Topics and Graduate Course Descriptions

Topics and Graduate Course Descriptions

Fall 2011
- Math 102: Foundations of Smooth Manifolds (Sutton)
  Prerequisites: Linear algebra (Math 24), point-set topology (math 54) and multivariable analysis (Math 73). It will also help to be familiar with covering spaces and the fundamental group.
  
  Description
  Manifolds provide mathematicians and other scientists with a way of grappling with the concept of "space." The space occupied by an object. The space that we inhabit. Or, perhaps, the space of configurations of a mechanical system. While manifolds are center stage in the study of geometry and topology, they also provide an appropriate framework in which to explore aspects of mathematical physics, dynamics, control theory, medical imaging, econometrics and robotics, to name just a few. This course will serve as an introduction to the basics of manifold theory and lay the foundations needed to explore problems in which "space" plays a fundamental role.
  
  Topics will include some of the following: manifolds & tangent bundles; vector fields & vector bundles; smooth maps and the inverse function theorem; cotangent bundle & differential forms; densities, integration on manifolds and the generalized theorem of Stokes; Whitney's imbedding theorem; Lie groups, group actions & homogeneous spaces; Tensor Fields & Riemannian metrics; Differential forms & orientation; integral curves & the existence and uniqueness theorem for ODEs; Distributions and Frobenius' Theorem.
- Math 105: Primes and polynomials (Pomerance)
  Prerequisites: An undergrad number theory course, as well as some abstract algebra. I'll be happy to try and fill in gaps for motivated students.
  
  Description/Grading
  This course will develop elementary estimates on the distribution of prime numbers, including elementary sieve methods. A recurrent theme will be the prime factorization of integer-polynomial values. We will follow the first 3 chapters of Pollack's "Not always buried deep", as well as Chapter 6, and also Chapter 3 of Montgomery & Vaughan's "Multiplicative number theory". I recommend acquiring Pollack's book, the second book is optional.
  
  Grading: There will be weekly written assignments, plus some class presentations. There will be no formal exams. Enrolled graduate students who have been admitted to candidacy will be excused from the written assignments.
- Math 108: Combinatorial Representation Theory (Orellana)
  Prerequisites: Linear algebra and algebra (Math 31, 71, or 101). No prior knowledge of combinatorics or representation theory is expected.
  
  Details
  This is an introduction to algebraic combinatorics, specifically symmetric function theory and its connections to representation theory, algebraic geometry, and tableaux combinatorics.
  Full description.
Winter 2012
- Math 17: An Introduction to Mathematics Beyond Calculus (Shemanske)
  Prerequisite: Math 8, or placment into Math 11.
  
  Details
  This year's offering will be "From Caculus to Elliptic Curve Cryptography in ten weeks"
  See the web site.
- Math 89: Set theory (Weber)
  Prerequisite: Math 39 or Math 69 or familiarity with the language of first-order logic and readiness for an upper level math course.
  
  Details
  Math 89 satisfies the culminating experience requirement for mathematics majors, and is appropriate for any graduate student who wants to take a course in logic and doesn't yet know a lot of set theory.
  
  We will study the axioms of set theory, what they tell us about the universe of sets, and how we can use the universe of sets to study the universe of mathematics.
- Math 112: Geometry (Gordon)
  Prerequisite: A course in differential topology, including vector fields and their flows. (The course that used to be called Math 124 and is called Math 102 this fall is ideal.)
  
  Details
  Lie groups and Lie algebras are ubiquitous throughout mathematics, e.g. in geometry, number theory, combinatorics, analysis and dynamics. The first part of the course will address the basic notions and results elucidating the Lie group -- Lie algebra correspondence. We will then introduce various types of Lie groups and Lie algebras (semisimple, solvable, nilpotent, compact). There are various directions that we could pursue after that, depending on time and the interests of the class.
- Math 122: The Atiyah-Singer index theorem and the heat kernel proof (van Erp)
  Prerequisites: the introductory graduate level analysis and topology sequences (103/113, 124/114).
- Math 126: Numerical analysis for PDEs and wave scattering (Barnett)
  Prerequisite: some programming experience (preferred: Matlab/octave, C, or fortran; esp. the first).
  
  Recommended background: some PDEs (could be at undergrad level, eg Math 46) and real analysis (Math 63 and some graduate-level functional analysis). However the background is flexible: a motivated advanced undergrad or other science/engineering/CS student (undergrad or grad) could pick up enough to learn a lot and do well.
  
  Details
  This will cover modern integral-equation-based methods for solving piecewise-constant coefficient PDEs (focusing on those arising in electrostatics and time-harmonic waves), including the theory, analysis, and coding experience required for proficiency. Much content will overlap with previous Math 116's offered by Barnett.
  
  We start with an introduction to numerical analysis, conditioning and stability, numerical interpolation and integration, followed by potential theory, Laplace's equation and Helmholtz equation. The latter leads to scattering problems, a flavor of fast algorithms (FFT, fast multipole), and final projects. Homework will be dominated by coding and implementing the algorithms, but also include some proofs. Projects can be coding-based or a deeper study of numerical analysis results.
Spring 2012
- Math 121: Current problems in algebra (Webb)
- Math 125: Reflection and Coxeter groups; Buildings and Classical Groups (Shemanske)
  Prerequisites: 101, 111 (suitable for first year students).
  Any number theory needed (not much) will be developed.
  
  Details
  This is bits of algebra, geometry, topology and a little combinatorics all rolled into one course. The goal is to say something about the theory of buildings, with slightly more interest in their combinatorial/topological structure and number theoretic applications than their original purpose, which was to understand the structure and classify classical p-adic groups (using the building as a representation space), in analogy to semi-simple Lie theory.
- Math 128: Topics in Combinatorics (Elizalde)
  Prerequisites: Math 118. If you have had an advanced undergraduate course in combinatorics and are interested, talk to Sergi about your preparation.
  
  Details
  The course will cover pattern-avoiding permutations, permutations in dynamical systems, bijections, counting methods, generating trees, and perhaps asymptotic analysis of generating functions, among other topics to be decided.
Winter 2011
- Math 17: Beyond Calculus
  W11 topic: Random Walks and Electric Networks
- Math 102: Topics in Geometry
  W11 topic: The soccer ball and the solution of equations of the fifth degree
Winter 2010
- Math 17 (An Introduction to Mathematics Beyond Calculus)
  W10 topic: Applications of algebraic structures in number theory and geometry
Winter 2008
- Math 17 (An Introduction to Mathematics Beyond Calculus)
  W08 topic: The Isoperimetric Problem
Fall 2007
- Math 53 (Chaos!)
Winter 2007
- Math 17 (An Introduction to Mathematics Beyond Calculus)
  W07 topic: The Geometry of the Fourth Dimension
Spring 2006
- Math 112 (Introduction to Riemannian Geometry)
Winter 2006
- Math 116 (Boundary Methods and Wave Asymptotics)