The moduli problem reduces to the study of conditions for the period mapping to be bijective. The principal boundary, counting problems and the siegelveech constants. The aim of this talk is to describe some of the moduli spaces that come up in dynamics, and to see how. Homogeneous flows, moduli spaces and arithmetic, pisa, italy. Dynamicists have studied interval exchanges and translation surfaces since the 60s as a source of.
The setting for homogeneous dynamics is the theory of lie groups. Dynamics, geometry, and the moduli space of riemann. Moduli spaces of isoperiodic forms on riemann surfaces curtis t. We refer to silvermans paper 56 for the moduli of dynamics on p1. More precisely, one studies the space of isomorphism classes of varieties having a speci ed structure, i. Monopole operators, moduli spaces and dualities article in journal of high energy physics 20123 august 2011 with 18 reads how we measure reads. It is an expanded version of the notes for a series of lectures delivered at a workshop on moduli spaces and the arithmetic of dynamical systems at the bellairs research institute, barbados, in 2010. Yan soibelman kansas state universitymoduli spaces of higgs bundles in mathematics and physicsnovember 19, 20 18 33 hitchin integrable systems and donaldsonthomas theory i start with mathematics and later discuss the related physics. To say that a ne moduli space exists, in this language, is to say. Monopole operators, moduli spaces and dualities request pdf.
Background let k be a complete, algebraically closed, nonarchimedean. Arithmetic of moduli stack of curves department of. Furthermore, moduli spaces themselves are rich geometric objects in their own right. Moduli spaces can be thought of as giving a universal space of parameters for the problem.
Princeton university library one washington road princeton, nj 085442098 usa 609 2581470. Moduli spaces for families of rational maps on p1 request pdf. Arithmetic coordinates on dynamical moduli spaces robert rumely special session on arithmetic dynamics seattle joint mathematics meetings january 6, 2016. More concretely a cover consists of an irreducible curve xde. Moduli spaces of higgs bundles in mathematics and physics. Kawa 2015 dynamical moduli spaces and elliptic curves 3 critical point is also conjugate to one of this form. He also conjectured that this quantity was commensurate to an ample weil height on the moduli space of. Moduli spaces of riemann surfaces benson farb, richard hain.
Mcmullen 6may2012 abstract this paper describes the intrinsic geometry of a leaf alofthe absolute period foliation of the hodge bundle. They are meaningful spaces, in that any statement about their geometry has a modular interpretation, in terms of the original classi. Let x k be the coarse moduli space of sl 2clocal systems on with prescribed boundary traces k2anc. One of the many motivations is that these moduli spaces provide an elementary and friendly setting to first encounter a number of important ideas useful all over mathematics. Moduli spaces and uniform boundedness in arithmetic dynamics, oklahoma state university, colloquium feb 3, 2020 moduli spaces for dynamical systems with level structure, joint mathematics meetings, special session on arithmetic dynamics, denver, co jan 18, 2020 moduli spaces for dynamical systems, miniworkshop on arithmetic dynamics, uni. Ows on homogeneous spaces, moduli spaces and their. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in teichmuller dynamics lead to connections with homogeneous spaces. Cycles on moduli spaces, geometric invariant theory, and. Moduli spaces associated to dynamical systems the geometry of dynamical moduli spaces dynamical moduli spaces further topics dynatomic polynomials and dynamical modular curves canonical heights postcritically finite maps field of moduli and field of definition. Cubic curves and totally geodesic subvarieties of moduli. Classically, discrete dynamics refers to the study of the iteration of selfmaps of the complex plane or real line. Msri geometric and arithmetic aspects of homogeneous dynamics.
Rtg seminar on geometry, dynamics and topology seminar. We study onedimensional algebraic families of pairs given by a polynomial with a marked point. Bookfi explains a elderly epub homogeneous flows, moduli spaces and arithmetic. Here different solutions are identified if they are isomorphic that is, geometrically the same. Hilbert modular varieties and the locus ofreal multiplication 288 4. Swarnava mukhopadhyay mathematics department university of maryland, college park the space of nonabelian theta functionsconformal blocks associated to a complex semisimple lie group g is a generalization of the space of classical theta functions. The presence of global injectivity for the period mapping is the socalled localglobal torelli problem.
The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of torelli groups, and arithmetic mapping class groups. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. In some sense, modular arithmetic is easier than integer artihmetic because there are only finitely many elements, so to find a solution to a problem you can always try every possbility. Moduli spaces of riemann surfaces about this title. Geometric invariant theory construction of moduli spaces. Arithmetic dynamics and dynamics on moduli spaces mathoverflow. Arithmetic properties of moduli spaces and topological. The following papers are among those that investigate xdyn. We prove an unlikely intersection statement for such pairs thereby exhibiting strong rigidity features for these pairs.
