The Basel Problem---finding the exact sum of the p-series with p = 2---baffled mathematicians for decades until Euler solved it in 1735. He later gave an alternate proof consisting of several steps, "each of which falls well within the scope of a modern calculus course" according to Dunham. Indeed, every step in the proof utilizes standard techniques from Calculus II. Furthermore, the proof broadly draws from diverse portions of the course, including integration by parts, improper integrals, and integration of power series, so that it serves as an elegant example of how these different concepts can be brought together and used in harmony. Calculus II students in my classes complete steps of the proof as "puzzle pieces" during the semester and then, as the final activity of the course, put the pieces together to obtain Euler's sum. The approach of assigning puzzle pieces building through the semester toward an ultimate course goal is transportable to other courses. At its best, in engenders in students feelings of curiosity and mystery, as well as a sense that the course material has a definite direction and larger purpose.
University Calculus II, as taught at SUNY Fredonia, is all about adding together infinitely many things. If the things added are real numbers, then such an infinite sum is a series. In this talk I describe some historical instances of summing series, then focus on one particularly challenging series, the sum of the reciprocals of the squares of all positive integers. Finding its exact value is one of the greatest problems in mathematics, known as the Basel Problem, and it was solved by Leonhard Euler in 1735. I outline two different proofs he gave, one of whose steps all involve topics covered in University Calculus II. Finally, I describe how I have University Calculus II students develop this beautiful result for themselves by completing steps of the proof as "puzzle pieces" during the semester and putting the pieces together in an activity at the end of the course.
This talk explores an interesting connection between the History of Mathematics and University Calculus II courses I'm teaching this semester and contains nothing new. In fact most of it is several thousand years old. But it's a way of thinking about mathematics that to an unfortunate extent has been lost: doing math by pictures. We begin by describing a suspected Babylonian method for approximating square roots, along with the geometry motivating it. Then we proceed to the Calculus II problem of proving the formula for area of a circle, and see how, in conjunction with other influences, it led to rediscovery of a geometric proof of the addition formula for sine.
Groups are important because they measure symmetry. After defining and providing examples of symmetries, this talk examines how transformation groups were historically used to encode the possible symmetries of physical entities. Then the transition from transformation groups to abstract groups through abstraction is described and motivated. Classification of groups is discussed, concluding with an overview of the classification of finite simple groups.
All modern calculus textbooks include formulas for the sum of the first n integers and the sums of their squares and cubes. These formulas are indispensable in evaluating integrals as limits of Riemann sums. The sum of integers formula can be explained in such a way that it is immediately obvious that the formula is correct. Why is the sum of squares formula valid? Textbooks that prove it do so using induction or an argument involving a telescoping sum with cubic terms, neither of which make very clear why the formula has the form that it does. In this talk we explore various alternative derivations of the sum of squares formula. While none of these seem to be quite as illuminating as the proof of the sum of integers formula, they offer an improvement over the standard arguments. [Several quite illuminating proofs were later found and included in this talk.]
A primary ideal I is one with the property that if ab is in I and a is not in I, then b^n is in I for some n. However, in general n can become arbitrarily large. What happens if we require n<=N for a fixed N? This talk gives some answers obtained in joint work with Hetzel. We have coined the phrase "uniformly primary of order N" to describe such an ideal. We will look at various conditions that guarantee an ideal is uniformly primary, as well as conditions that allow comparison of orders of such ideals. We will explore how the uniformly primary notion compares with that of primary and strongly primary in various types of rings. Finally, we will address why an algebraic geometer might care about such a topic.
Algebraic geometry is the study of the interactions between algebra and geometry and the use of these connections to solve problems. In particular, enumerative geometry involves trying to count the number of some type of geometric objects satisfying certain conditions. We will introduce the basic issues and techniques of enumerative geometry by attempting to count the number of roots of a polynomial in one variable. String theory has started what may be the next great revolution in physics, providing a possible starting point for a coherent, fundamental description of the universe. We will explore surprising interactions between string theory and enumerative geometry. This talk is appropriate for undergraduates, and a student who has had one semester of calculus should be able to follow most of it.
A primary ideal I is one with the property that if ab is in I and a is not in I, then b^n is in I for some n. However, in general n can become arbitrarily large. What happens if we require n<=N for a fixed N? This talk gives some answers obtained in joint work with Hetzel.
String theory has started what may be the next great revolution in physics, providing a viable possibility for a coherent, fundamental description of the universe. We will explore surprising interactions between string theory and the mathematical field of algebraic geometry. Most impressively, string theory has provided the inspiration for solving century old problems about counting curves in the plane. In return, algebraic geometry is working to put the intuitive calculations of string theorists on a solid mathematical footing.
We describe how algebraic curves arise naturally in the study of superstring theory in physics and then explain how moduli theory and intersection theory are used to compute the number of curves satisfying certain conditions.
Continuing from last week, we will finish the proof that isomorphism classes of elliptic curves are in one-to-one correspondence with the points of the affine line. Then we will give the more modern definition of a moduli problem in terms of a moduli functor and examine different types of moduli spaces that can result, including fine moduli spaces, coarse moduli spaces, and moduli stacks. We will briefly explore the concept of stacks as time permits.
We will begin describing in rough terms what a moduli space is. Then, after reviewing selected foundational material from algebraic geometry, we will focus on a classical example of a moduli space, the variety of moduli for elliptic curves.
Abstract (Talk 1): Gromov-Witten invariants, moduli spaces of stable maps, and using localization on the latter to compute the former: in some sense they all got their (mathematical) start in this paper. Many of the papers we have covered this semester and previously have been "descendents" of this landmark masterpiece. Today we will focus on Kontsevich's motivation for introducing these concepts, look at three examples of enumerative problems where these methods bear fruit, and possibly start to describe the setup for localization computations. Join us as we look back to where it all began!
Abstract (Talk 2): We will continue to study this groundbreaking paper of Kontsevich. Today's talk will be concerned with the details of the localization computations, especially as they apply to the three enumerative examples in the paper. We will describe the normal bundles of each fixed component under the torus action, as well as the restriction of the appropriate bundles to these components. Once we have reduced the integrals to sums over fixed components using localization, we will examine Kontsevich's (not quite successful) attempts to evaluate these sums using critical values.
Abstract (Talk 3): Filling in a gap from last week's talks, we will explain in detail various examples of the enumerative geometry of rational curves. In particular, we will show how the enumerative numbers in each example correspond to integrals of polynomials in Chern classes. The three examples are degree d rational plane curves through 3d-1 general points, degree d rational curves on a quintic threefold, and the contribution of multiple covers of a rational curve to a Gromov-Witten invariant of a Calabi-Yau threefold. If time permits, we will also discuss how to add up the sums resulting from localization of the above integrals. This is done by identifying each sum with the critical value of a certain functional.
Abstract (Talk 4): Today we will continue the localization computations from last Friday. We have seen how the contribution of the normal bundle of a fixed component breaks up into several pieces. With help from the intersection theory of stable curves, we will compute each piece. Then, for each of the three examples described last time, we will compute for each fixed component the contribution of the relevant polynomial in Chern classes to the integral. The final result is a sum over fixed components that can be added up as described last time.
Algebraic geometry involves solving geometry problems via converting them into problems of algebra. The talk begins with examples showing how algebraic geometry is permeates mathematics from the earliest algebra course. Enumerative geometry counts how many geometric objects there are satisfying certain conditions. We will introduce the basic issues and techniques of enumerative geometry by attempting to count the number of roots of a polynomial in one variable in such a way that every polynomial of degree d has exactly d roots. This talk is appropriate for an undergraduate audience, and a student who has had one semester of calculus should be able to follow most of it.
Abstract(Talk 1): Last week we saw how Hurwitz numbers can be expressed in terms of certain tautological integrals on moduli spaces of stable curves. Today we will look at the other side of the story in this paper of Okounkov and Pandharipande. After giving another definition of Hurwitz numbers as a weighted count of branching graphs, we will find that considering asymptotics of Hurwitz numbers under both definitions gives a new proof of Kontsevich's Theorem on the tautological integrals. Details of the proof will be put off until Friday. Some discussion of matrix models and integrable hierarchies will also be included to give an idea of the approach Kontsevich used in his original proof.
Abstract(Talk 2): We will undertake asymptotic analysis of weighted numbers of branching graphs in order to see where the identities used in the new proof of Kontsevich's Theorem come from. This will involve a a good amount of probability, as it requires a study of random trees. This is a continuation of Wednesday's talk.
