MATH 704 SENIOR THESIS PRESENTATIONS
Warner 202
Monday, December 6, 3:15 p.m.
Emily Berg '05
The Completion of Fourier's Analysis
Near the turn of the 19th Century, the French mathematician Joseph Fourier proposed that any complex-valued function defined on the unit circle may be represented by a trigonometric series. Although the roots of this bold conjecture lie in solutions to physical problems, Fourier analysis is of mathematical interest largely for its synthesis of the theories of Hilbert spaces and Lebesgue integration. This talk will explore Fourier's conjecture and discuss the context in which it is, essentially, true.
Tuesday, December 7, 3:00 p.m.
Uche Opara '05
Application of Differential Equations to Oscillatory Motion
Oscillatory motion, in its simplest form, is periodic motion to and from a fixed position. Variables for oscillatory motion, such as time and displacement, and related parameters, such as angular frequency, give rise to differential equations that describe the motion. The introduction of additional parameters into a differential equation can alter the equation; oscillatory motion provides realistic examples by using factors such as damping and driving. This talk will consider the differential equations of several oscillatory systems and outline their solutions at any given time. For equations that cannot be solved in closed form, appropriate approximation techniques are helpful. Vivid applications from the physical world will help to illuminate practical purposes of differential equations.
Tuesday, December 7, 3:30 p.m.
Amrita Sarkar '05
Galois Theory and Geometric Impossibilities
Founded by Evariste Galois,Galois theory is that branch of abstract algebra which studies the symmetries of the roots of polynomials. Although substantially theoretical, Galois theory has applications to classical mathematical problems. These include definitive answers to the geometric questions that the ancient Greeks used to ponder:
"Is it possible to double the volume of a cube?"
"Can all angles be trisected using straight edge and compass?"and "Given a circle of a certain area, is it possible to construct a square with the same area?"
This talk will focus on the basics of Galois theory and its role in answering the three questions posed above.
Tuesday, December 7 at 4:00 p.m.
Betsy Sullivan '05
Dynamical Systems with Linear Algebra
In realistic applications, measured interactions are almost always positive and always changing. For example, researchers may ask about the time it takes to drive from warehouse i to warehouse j, or about the percentage of people in age group i that survive into age group i+1. Applications like these can often be modeled using non-negative square matrices that describe the transitions. Applying the simple results of the Perron-Frobenius theorem to these non-negative square matrices, allows us to see how these systems will behave over time.