The goal of this colloquium is to encourage interaction among graduate students, specifically between graduate students who are actively researching a problem and those who have not yet started their research. Speakers will discuss their research on a level which should be accessible to nonspecialists. The discussions will be geared toward graduate students in the beginning of their program, but all are invited to attend. (This invitation explicitly includes undergraduate students.)Talks will be held on Wednesdays at 4:30 PM in JFB 103, unless otherwise noted.
ABSTRACT: The main ideas of representation theory of Lie groups developed in an interaction of algebra, geometry, harmonic analysis and quantum physics. We are going to discuss the historical development of the subject, some of its major successes and some directions of the current research.
ABSTRACT: Ancient Greeks knew that for any two polygons of equal area there is a way to cut one into subpolygons and reassemble the pieces to construct the other one (just recall how the Pythagorean Theorem is proved). In his famous list of 23 problems, Hilbert asked if the analogous statement is true in dimension 3 (he conjectured no). This was proved by Max Dehn. In the talk I will try to explain his argument.
ABSTRACT: One dimensional second-order parabolic differential equations with barrier have been widely used as models for the spread of genes across a barrier in a continuous habitat in population genetics. The generalization of these models to higher dimensions is considered here, and the existence, uniqueness, and regularity of a solution of the initial value problems are established.
ABSTRACT: The paper considers various structures designed to shield temperature sensitive devices from the heat generated by a given distributed source. These structures should redistribute the heat and control the total heat dissipation, in order to maintain a prescribed temperature profile in a certain region of the domain. We assume there are several materials available, each with different constants of heat conductivity, and that we are allowed to mix them arbitrarily. It is known that optimal structures consist of laminate composites that are allowed to vary within the design domain. The paper discusses optimal distributions of these composites for different settings of the problem that include: prescribed volume fractions and various types of boundary conditions. We solve the problem numerically using the method of finite elements.
ABSTRACT: Nonstandard analysis was invented in the 1960's and helped explain some of the ideas of Archimedes, Newton, Leibniz, Euler and more. For example, what is an infinitesimal quantity $dx$ as discussed by Leibniz? I will talk about this (briefly) and then will try to get to some applications and some other related constructions.
ABSTRACT: Given some complex-valued polynomial, P[z1,z2], we find the roots, numerically of course, using a homotopy proceedure as follows. Find some polynomial with known roots, G[z1,z2], for example the roots of unity (roots on S1 in the complex plane). Then create the family of polynomials:
ABSTRACT: I will discuss some of the motivations and basic ideas in the theory of harmonic maps, trying to stay away from technicalities. Starting with some intuitive examples, I will give you a flavor for how to look at more general situations, and present some applications in geometry, physics, etc.
Groups given by presentations (i.e., generators and relations) are the fundamental objects of combinatorial group theory. I will discuss some of the main questions in this field, such as the isomorphism problem and the word problem, as well as some topological and geometric techniques.
ABSTRACT: Oscillatory behavior is a natural occurrence in the every day world. It is common in predator-prey models, mechanical systems, chemical interations, as well as cyclical behavior of economical environments. Any natural scientist will inquire as to the period of the oscillating system and also to the relationship of this period to the amplitude. This talk examines limits of common linearization techniques. A particular second order system is analyzed, and finally remarks are made about the motivating example.