Abstract (Talk 1): We will study an approach recently developed by Okounkov and Pandharipande to connect Gromov-Witten theory to matrix models. Such a connection, first conjectured by Witten and motivated by the study of two-dimensional quantum gravity in physics, was a beautiful and unexpected surprise to mathematicians. Kontsevich gave a mathematical proof of Witten's conjecture in 1992, but here the authors give a more comprehensive framework and a new proof in which Hurwitz numbers form the bridge between Gromov-Witten theory and the theory of matrix integrals. This week we will give the background on Gromov-Witten theory and Hurwitz numbers. Next week, matrix models and integrable hierarchies will be introduced and their connections with Gromov-Witten theory via Hurwitz numbers will be made.
Abstract (Talk 2): We will continue to develop the concepts of Gromov-Witten theory, Hurwitz numbers, and their connections as expounded by Okounkov and Pandharipande. In particular, we will explore the Gromov-Witten theory of P^1, briefly review virtual localization, and describe the path between Hurwitz numbers and Hodge integrals. This is a continuation of Wednesday's talk.
We introduce enumerative geometry by considering the example of degree d rational curves through 3d-1 general points in the projective plane. Generalizing this example, we explain how enumerative geometry is carried out using intersection theory on parameter spaces, or moduli spaces, of the geometric objects being studied. Next we define the moduli space of stable maps and survey the cases in which their cohomology rings have previously been computed. Finally, we state a new result: the first presentation for the cohomology ring of a moduli space of stable maps of degree greater than one with more than one marked point.
Abstract: A presentation for the Chow ring of a space \bar{M}_g,n(X,\beta) gives a full understanding of its intersection theory, and also give insight into computations of physically interesting numbers like Gromov-Witten invariants and gravitational correlators. I will give a presentation for the Chow ring of \bar{M}_0,2(P^1,2), the first example of such a presentation for n>1 and \beta=d>1. This presentation is almost completely geometric in the sense that all the relations (are expected to) have geometric explanations. At present, a more computational method involving localization and linear algebra is still needed in a couple places. We will start from the definition of a moduli space of stable maps. This is a talk on my thesis research.
Abstract (Talk 1): A presentation for the Chow ring of a space \bar{M}_g,n(X,\beta) gives a full understanding of its intersection theory, and could also give insight into Gromov-Witten invariant computations. I will give two presentations for the Chow ring of \bar{M}_0,2(P^1,2), the first examples of such presentations for n>1 and d>1. The first presentation is computed via localization and linear algebra. Recently, a second presentation has been formulated in a more geometric fashion. Basics and intuition of computing intersections on the moduli space of stable maps will be explained. This is a talk on my thesis research
Abstract (Talk 2): I will give a more detailed account of the methods used to compute the presentations. This is a continuation of Wednesday's talk.
Abstract (Talk 1): We will discuss this paper, in which Graber and Pandharipande extend the Atiyah-Bott localization formula to singular schemes and stacks which admit an embedding into a nonsingular object. They apply this formula to the moduli stack of stable maps, which allows them to extend the genus zero Gromov-Witten computations of Kontsevich and others to the positive genus case. We will focus in particular on their computation of the contribution of multiple covers of a fixed curve in a Calabi-Yau 3-fold to positive genus Gromov-Witten invariants.
Abstract (Talk 2): Many subtleties are involved in extending the Atiyah-Bott localization formula to singular Deligne-Mumford stacks admitting an embedding into nonsingular Deligne-Mumford stacks. We will investigate how Graber and Pandharipande tackle some of these issues in their proof of this more general localization formula. We will also explore more details of their Gromov-Witten computations. This is a follow-up to Wednesday's introductory talk.
Abstract: Over the past several weeks, we have looked at pieces of the program for construction of Gromov-Witten invariants developed by Konstevich, Manin, Behrend, Fantechi, et. al. We will try to tie together these pieces to give an overall picture of the development and construction of Gromov-Witten invariants and related concepts.
Abstract: We will develop concepts from "Localization," Chapter 9 in Mirror Symmetry and Algebraic Geometry by Cox and Katz. I will take an algebraic approach to equivariant intersection theory instead of the topological approach taken in the text. I will concentrate on building up enough background and localization technique to be able to give a simple example of how I use localization in my research. In particular, I will demonstrate how to get a presentation for the Chow ring of the moduli space of 1-pointed stable maps of degree 2 to the projective line.
Abstract: In this paper, Andrew Kresch defines Chow groups of Artin stacks. He also shows how intersection theory with integer coefficients can be carried out on Deligne-Mumford stacks. So we will be expanding in two directions on the previous talks about intersection theory with rational coefficients on Deligne-Mumford stacks, as well as looking ahead toward intrinsic normal cones, which are Artin stacks. However, intersection theory on Artin stacks is much more subtle, and requires different techniques.