P1, parameterized by t in a riemann surface x, and the arithmetic dynamics of f t on rational points p1k where k cx or qx. Pn, a map y x, and an assignment of weights to the points in y. The school was designed to serve as a comprehensive introduction to the theory of. Download our spring pdf catalog for a look at our latest releases. Geometry and dynamics on moduli spaces clay mathematics. In these notes, we present a connection between the complex dynamics of a family of rational functions f t. Sean lawton mason experimental geometry lab geometry labs united conference, august 2830, 2015 statement of the problem in this project, we study the dynamics of the action of. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in teichmuller dynamics lead to connections with. For another perspective, recall that m gembeds into the moduli space of abelian varieties a g h gsp 2 z, a locally symmetric space amenable to the methods of homogeneous dynamics. Epub homogeneous flows, moduli spaces and arithmetic. Cubic curves and totally geodesic subvarieties of moduli space pages 957990 from volume 185 2017, issue 3 by curtis t.
Postcritically finite maps in complex and arithmetic dynamics aimpl. Dynamics on the moduli spaces of curves i institute for. Questions tagged modulispace mathematics stack exchange. An introduction to mathematical cryptography, with jill pipher and jeffrey hoffstein, springerverlag, utm, 2008. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. An important property of the compactified moduli spaces. We infer from this result the dynamical andreoort conjecture for curves in the moduli space of polynomials, by describing onedimensional families in this parameter. Cubic curves and totally geodesic subvarieties of moduli space pages 957990 from. Benson farb, university of chicago, chicago, il, richard hain, duke university, durham, nc and eduard looijenga, tsinghua university, beijing, china, editors.
The study of transversality is based on the virtual fundamental chain techniques the theory of kuranishi structures and their multisections and chain level intersection theories. Moduli spaces are spaces of solutions of geometric classification problems. Periods on arithmetic moduli spaces robert argus, patrick brown, jermain mcdermott, graduate student adviser diaaeldin taha, faculty adviser dr. Minicourse on moduli spaces university of michigan.
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. The minicourses will be aimed primarily at nonexperts and will benefit graduate students and early career researchers in related areas, who are particularly encouraged to apply to participate. Mar 30, 2012 this monograph studies moduli problems associated to algebraic dynamical systems. Moduli spaces and arithmetic dynamics, crm monograph series 30, ams, 2012. Workshop on dynamics and moduli spaces of translation surfaces. Pdf the arithmetic of dynamical pairs semantic scholar. Robert rumely arithmetic coordinates on dynamical moduli spaces. In arithmetic geometry, rather than studying one variety, it is fruitful to look at the space of all varieties. Arithmetic properties of moduli spaces and topological string partition functions of some calabiyau threefolds. Clay mathematics proceedings volume 10 homogeneous flows. The arithmetic of dynamical systems, springerverlag, gtm 241, 2007. Msri dynamics on moduli spaces of geometric structures. A selfcontained account of the general theory of kuranishi structures is also included in the appendix of this volume.
The introductory courses treat mapping class groups and teichmuller theory. Variation ofhodgestructures and realmultiplication 284 4. The second example that will play an important role is 1. This is a continuation of our diophantine study 19 of moduli spaces for local systems on surfaces and their mapping class group dynamics. Moduli spaces of isoperiodic forms on riemann surfaces.
Silverman defined the critical height of a rational function f z of degree d. Homogeneous flows, moduli spaces and arithmetic clay. Arithmetic dynamics is the study of the numbertheoretic properties of integer, rational, padic, andor algebraic points under repeated application of a polynomial or. A detailed analysis comparing the orientations of the moduli spaces and their fiber products is carried out. The arithmetic of dynamical systems, springerverlag, gtm 241, 2007, theres also the following monograph that discusses dynamicalrelated moduli spaces from an algebraic and arithmetic viewpoint. Fields medalist maryam mirzakhani, former member in the school of mathematics, gives the first of three 2012 marston morse lectures on dynamics on the moduli spaces of curves. I am working with moduli spaces of curves and i am not used to work with infinite dimensional spaces. That is, the points of a moduli space correspond to solutions of geometric problems.
Oct 17, 2017 the moduli space of riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. Spaces and topological string partition functions of some calabiyau threefolds the harvard community has made this article openly available. Clay mathematics proceedings volume 10 homogeneous. Nt 24 dec 2018 moduli spaces for dynamical systems with portraits john r. Dynamics, geometry, and the moduli space of riemann surfaces. Moduli spaces of riemann surfaces benson farb, richard. This monograph studies moduli problems associated to algebraic dynamical systems. Geometric invariant theory construction of moduli spaces of. Workshop on dynamics and moduli spaces of translation. Pdf riemann surfaces oxford graduate texts in mathematics. Yan soibelman kansas state university moduli spaces of higgs bundles in mathematics and physicsnovember 19, 20 18 33 hitchin integrable systems and donaldsonthomas theory i start with mathematics and later discuss the related physics.