduli spaces of curves via hyperbolic geometry The University of Melbourne In the past few decades, moduli spaces of curves have become the centre of a rich confluence of rather

### Geometry of the Calabi-Yau Moduli

Geometry of the Calabi-Yau Moduli Zhiqin Lu 2012 AMS Hawaii Meeting Department of Mathematics, UC Irvine, Irvine CA 92697 March 4, 2012 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 1/51

### Monodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H.

Monodromy of the Dwork family, following Shepherd-Barron 1. The Dwork family. Consider the equation (f λ ) f λ (X 0, X 1,..., X n ) = λ(x n+1 0 + + X n+1 n ) (n + 1)X 0... X n = 0, where λ is a free parameter.

### Mirror Symmetry: Introduction to the B Model

Mirror Symmetry: Introduction to the B Model Kyler Siegel February 23, 2014 1 Introduction Recall that mirror symmetry predicts the existence of pairs X, ˇX of Calabi-Yau manifolds whose Hodge diamonds

### Mathematical Results Inspired by Physics

ICM 2002 Vol. III 1 3 Mathematical Results Inspired by Physics Kefeng Liu Abstract I will discuss results of three different types in geometry and topology. (1) General vanishing and rigidity theorems

### INTERPOLATION PROBLEMS: DEL PEZZO SURFACES

INTERPOLATION PROBLEMS: DEL PEZZO SURFACES AARON LANDESMAN AND ANAND PATEL Abstract. We consider the problem of interpolating projective varieties through points and linear spaces. After proving general

### Riemann surfaces. Paul Hacking and Giancarlo Urzua 1/28/10

Riemann surfaces Paul Hacking and Giancarlo Urzua 1/28/10 A Riemann surface (or smooth complex curve) is a complex manifold of dimension one. We will restrict to compact Riemann surfaces. It is a theorem

### On the BCOV Conjecture

Department of Mathematics University of California, Irvine December 14, 2007 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called

### THE LANDAU-GINZBURG/CALABI-YAU CORRESPONDENCE

THE LANDAU-GINZBURG/CALABI-YAU CORRESPONDENCE EMILY CLADER WEDNESDAY LECTURE SERIES, ETH ZÜRICH, OCTOBER 2014 Given a nondegenerate quasihomogeneous polynomial W (x 1,..., x N ) with weights c 1,..., c

### Cellular Multizeta Values. Sarah Carr

Cellular Multizeta Values Sarah Carr August, 2007 Goals The following is joint work with Francis Brown and Leila Schneps based on a combinatorial theory of pairs of polygons. 1. Objects: Periods on M 0,n

### Aspects of (0,2) theories

Aspects of (0,2) theories Ilarion V. Melnikov Harvard University FRG workshop at Brandeis, March 6, 2015 1 / 22 A progress report on d=2 QFT with (0,2) supersymmetry Gross, Harvey, Martinec & Rohm, Heterotic

### Enumerative Invariants in Algebraic Geometry and String Theory

Dan Abramovich -. Marcos Marino Michael Thaddeus Ravi Vakil Enumerative Invariants in Algebraic Geometry and String Theory Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 6-11,

### New worlds for Lie Theory. math.columbia.edu/~okounkov/icm.pdf

New worlds for Lie Theory math.columbia.edu/~okounkov/icm.pdf Lie groups, continuous symmetries, etc. are among the main building blocks of mathematics and mathematical physics Since its birth, Lie theory

### RIMS-1743 K3 SURFACES OF GENUS SIXTEEN. Shigeru MUKAI. February 2012 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES. KYOTO UNIVERSITY, Kyoto, Japan

RIMS-1743 K3 SURFACES OF GENUS SIXTEEN By Shigeru MUKAI February 2012 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan K3 SURFACES OF GENUS SIXTEEN SHIGERU MUKAI Abstract. The

### Linear connections on Lie groups

Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right)

### Then the blow up of V along a line is a rational conic bundle over P 2. Definition A k-rational point of a scheme X over S is any point which

16. Cubics II It turns out that the question of which varieties are rational is one of the subtlest geometric problems one can ask. Since the problem of determining whether a variety is rational or not

### Stable maps and Quot schemes

Stable maps and Quot schemes Mihnea Popa and Mike Roth Contents 1. Introduction........................................ 1 2. Basic Setup........................................ 4 3. Dimension Estimates

### where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism

8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the

### CALIBRATED FIBRATIONS ON NONCOMPACT MANIFOLDS VIA GROUP ACTIONS

DUKE MATHEMATICAL JOURNAL Vol. 110, No. 2, c 2001 CALIBRATED FIBRATIONS ON NONCOMPACT MANIFOLDS VIA GROUP ACTIONS EDWARD GOLDSTEIN Abstract In this paper we use Lie group actions on noncompact Riemannian

### A RECONSTRUCTION THEOREM IN QUANTUM COHOMOLOGY AND QUANTUM K-THEORY

A RECONSTRUCTION THEOREM IN QUANTUM COHOMOLOGY AND QUANTUM K-THEORY Y.-P. LEE AND R. PANDHARIPANDE Abstract. A reconstruction theorem for genus 0 gravitational quantum cohomology and quantum K-theory is