Abstract: From an algebraic geometry viewpoint, Gromov-Witten invariants are degrees of intersections on moduli stacks of stable curves or stable maps. Thus for a rigorous treatment of the Gromov-Witten theory, we need to have a good understanding of how intersection theory works on stacks. We will summarize the first three sections of paper of this title by Vistoli, which develops intersection theory in the spirit of Fulton for Deligne-Mumford stacks. Time restrictions will again largely limit us to stating important concepts and results.
Abstract: The theory of stacks is the foundation upon which the algebro-geometric theory of Gromov-Witten invariants is built. We will focus on motivation, definitions, and examples. The talk will be based primarily on the appendix to Vistoli's paper "Intersection theory on algebraic stacks and on their moduli spaces."
Abstract: The sheaf O(1) on a projective variety is of central importance in algebraic geometry. In Chapter II, Section 5 of Hartshorne's Algebraic Geometry, the sheaf O(1) is introduced as "the twisting sheaf of Serre." While "twisting" has an explicit algebraic meaning, a name like that should certainly have geometric motivation as well. To see this, we use the language of bundles as well as "translations" between it and the language of sheaves. I will assume the definition of a line bundle, both geometrically and in terms of transition functions. References for these definitions can be found on my web site. I also will assume various results from the above mentioned section of Hartshorne. In particular, I will assume the standard description of global sections of the sheaves O(n). This follows from the above mentioned section of Hartshorne, and is a good exercise to do beforehand if you haven't worked through it before.
Abstract: This fundamental paper of K. Behrend and Yu. Manin will be discussed. The paper proves important basic facts about moduli stacks of stable maps and, in special cases, the "virtual fundamental class" and the Gromov-Witten Invariants.
Abstract:We will discuss this paper of Rahul Pandharipande. In a previous talk we proved a few facts about these Q-divisors. The definitions and facts will be restated. This time we will actually compute intersections of Q-divisors and demonstrate applications to enumerative geometry. This is an important beginning step in understanding the Chow ring of the moduli space of genus 0 stable maps.
Abstract: We will explore results from Pandharipande's paper "Intersections of Q-divisors on Kontsevich's Moduli Space \bar{M}_0,n(P^r,d) and Enumerative Geometry." Basically, he tells us how to do intersection theory of divisors on the moduli space of stable maps, assuming we restrict to genus zero and take rational coefficients. The results allow many enumerative computations. However, in this talk we will cover only Section 1, which focuses on the vector space structure of the Picard group. Intersecting divisors with curves in the moduli space is a key technique.
Abstract: What happens to a bundle when you push it forward? Riemann-Roch theorems answer this question. This information, together with appropriate exact sequences of bundles, gives relations in Chow ring of a space which then allow computation of certain integrals. The integrals give enumerative information about the space. I will give examples for Grassmannians and, at least conjecturally, for moduli spaces of stable curves.
Abstract: We will begin with a brief seminar organizational meeting for about five minutes. We then review the construction from the paper "Notes on Stable Maps and Quantum Cohomology," focusing on sections 3 and 4. (Relevant definitions will be recalled.) These spaces are important in mirror symmetry (subject of a course this semester) and string theory (subject of a RAP this semester).
Abstract (Talk 1): The basic ideas from this paper of Fulton and MacPherson will be presented. Emphasis will be placed on determining the Chow rings of these spaces. Connections will be made with moduli spaces of stable maps
Abstract (Talk 2): We will continue looking at various descriptions of the Fulton-Macpherson compactifications, from the intuitive to the functorial to an explicit construction by blowups. Time permitting, we will begin calculating the Chow rings of these spaces.
Abstract (Talk 3): We will inductively construct the Fulton-MacPherson compactification X[n] of configuration space using an explicit sequence of blowups. We will state some further properties of X[n] that follow from this construction. We may start laying the foundations for calculating the intersection ring of X[n].
Abstract (Talk 1): A highly condensed summary of the paper will be presented. We will fill in details in future talks as time and understanding permit. I will outline the method used to describe the "cohomology ring of degree 3 maps into infinite-dimensional projective space."
Abstract (Talk 2): We will extend to P^n the constructions done for P^1 in the first talk. Hopefully we will also compute the cohomology ring of stable maps into infinite dimensional projective space with image of degree 3. We may also consider the degree 2 case.