The following question is more of a request for pointers to suitable literature on introductory material for arithmetic dynamics and dynamics on moduli spaces. Modulus space line bundle algebraic group global section hilbert. If moduli v is a separated scheme, then there exist at most one diagonal arrow completing the following. Much of this is presented in analogy with the space of lattices sl n. Moduli and git a brief history geometric invariant theory git is an important tool in the study of moduli. Bombay, 1978 out of print but many people have copies. Geometric invariant theory peter newstead to cite this version. Newstead author of introduction to moduli problems. Buy introduction to moduli problems and orbit spaces by p. Continuous frames, coorbit spaces and the discretization problem. Acrobat reader introduction to moduli problems and orbit spaces. The moduli space of lines in the plane let us begin with a problem that looks rather simple, but that nevertheless illustrates many of the important ideas of moduli spaces. Some notes containing a brief survey of the cohomology of the moduli space of curves and the harerzagier formula for the orbifold euler characteristic.
The study of moduli spaces is an old branch of algebraic geometry with. Minicourse on moduli spaces emily clader june 2011 1 what is a moduli space. All results work for the more general moduli space mg. An introduction to the topology of the moduli space of stable. Newstead, introduction to moduli problems and orbit spaces, tifr. With this definition, it is not necessary for y to be an orbit space and so. What is a good introductory text for moduli theory. I added a few words outlining other work of harer on the stability of the cohomology of the moduli space of curves. Introduction to moduli problems and orbit spaces, tifr, bombay, 1978 out of print but many people have copies. An ideal solution to a moduli problem is a ne moduli space, which is a scheme that represents this functor. Moduli spaces and their birational geometry hanbom moon department of mathematics university of georgia february 25, 20. Hence, aside from foundational subtleties, it exists as a set.
There are numerous instances of this principle in the current literature on moduli spaces, but we choose to mention here only two examples familiar to the author. One can consider moduli spaces of vector bundles, coherent sheaves, etc. Our aim is to understand the topological nature of. Its principal application is to the construction of various moduli spaces. In the second part we construct such moduli spaces in the case where the riemann surface has boundary, and study a cobordism category built out of such moduli spaces.
Lectures on introduction to moduli problems and orbit. On topics in finite element solution of elliptic problems. Volumes 186 are being made available for free download from this site. Theory and practice so far, we have worked with vectors of length n and performed basic operations on them like scaling and addition. Universal moduli spaces of surfaces with flat bundles and. An example of a different kind is the jacobian, which is a moduli space that classi. But the main strategy is to replace polynomial divergence by the \exponential drift idea of benoistquint bq. Newstead is the author of introduction to moduli problems and orbit spaces 0. Some problems on mapping class groups and moduli space benson farb. Several topics in these notes will not be covered and some may be discussed only brie y in the.
Describe the collection of all lines in the real plane r2 that pass through the origin. Integrable systems in this section i will try to give, in a few pages, a precise idea of what an integrable system is, in the context of evolutionary hamiltonian pdes. In this series of lectures, we describe some counting problems in moduli space and outline their connection to the dynamics of the sl 2. Sep 03, 2008 having just created my first pdf file and a big one at that i noticed that spaces had randomly appear inside of words. That is, the points of a moduli space correspond to solutions of geometric problems. If a moduli problem can be expressed in terms of algebraic geometry then. Published for the tata institute of fundamental research, bombay by narosa pub. An introduction to the topology of the moduli space of. Formal moduli problems november 5, 2011 contents 1 deformation theories. People tend to say classifying space when in the context of topology, and they tend to say moduli space when in a context of complex geometry or algebraic geometry. Then the problem is to give it more structure and to prove it has some nice properties. The moduli spaces of holomorphic maps elenynicoleta ionel abstract this is a draft of some informal notes that are meant to accompany the second week of the graduate lectures at the 2012 ias women and mathematics program.
On introduction to moduli problems and orbit spaces. For simplicity, we will concentrate on the moduli space mg of the compact riemann surfaces. The file was converted from a word document with justified text. Lectures on introduction to moduli problems and orbit spaces lectures on mathema volume 51 of lectures on mathematics and physics. An introduction to moduli spaces of curves 2 parameters. In mathematics, in particular algebraic geometry, a moduli space is a geometric space usually a scheme or an algebraic stack whose points represent algebrogeometric objects of some fixed kind, or isomorphism classes of such objects. Suppose that we want to classify some kind of object, for example. Moduli spaces are spaces of solutions of geometric classification problems. We will introduce the minimal notions that will be used in what follows and assume a certain familiarity. To include a comma in your tag, surround the tag with double quotes. The construction of moduli spaces and geometric invariant. Some problems on mapping class groups and moduli space. In section 1 we discuss the general problem of constructing.
To save writing, we are using the word line to mean. Still the theory for genus greater than 1 is much less explicit than the elliptic curves case, where we can represent the moduli space as the quo. Introduction to moduli problems and orbit spaces welcome to. One of the multiple factors that led us to write this article is that the origin and history of teichmu. The theorys principal application is to the construction of various moduli spaces.
Newstead, 9788184871623, available at book depository with free delivery worldwide. Pdf the determination of the orbit spaces of compact. Huisgenzimmermann dedicated to ragnarolaf buchweitz on the occasion of his seventieth birthday abstract. Nevertheless, something is known about the type of singularities they can attain. Maryam mirzakhani on dynamics on the moduli spaces of. Such spaces frequently arise as solutions to classification problems. Indeed, if x n and is a counting measure, we acquire a discrete frame. Geometric invariant theory and applications to moduli problems. June 18, 2006 abstract this paper presents a number of problems about mapping class groups and moduli space. Introduction to moduli problems and orbit spaces by p.
