Harvard Mathematics Logo Summer Tutorials 2001

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NOTE: This is an archived page from a previous tutorial. Information about the current tutorials can be found here.

Welcome Message


Now that you have some days to think about the summer tutorials, can those of you who are definite about signing up, or even just interested, please email to Svetlana (svetlana@math) your level of interest (definite, highly possible, maybe) with an indication of which tutorial you want? We would like to get a rough head count of the level of participation.
Please do this ASAP (after some aimless procrastination) to help out with our planning. We haven't set a final deadline for joining up, but even so, we need to be able to make plans, so help us out. However, if you are definite, then say so and you will be guaranteed a slot.
Assuming sufficient interest, we will be running three tutorials this summer.
  • Special functions and their many guises in number theory and geometry: by Izzet Coskun (coskun@fas)
  • Complex dynamics: led by Laura DeMarco (demarco@math)
  • Hyperbolic geometry in 2 and 3 dimensions: led by David Dumas (ddumas@fas)

They will meet in the evenings so as not to interfere with any day time obligations that you might have; they will most probably meet twice per week for six weeks or so, starting the week of July 1. As with last year's tutorials, they will not count for Harvard University credit nor Math Concentration credit, but the final paper can be used for the 5 page, junior paper requirement. You will also receive a 'stipend' of $600 for attending the tutorial. This is not enough to live on (unless you sleep in the Cambridge Common and eat only oatmeal), so you should arrange some other activities for the summer in Cambridge. Also, you will be responsible for your own lodging, etc. Our tutorial program has no official university affiliation and no obligation to take care of you during the summer.
The format of the tutorial is the same as during the semester, which is to say that the tutorial leader will lecture at the beginning, but the participants will lecture towards the end on topics that they have researched during the program. The final paper will be an exposition of the researched topic.
With the preceding understood, what follows are the tutorial topics and the tutorial leaders. Attached below are syllabi for the tutorial. Also, feel free to email the leaders to get more information about the subject matter and prerequisites.


Tutorial: 'Special Functions and Their Many Guises in Number Theory and Geometry'

Led by Izzet Coskun (coskun@math)
The aim of my tutorial will be to introduce the students to special functions in complex analysis by showing them a sample of "neat" applications. I hope in the process I can get them to appreciate and enjoy classical analysis. A first course in complex analysis should be enough preparation for the tutorial. I am planning to review the main theorems as I go along, so I hope background will not be such an issue. The main functions I am planning to talk about are the Riemann zeta function, the Gamma function, the Theta functions, the elliptic functions and (if there is time) a little bit about the geometry of univalent functions. I am planning to introduce these functions in the context of a problem. I will discuss the zeta function in conjunction with distributions of primes. I will prove the prime number theorem. I might talk a little bit about the Riemann hypothesis. Depending on how the discussion is going I might either talk about L-functions and the Dirichlet density theorem or about special values of the zeta function. For example, the irrationality of zeta of 3 might be interesting to talk about. (Of course, both of these might be project topics in case we are running short on time). I will probably need to introduce the Gamma function for my discussion of the zeta function. It would be interesting to illustrate how the gamma function and its close cousins like the beta function come up in counting problems. A potential problem here would be counting the number of ways of identifying the sides of a polygon to obtain a genus g surface. Next I would like to talk about partitions. Especially Euler's pentagonal number formula, the Roger-Ramanujan identities and a little bit about the theta functions. The Jacobi identity would be nice to discuss. Next I would like to turn to geometry. I will present the Riemann mapping problem. Then see that the elliptic functions map the rectangle to the disc. I can talk about their addition formulas and the differential equations they satisfy and briefly tie this in to the theory of Riemann surfaces and elliptic curves. (I would not want to spend so much time on this since other courses at Harvard seem to discuss the elliptic functions in depth. The point would be for the students to see that some of the identities that were proven using classical analysis can be reinterpreted in terms of algebraic geometry and representation theory.) Finally, I would like to briefly discuss the theory of univalent functions. The Bieberbach conjecture would be fun to talk about. This might be a good source of project topics. Students can think about quasiconformal maps and so on. If somebody is interested, a topic in Teichmuller theory might be fun to work on. Maybe I can get somebody to do a project on hypergeometric functions.

