FALL 2007 graduate courses Special topics courses with an asterisk have a descriptive paragraph listed below the schedule. MWF T TH 9 MATH 630 (Fitzpatrick) 9:30 MATH 670 (Balan) MATH 603 (Schafer) AMSC 600 (O'Leary) 10 MATH 600 (Haines) MATH 748R (Rosenberg) STAT 770 (Smith) 11 MATH 648G (Grillakis) 11 MATH 712 (Laskowski) STAT 740 (Kedem) MATH 730 (Cohen) MATH 634 (Warner) MATH 621 (Washington) AMSC 660 (Elman) 12 MATH 642 (Forni) MATH 718L (Goodrick) MATH 748G* (Goldman) MATH 673 (Machedon) 12:30 MATH 660 (Jakobson) MATH 648K* (Kaloshin) MATH 608T* (Tamvakis) STAT 600 (Koralov) 1 AMSC 666 (TVP) MATH 608R* (Ramachandran) 2 MATH 740 (Novikov) 2 MATH 648B (Benedetto) STAT 710 (Slud) 3 Faculty meeting hour 3:30 5 AMSC 663 (Balan) STAT 700 (Kagan) MAIT MAIT 623 (Konstantinidis) Mon. 6-8:45pm MAIT 633 (Healy) MAIT 679E (Easley) MAIT 679T Special Topics Courses MATH 608R (Ramachandran): Etale cohomology and the Weil conjectures The conjectures of Weil have influenced (or directed) much of 20th century algebraic geometry. This course will provide an overview of the methods and ideas which have led to the formulation and the proof of the Weil conjectures (due to Grothendieck, Deligne). In particular, the basic concepts of etale cohomology (and why it is a generalization of Galois cohomology) will be presented. Textbooks are the book by E. Freitag and R. Kiehl (whose title is the title of the course) and the course notes on etale cohomology by Jim Milne available at www.jmilne.org. For the French literati, the excellent notes of Deligne in SGA 4 1/2 (scans available online) called Cohomologie etale cannot be recommended enough. The course should be of interest to aspiring number theorists and algebraic geometers. Basic material from commutative algebra, homological algebra and manifold theory will be assumed. MATH 608T (Tamvakis): Grassmannians, Flag Manifolds, and Schubert Calculus Our goal in this course is to study Grassmannians and flag manifolds from many different points of view, all related to each other: - Geometry: Homology and cohomology theories for these manifolds; Schubert varieties, Chern classes and Schubert calculus. Applications to classical enumerative geometry. - Algebra: The structure of the cohomology rings of these spaces is described by the beautiful theory of Schur polynomials (for Grassmannians) and Schubert polynomials (for flag varieties). We will make connections with the representation theory of the symmetric and general linear groups. - Combinatorics: Partitions: Young diagrams and tableaux, calculus of tableaux. Permutations: the Bruhat order, divided difference operators, combinatorics of Schur and Schubert polynomials. - Quantum cohomology: In the latter part of the course, we will be in a position to study recent work on the enumerative geometry of rational curves in the above spaces. This leads to an elementary treatment of the "quantum" versions of all the above questions, and an understanding of the "quantum Schubert calculus". Background necessary: I will attempt to keep the discussion as self-contained as possible. This means I will NOT assume familiarity with algebraic geometry, cohomology theories, characteristic classes, etc. Rather than give a full exposition of all the background needed (as this is impossible to do in one semester), I will tell you enough about each topic so that you can understand what is going on. I hope that what people see in the course motivates them to learn more about each individual subject (it worked for me!). In the first part of the course we will be studying geometry, so it will help to have a general familiarity with geometric reasoning. MATH 648B (Benedetto): Harmonic Analysis and Waveform design MATH 648G (Grillakis): Applied Analysis MATH 648K (Kaloshin): Instabilities in the 3-body problem The course will be mainly devoted to the (Newtonian) 3 body problem, where point masses are mutually attracted with a force proportional to the product masses and inverse proportional to the distance between them. We start with a description of motions of the 2 body problem. Then we will discuss various fascinating examples of motions in the 3 body problem including KAM quasiperiodic motions, Sitnikov example of oscillatory motions, choreography, and etc. Time permitting we shall talk about non-collision singularities and ergodic theorems of celestial mechanics. MATH 748G (Goldman): Geometric Structures on Manifolds Given a topology, a loose organization of points, can you rigidify it by geometry? A geometry here means a homogeneous space of a Lie group, with such quantitative notions as distance, angle, parallelism, linearity, circularity, and so on. Even geometries lacking an invariant notion of distance are rich and fascinating. A locally homogeneous geometric structure (in the sense of Ehresmann and Thurston) is defined by a system of local coordinate systems on the topology, taking values in the geometry. Different local coordinates on overlapping patches relate by geometric automorphisms For example the sphere S^ admits no structure modeled on Euclidean geometry R^n: every world atlas must be metrically inaccurate somewhere. This subject naturally leads into the theory of moduli of representations of fundamental groups of surfaces. Full of rich examples, it relates to Differential Geometry, Geometric Topology, Algebraic Geometry, Lie Groups, Dynamical Systems, and PDE. We will loosely follow selected publications and preprints, my notes Projective geometry of manifolds, (which I am expanding into a book) and Thurston's monograph, Three-dimensional geometry and topology. The class meets Monday-Wednesday-Fridays at 12:00. MATH 748R (Rosenberg): Noncommutative Geometry and Topology MATH 748T (Novikov): Theory of knots Nontrivial knots were observed over 2 thousand years ago. Algebraic topology made an important breakthrough here in 1950-1970s. New type unexpected ideas appeared in this area during last 20 years. Topological invariants with remarkable properties have been found. It led to solution of some classical problems of knot theory . These ideas have original sources in math physics (Yang-Baxter Equations, Topological Quantum Fields Theories). Literature:1. L.Kaufmann ''Knots and Physics'', Part I, 2. D. Bar Natan. Lectures in the theory of Vassiliev invariants. Very recent new approach in the theory of knots will be also presented. No knowledge of Classical Algebraic Topology is required.