Curves of genus g, onedimensional subspaces of rn, finite sets. Lectures on introduction to moduli problems and orbit spaces by p e newstead, 9780387088518, available at book depository with free delivery worldwide. A moduli space is usually the set of isomorphism classes of objects of a given type. When studying geometric objects, it is desirable to classify them according to different criteria in order to be able to distinguish the equivalent classes in this category. On numerical methods for time dependent equations applications to fluid flow problems. In addition, the introduction of methods from topology, mainly sheaves and their cohomology, rst by serre. For xed x, r, d, there exists a connected moduli space mg of sequivalence classes of rank r, degree dsemistable bundles over x, which is a complex projective variety, having dimension r2g 1. An introduction to moduli spaces of curves and its. Since the subject is huge, it helps also to know which aspect is of interest. Some of the ideas came from discussions with amir mohammadi. Concerning the geometry of moduli spaces of sheaves of higher rank, there are two guiding principles for the investigation. We can construct moduli spaces of curves and we know their basic properties.
Next, we looked at solving linear systems via gaussian elimination and lu factorization. The geometry of the moduli space tells you things about the geometry of the spaces points in it. View the article pdf and any associated supplements and figures for a period of 48 hours. Newstead gave a series of lectures in 1975 at the tata institute of fundamental research, mumbai, on git and its application to the moduli of vector bundles on curves. Our aim is to construct categorical quotients with good geometrical properties. However, there are many simple moduli problems which do not admit such a solution. Mathematics tata institute lecture notes tata institute of fundamental research volume 51 of tata institute of fundamental research. Lectures on introduction to moduli problems and orbit spaces. Maryam mirzakhani on dynamics on the moduli spaces of curves. Introduction to moduli problems and orbit spaces p e. Instanton moduli spaces and bases in coset conformal. Notes on the construction of the moduli space of curves dan edidin the purpose of these notes is to discuss the problem of moduli for curves of genus g. The volumes are being made available for free download only for individual use. Using the simplest methods, and proving as much as possible, formulas are derived for the betti numbers.
We prove that the moduli space of trilinear forms onu. We usually assume that g 2 so that each riemann surface gadmits a canonical metric conformal to the complex structure, and g. There are other methods using stacks or algebraic spaces or by direct construction example 1. The construction of moduli spaces and geometric invariant theory by dinamo djounvouna in algebraic geometry, classi. Already, we ran into the problem of what to do if a zero pivot is encountered. Minicourse on moduli spaces university of michigan. If xis a moduli space over a eld kof characteristic zero, then a formal neighborhood of any point x2xis controlled by a di erential graded lie algebra. Introduction to moduli problems and orbit spaces reprint edition by p. The primary example is the moduli space of elliptic curves. Letu, v andw be three dimensional vector spaces over. Find materials for this course in the pages linked along the left. On spaces of maps from riemann surfaces to grassmannians and applications to the cohomology of moduli of vector bundles.
Moduli spaces can be thought of as giving a universal space of parameters for the problem. There are many excellent and thorough references on the subject, ranging from the slightly archaic igu59 and shi94 to the more di. Thus there is no orbit space but there is a categorical quotient consisting of a single point. I have lots of links set up within the pdf doc so i do not wish to have to start all over and recreate all the links. The paper will appear in the book problems on mapping class groups and related topics, ed. Introduction to moduli problems and orbit spaces reprint. Introduction let x be an artin stack always assumed to have quasicompact and separated diagonal over specz. Here different solutions are identified if they are isomorphic that is, geometrically the same. Moduli problems and geometric invariant theory 3 uniquely through. More precisely, these moduli spaces are constructed as a git quotient of a reductive group g q.
Lectures on introduction to moduli problems and orbit spaces tata institute lectures on mathematics and physics 9783540088516. Arithmetic dynamics and dynamics on moduli spaces mathoverflow. If one can show that a collection of interesting objects e. Mar 26, 2015 the determination of the orbit spaces of compact coregular linear groups. Introduction to moduli problems and orbit spaces is supposed to be easier to read than mumfords book on git. This article surveys some developments in pure mathematics which have, to varying degrees, grown out of the ideas of gauge theory in mathematical physics. The rough idea is but see the caveat below that the term moduli space is essentially a synonym for representing object and for classifying space. Peter newstead to cite this version hal archive ouverte. Moduli spaces of sheaves of rank 2 on a surface are not smooth, unless we consider sheaves with special invariants on special surfaces. Geometric invariant theory construction of moduli spaces. Tata institute of fundamental research lectures on mathematics and physics, 51.
Moduli spaces and their birational geometry hanbom moon. Mumford used git to show that the moduli space of curves mg is quasiprojective. Newstead author of introduction to moduli problems and. Concerning the geometry of moduli spaces of sheaves of higher rank, there. Peter newstead gave a series of lectures in 1975 at the tata institute of fundamental research, mumbai on git and its application to the moduli of vector bundles on curves. Maryam mirzakhani on dynamics on the moduli spaces of curves, i. Introduction the following thesis plays a central role in deformation theory. Moduli spaces of curves and maps stanford university. Our method of doing this is geometric invariant theory git. Introduction the purpose of these notes is to provide a quick introduction to the moduli of elliptic curves. Verlag for tata institute of fundamental research 1979. Given a collection of geometric objects, and an equivalence relation among them, we.
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