Tutorial: 'Complex Dynamics'

Led by Laura DeMarco (demarco@math).
Have you seen pictures of the Mandelbrot set, but aren't really sure what it is? Have you wondered why Julia sets look so cool? Did you enjoy your complex analysis class but don't know what it's good for? Do you like Milnor's writing style? and the company of friends to enjoy a seminar-like exploration of a fun subject? Join us!
Complex dynamics is a beautiful subject which is an active area of current research, incorporates many areas of mathematics, and is easily accessible to students of all levels. The principal goal of the tutorial is to present a picture of what dynamics of one complex variable is all about - illustrating tools primarily from complex analysis and a bit of hyperbolic geometry. The main text will be Milnor's Dynamics in One Complex Variable.
The fun of the subject is that, depending on time and student interest, there is freedom to delve into any of the following areas: (1) ergodic theory, by examining invariant measures and entropy, (2) potential theory, by examining the role of equilibrium (charge) distributions, (3) number theoretic properties of rotation numbers and the relation to local dynamics, (4) computer experiments to visualize the wild geometry of invariant sets, (5) history of the subject (a personal favorite), and of course, (6) the relation to the study of real dynamics, expansive behavior, stability, and chaos.
My vision for the course is that we begin with informal lectures following Milnor, move into student presentations of sections from the text, and finally develop independent projects in any of the related areas. I expect students to present their research projects to the group as well as submit a short written summary of what they have learned.
Prerequisite - complex analysis on the level of 113

Tutorial: 'Hyperbolic Geometry in Two and Three Dimensions '

Led by David Dumas (ddumas@math)
Abstract: We will discuss hyperbolic geometry in dimensions two and three at an introductory level, starting with the basics of geometry in the hyperbolic plane and hyperbolic 3-space. The intrinsic geometry of closed hyperbolic surfaces will be emphasized, with detailed investigation of specific examples. We will also give a flavor of the vast field of hyperbolic 3-manifolds through gluings of polyhedra and link complements. Outline of topics to be covered:
  • The hyperbolic plane, its metric, various models thereof
  • Hyperbolic geodesics, angles, areas, polygons
  • The group of hyperbolic isometries, classification
  • Construction of hyperbolic surfaces
  • Comments on Hyperbolic vs. Riemann surfaces vs. algebraic curves
  • Geodesics and isometries of hyperbolic surfaces
  • Examples of hyperbolic surfaces
  • Hyperbolic 3-space, isometries, geodesics, hyperplanes
  • The sphere at infinity, connection with conformal geometry
  • Examples of hyperbolic 3-manifolds
Some possible student project topics:
  • Construction of hyperbolic polyhedra (Andreev's theorem)
  • Tilings of the hyperbolic plane and their symmetry groups
  • Geodesics on hyperbolic punctured tori
  • Hyperbolic surfaces with many automorphisms
  • Volumes of knot complements
  • Convex geometry in hyperbolic space
  • Hyperbolic manifolds with boundary
Prerequisites: Some familiarity with topology and differential geometry of smooth manifolds (especially closed surfaces). Previous exposure to Riemannian geometry would be useful, but is not necessary.
As summer tutorials are typically small, there is a great opportunity to tailor the level, pace, and content to the common background of a group of students, working from the basic outline and ideas above. References:
  • Three dimensional geometry and topology, W. Thurston, Princeton Mathematical Series, 35. Princeton University Press, 1997
  • Lectures on hyperbolic geometry, R. Benedetti and C. Petronio, Universitext, Springer-Verlag, 1991
  • Foundations of hyperbolic manifolds, J. Ratcliffe, Graduate texts in mathematics 149, Springer-Verlag, 1994
Archived in March 2002, from here