### Homological mirror symmetry

Homological mirror symmetry HMS (Kontsevich 1994, Hori-Vafa 2000, Kapustin-Li 2002, Katzarkov 2002,... ) relates symplectic and algebraic geometry via their categorical structures. A symplectic manifold

### Billiards and Teichmüller theory

Billiards and Teichmüller theory M g = Moduli space moduli space of Riemann surfaces X of genus g { } -- a complex variety, dimension 3g-3 Curtis T McMullen Harvard University Teichmüller metric: every

### Counting surfaces of any topology, with Topological Recursion

Counting surfaces of any topology, with Topological Recursion 1... g 2... 3 n+1 = 1 g h h I 1 + J/I g 1 2 n+1 Quatum Gravity, Orsay, March 2013 Contents Outline 1. Introduction counting surfaces, discrete

### Stringy Topology in Morelia Week 2 Titles and Abstracts

Stringy Topology in Morelia Week 2 Titles and Abstracts J. Devoto Title: K3-cohomology and elliptic objects Abstract : K3-cohomology is a generalized cohomology associated to K3 surfaces. We shall discuss

### Virasoro and Kac-Moody Algebra

Virasoro and Kac-Moody Algebra Di Xu UCSC Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 1 / 24 Outline Mathematical Description Conformal Symmetry in dimension d > 3 Conformal Symmetry in dimension

### Singularities, Root Systems, and W-Algebras

Singularities, Root Systems, and W-Algebras Bojko Bakalov North Carolina State University Joint work with Todor Milanov Supported in part by the National Science Foundation Bojko Bakalov (NCSU) Singularities,

### Moduli space of curves and tautological relations

MSc Mathematical Physics Master Thesis Moduli space of curves and tautological relations Author: Farrokh Labib Supervisor: prof. dr. Sergey Shadrin Examination date: July, 017 Korteweg-de Vries Institute

### Titles and Abstracts

29 May - 2 June 2017 Laboratoire J.A. Dieudonné Parc Valrose - Nice Titles and Abstracts Tamas Hausel : Mirror symmetry with branes by equivariant Verlinde formulae I will discuss an agreement of equivariant

### A NEW FAMILY OF SYMPLECTIC FOURFOLDS

A NEW FAMILY OF SYMPLECTIC FOURFOLDS OLIVIER DEBARRE This is joint work with Claire Voisin. 1. Irreducible symplectic varieties It follows from work of Beauville and Bogomolov that any smooth complex compact

### An introduction to heterotic mirror symmetry. Eric Sharpe Virginia Tech

An introduction to heterotic mirror symmetry Eric Sharpe Virginia Tech I ll begin today by reminding us all of ordinary mirror symmetry. Most basic incarnation: String theory on a Calabi-Yau X = String

### Moduli Space of Riemann Surfaces

Moduli Space of Riemann Surfaces Gabriel C. Drummond-Cole June 5, 2007 1 Ravi Vakil Introduction to the Moduli Space [Tomorrow there is a reception after the first talk on the sixth floor. Without further

### Oral exam practice problems: Algebraic Geometry

Oral exam practice problems: Algebraic Geometry Alberto García Raboso TP1. Let Q 1 and Q 2 be the quadric hypersurfaces in P n given by the equations f 1 x 2 0 + + x 2 n = 0 f 2 a 0 x 2 0 + + a n x 2 n

### Cohomology jump loci of quasi-projective varieties

Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)

### The Pennsylvania State University The Graduate School Eberly College of Science AN ASYMPTOTIC MUKAI MODEL OF M 6

The Pennsylvania State University The Graduate School Eberly College of Science AN ASYMPTOTIC MUKAI MODEL OF M 6 A Dissertation in Mathematics by Evgeny Mayanskiy c 2013 Evgeny Mayanskiy Submitted in Partial

### Congruence sheaves and congruence differential equations Beyond hypergeometric functions

Congruence sheaves and congruence differential equations Beyond hypergeometric functions Vasily Golyshev Lille, March 6, 204 / 45 Plan of talk Report on joint work in progress with Anton Mellit and Duco

### Broken pencils and four-manifold invariants. Tim Perutz (Cambridge)

Broken pencils and four-manifold invariants Tim Perutz (Cambridge) Aim This talk is about a project to construct and study a symplectic substitute for gauge theory in 2, 3 and 4 dimensions. The 3- and

### Exotic nearly Kähler structures on S 6 and S 3 S 3

Exotic nearly Kähler structures on S 6 and S 3 S 3 Lorenzo Foscolo Stony Brook University joint with Mark Haskins, Imperial College London Friday Lunch Seminar, MSRI, April 22 2016 G 2 cones and nearly

### A wall-crossing formula for 2d-4d DT invariants

A wall-crossing formula for 2d-4d DT invariants Andrew Neitzke, UT Austin (joint work with Davide Gaiotto, Greg Moore) Cetraro, July 2011 Preface In the last few years there has been a lot of progress