Mathematics Department Graduate Seminar - Spring 2005

Instructor: John Roe

Meeting: Tuesdays and Thursdays, 11.15 a.m. in 312 Boucke

Aims of the Seminar:

- to learn some basic principles about writing mathematics
- to understand how to search the mathematical literature
- to gain experience in preparing and making mathematical presentations
- to be aware of the advantages and pitfalls of various communication technologies
- to learn about the 'etiquette' of mathematical writing and presenting
- to meet various faculty members
- to see some beautiful nuggets of mathematics, and
- to have fun.

Seminar Timetable. (This will be modified as the semester progresses.)

Week | Tuesday | Thursday |

2 | 1/18 Course Introduction; Mathematical Writing I; Homework Assignment | 1/20 Faculty Presentation: What is a C*-algebra?
(John Roe); Giving and Receiving Feedback |

3 | 1/25 Mathematical Writing II; Mathematical Writing III | 1/27 Faculty Presentation: The Fourier Transform
(Victor Nistor); Review of Homework Assignment
(mini-definitions) |

4 | 2/1 Faculty Presentation An Application of the
Borsuk-Ulam Theorem (Dima Burago); Using Slides; Homework Assignment |
2/3 Student Presentations: Steve Hair, Jonas Kibelbek |

5 | 2/8 Faculty presentation What is a Computable Function?
(Steve Simpson); Mathematical Speaking |
2/10 Student Presentations: Lei Zhang, Nikita Zinovyev |

6 | 2/15 Faculty Presentation What is a representation>
(Adrian Ocneanu); Technological Tools; Homework
Assignment |
2/17 Faculty Presentation (Nigel Higson) The Cayley
graph of a group ; Student
Presentation (Uuye Otgonbayer) |

7 | 2/22 Faculty Presentation What Is a Latin Square?
(Gary Mullen); Student Presentation (Ho Wing Kai) |
2/24 Faculty Presentation Classical Mechanics and
Poisson Algebras (Aissa Wade); Student Presentation (Andriy Gogolyev) |

8 | 3/1 Faculty Presentation Billiards (Sergei
Tabachnikov); Student Presentation(Vivek Srikrishnan) |
3/3 Student Presentations (Hu Yong) and Questions |

9 | 3/15 | 3/17 |

10 | 3/22 | 3/24 |

11 | 3/29 Presentation: Andriy, Geodesic flows.
Chair: Vivek |
3/31 Presentation: Lei, Boundary Problems and
Evolution Equations. Chair: Nikita |

12 | 4/5 Presentation: Steve, Elliptic curves and
cryptography. Chair: Otogo |
4/7 Presentation: Vivek, Degree and the Poincare-Hopf
theorem on vector fields.
Chair: Hu Yong |

13 | 4/12 Presentation: Wing Kai, Integral Geometry.
Chair: Jonas |
4/14 Presentation: Nikita, Nash's equilibrium point
in zero sum games.
Chair: Steve |

14 | 4/19 Presentation: Otogo, Compact Lie Groups.
Chair: Andriy |
4/21 Presentation: Jonas, The logistic
equation, bifurcations and chaos. Chair: Wing Kai |

15 | 4/26 Presentation: Hu Yong, C*-Algebras.
Chair: Lei |
4/28 Wrap-up session |

The sessions before Spring Break will be structured as a sandwich: two 20-minute mini-presentations will form the bread, and a free discussion/response time will form the filling.

The student presentation sessions after Spring Break will be structured as a 50-minute presentation followed by a discussion time. Each session will have a student chairperson, and a student presenter. A third student will be designated to give feedback on the presentation.

Homework will be assigned for some sessions.

List of topics suggested by participating faculty members.

Each student in the course is required to give a 50-minute presentation. The topic should be a "jewel of mathematics" which can be appreciated and enjoyed by all the other class participants. You are advised to consult a relevant faculty member while you are preparing your talk. Some faculty members who have agreed to participate are listed below, together with some suggested topics; but you are welcome to choose a topic (or indeed a faculty adviser) not appearing on the list below!

Faculty Member | Possible Topics |

Mark Levi | Gibbs' phenomenon; Stokes' theorem; Gauss-Bonnet theorem and mechanical analogies. |

Steve Simpson | The continuum hypothesis; Uncountable cardinal numbers; Turing machines and computability; Kolmogorov complexity and randomness; Polynomial time computation and P=NP; Unsolvability of the Halting Problem; Unsolvability of the word problem for semigroups and/or groups; Degrees of unsolvability. |

Victor Nistor | Spectral theory and evolution equations; Boundary value problems on nonsmooth domains; Ellipticity and pseudodifferential operators. |

Sergei Tabachnikov | Skew and totally skew embeddings and immersions; Tire track geometry and flotation problems; Around Hilbert's fourth problem and magnetic analogs; Multidimensional outer billiards; Topological robotics and motion planning. |

Howie Weiss | 1) Proof of Perron-Frobenious theorem via Hilbert Projective Metric and application to Leslie population models or betting on football games. 2) Continued fractions and coding of geodesics on the modular surface 3) What is a fractal (a introduction to Hausdorff dimension and box dimension)? 4) How to solve the dilatation equation? This is the hard part of constructing wavelets. 5) Poincare-Bendixson theorem 6) What is the index of a vector field and the Poincare-Hopf formula 7) Heisenberg uncertainty principle 8) How is complex analysis for functions of several variables different from one variable? 9) The Pompeiu intregral formula and solving the delta-bar equation 10) Does Moreira's theorem hold for one triangle which is rotated and translated around the plane? How about for all translations of two circles with different radii? 11) What is a weak solution of a PDE? 12) The Cauchy-Kovalevsky theorem 13) The Ising model in one dimension 14) How to obtain an operator algebra from a finite mp dynamical system. 15) Ricci flow on surfaces of higher genus. On spheres, 16) What is Cech cohomology? 17) What is chaos? 18) The eight(ish?) homogeneous geometries on three manifolds. 19) Minkowski's theorem 20) How many lattice points are there in a circle of radius r? |

Nigel Higson | 1) Weyl's theorem on eigenvalue asymptotics (for domains) 2) Representations of U(n) (classification via characters) 3) Representations of compact groups and Hilbert's fifth problem 4) Traces of Hilbert space operators; Liddski's theorem 5) The Schrodinger equation for the hydrogen atom 6) Property T (say finite generation property plus the example of SL(3)) |

Aissa Wade | |

John Roe | |

Dima Burago |

- Alice Niemeyer's Scientific Communication course. I have made use of many ideas from this website in designing our own course.