**Junior Colloquium**

The Junior Colloquium is a series of talks intended for students interested in mathematics or related subjects, started in the fall of 2002. The JC takes place roughly every other Thursday at 3:30 in the fourth floor colloquium room of Ayres Hall. The JC attracts a large and diverse audience, and students at all levels (and even faculty) are invited to attend. Anyone interested in receiving e-mail announcements about the JC (who is not already on the UTKMATH, seminarlist or pmail e-mail lists) will find information on the Tennessee Today web site or on our weekly seminar list.

For those interested in speaking, here are some hints about what is expected:

1. Talks should be accessible to anyone with a good understanding of basic calculus. If substantial portions of the talk require a higher level of mathematics then the necessary background should be mentioned in the abstract.

2. Ideally, talks should appeal to a wide audience, which often includes engineering and other non-math majors.

3. Faculty may give talks as often as they wish--keep your notes/slides for future use! However, the same talk may be given at most once in any two consecutive years.

4. It is OK to use a talk to advertise an area of mathematics or a career field, but the main purpose of the talk should be to to tell an interesting story about problem(s) in pure or applied mathematics.

Anyone who would like to receive notices about the JC should go to listserv.utk.edu and add his/her e-mail address to the JRCOLL listserv.

Previous subjects have ranged from quaternions to soap bubbles to tornadoes, and previous speakers have included UT faculty and invited visitors from other universities. Potential speakers should contact Dr. Steve Wise in the Math Department for more information.

**Thursday, October 18**

TITLE: Old analogies in the Calculus of Variations - The Brachistochrone

**SPEAKER: Marco Mendez, University of Chicago**

TIME: 3:40 PM-4:35 PM

ROOM: Ayres 405

I will tell the story of the Brachistochrone problem. This is one of the oldest variational problems and can be solved only using elementary calculus. I will present in detail Johann Bernoulli's clever solution, which is based on Fermat's principle and a variational analogy between mechanics and optics. If time permits, I will briefly discuss a more recent analogy between the theory of phase transitions and minimal hypersurfaces, which will be the subject of my talk in the Geometric Analysis Seminar.

**Thursday, October 11**

TITLE: Introduction to L-functions

**SPEAKER: Daniel Shankman, Purdue University**

TIME: 3:40 PM-4:35 PM

ROOM: Ayres 405

L-functions are mysterious complex analytic functions whose zeroes and poles are connected to the behavior of prime numbers. The most famous L-function is the Riemann zeta function. First, I will prove some things about the Riemann zeta function in order to investigate the convergence of the infinite sum 1/2 + 1/3 + 1/5 + 1/7 + etc. of the reciprocals of the prime numbers. Then I will define a Dirichlet L-function and use it to prove that there are infinitely primes of the form 4n+1. There is a much easier way to prove this, without using any analysis, but the method which I will present, using L-functions, can be generalized to show that there are infinitely many prime numbers of the form an + b, where a and b are integers without any prime factors in common. This generalization is a theorem originally proved by Dirichlet in 1837, and there is no known proof of this which does not use analysis.

Previous Junior Colloquiums: