Spring 2008 graduate courses Courses with an asterisk are special topics courses. Title and sometimes a description are below. MWF T TH 8 MATH 631 (Fitzpatrick) 9:30 MATH 648M* (Margetis) 10 MATH 601 (Haines) AMSC 614 (Nochetto) MATH 748D* (Schafer) STAT 798L* (Slud) 11 MATH 734 (Rosenberg) 11 MATH 648I* (Okoudjou) STAT 741 (Kedem) AMSC 661 (Elman) AMSC 667 (Levy) MATH 648J* (Tamvakis) MATH 643 (Forni) 12 MATH 636 (Adams) MATH 671 (Kaloshin) meets WF for 1h 15min. 12:30 MATH 661 (Jakobson) STAT 601 (Koralov) MATH 674 (Machedon) 1 STAT 601 (Koralov) 2 MATH 742 (Novikov) 2 AMSC 698T* (Tadmor) STAT 705 (Slud) MATH 608E* (Prasanna) MATH 713 (Laskowski) 3 Faculty meeting hour 3:30 5 STAT 650 (Dolgopyat) 5 STAT 701 (Kagan) AMSC 664 (Balan) MAIT MAIT 623 MAIT 633 MAIT 679E (Benedetto) Wed. 6 - 8:45 pm MAIT 679M/AMSC698M (Balan) Wed. 6 - 8:45 pm MAIT 679T Special Topics Courses MATH 608E (Prasanna) Topics in Iwasawa theory About the subject: Iwasawa theory is a central area of research in algebraic number theory that has played a very important role in some of the most spectacular results of the past two decades, most notably the proof of Fermat's last theorem. The main objects of study in Iwasawa theory are certain arithmetic analogues of the Tate module in algebraic geometry. The "Iwasawa main conjecture" seeks to provide a description of these modules in terms of analytic objects called p-adic L-functions. It may thus be considered part of a long tradition in number theory that seeks to relate arithmetic (or algebraic) objects with analytic ones. Outline of the class: After reviewing the necessary background from class field theory, we will discuss p-adic L-functions and the Iwasawa main conjecture in various contexts. Finally, we will discuss the method of "Euler systems" and its application to proving some cases of the main conjecture. Prerequisites: Basic algebraic number theory, some familiarity with class field theory, elliptic curves and Galois cohomology. MATH 648M (Margetis) Advanced analytic methods with applications (back by popular demand) The course will include material on asymptotics for integrals, ODEs PDEs and integral equations; perturbation theory; some introduction to probability and stochastic processes; applied stochastic ODEs and PDEs; renormalization group and other methods of statistical mechanics; and introduction to multiscale analysis. Applications would include current problems in physical science and engineering. MATH 648I (Okoujdou) Analysis on Fractals description: The goal of this course is to introduce the students to some analytical tools on a class of fractals that includes the Sierpinski gasket. Topics include: Measure, Energy, and Metric on the Sierpinski gasket; Weak and Pointwise formulations of the Laplacian on the Sierpinski gasket; Spectrum of the Laplacian; Postcritically Finite Fractals. Moreover, selected topics such as Polynomial, and Power series; heat kernel estimates; and convergence of Fourier Series will be explored based on the interest of the audience. (I'll be using the following book: Differential Equations on Fractals. A Tutorial, by Robert S.~Strichartz, Princeton University Press, Princeton, NJ, 2006). MATH 648J (Tamvakis) Riemann surfaces This is a course on Riemann surface theory from the point of view of complex geometry, and can be taken as a second course after complex variables. We will only assume a rigorous background in real and one variable complex analysis, and cover any additional prerequisites during the lectures. The topics to be covered are a classic meeting ground of complex analysis and algebraic geometry, but the algebraic aspect of the theory will lie mostly in the background. After discussing the basic facts and examples of Riemann surfaces, we will proceed to more advanced topics: the Riemann surface of an algebraic function, cohomology of line bundles, divisors and the Riemann-Roch theorem, the canonical bundle and Serre duality. Our proofs will be analytic, for example Serre duality will be proved using a regularity theorem for the d-bar operator. We will then discuss Abel's theorem and the Jacobian, and continue with a treatment of theta functions, the theta divisor, and Riemann's beautiful theorem about meromorphic functions and theta. If time permits we will prove Torelli's theorem and talk about the Dirichlet problem on Riemann surfaces and the uniformization theorem. MATH 748D (Schafer) Homotopy Theory AMSC 698T (Tadmor) Time dependent nonlinear PDE's