The development of Morse Theory started in the 1920's with M.Morse's interest in critical point theory. Since then, Bott, Smale and Witten, among others, have extended Morse's original ideas in insightful ways to a variety of areas in mathematics. In this talk, I will state and show examples of Morse's basic results concerning non-degenerate functions on a manifold. In a second talk, I will show examples of Morse functions `in the sense of Bott' and mention some applications.
In this second talk on the subject of Morse Theory, I will define and show examples of Morse functions in the sense of Bott. In particular, I will mention how Morse-Bott functions can be useful for understanding the local structure of the phase space of certain integrable Hamiltonian systems.
In this talk, a new bivariate exponential distribution will be introduced. We describe this new distribution by constructing a new shock model; this new distribution is the joint distribution of the resulting lifetimes. Some aging properties e.g. increasing failure rate, increasing failure rate average, and some anti-aging properties, e.g. decreasing failure rate, decreasing failure rate average, will be studied.
In this talk I will introduce a system of PDE's, called the Reduced Maxwell-Bloch equations (RMB-equations), and show how we can find a Lax pair for this particular system. A Lax pair is a pair of differential operators that commute if and only if our system of PDE's is satisfied. We are interested in finding the Lax pair because it enables us, through some other techniques (i.e. Backlund transforms etc.), to find special solutions of the RMB-equations and from those build up more complicated ones.
We will introduce Riemann-Hilbert problems, show how some results on Singular Integral Equations give us simple solutions of the scalar case. Then we will give some results on the solution of matrix Riemann-Hilbert problems and related factorization problems in matrices over Banach algebras.
I will introduce Wavelet Transforms and Multiresolution Analysis. I will also give examples to show what makes wavelets so useful in the fields of data compression, image analysis and signal processing.
In this talk, I will introduce some of the beginning machinery of algebraic number theory, and use the results to present Kummer's proof of the first case of Fermat's Last Theorem. In particular, I will be discussing a particular invariant of number fields, known as the class number, and how it applies to this particular proof.
In the 1920's the physicist P.A.M. Dirac was searching for a relativistic theory of the electron. He was, roughly speaking, faced with the problem of finding a first order differential operator D whose square is the Laplacian. In my talk I will explain how this problem evolves to a whole theory, culminating in the index theorem for elliptic operators on compact manifolds.
Let C and D be two plane conics (a circle and an ellipse for example). When is it possible to find an n-sided polygon which is simultaneously inscribed in C and circumscribed around D? In 1822 Jean-Victor Poncelet proved that if there is one such polygon, then there are infinitely many. We will look at some of the pre-history of this problem, then sketch a modern proof (due to Griffiths) utilizing elliptic curves.
What is mathematics?
What is mathematical truth?
Amongst our everyday busy lives in mathematics, many of us have forgotten, or chosen to disregard such fundamental questions as these. I will introduce some of the more common philosophies behind mathematics, the history behind these philosophies, and how they affect us as mathematicians. I will also discuss some of my own theories for how these philosophies came to be. To finish, I will introduce perhaps one of the greatest philosophers who ever wrote, Immanuel Kant, and discuss how he came to the conclusion that the only type of geometry is Euclidean.
Oh, and I have no intention of answering the above two questions!
In 1931, Kurt Gödel published a paper that forever changed the way we view the foundations of mathematics. In this talk, we will discuss the relevant notions necessary for understanding the ideas of the theorem and its proof, and hopefully gain insight into why it is considered by some to be one of the most important results of the 20th century.
Given a ring A, one can define the notion of quaternions over A. Not only this notion is interesting in its own right, but it also has nice applications. For example, in the theory of numbers, this notion can be used to show that every natural number is a sum of four squares. I will define the notion of quaternions over a ring A and use it to prove this result.
This talk will be two-fold. First, I will include a discussion on the Triangle Peg Game: starting with one hole empty, move the pegs in a series of jumps so that only one peg remains on the board. I will include some history, mathematics, and possible solutions. Second, I will talk about the Tower of Hanoi and extensions of the Tower. I will include some history and the role of mathematics for this as well. Come all to see what great toys I will bring.
In 1982, A. K. Lenstra, H. W. Lenstra Jr., and L. Lovasz described an algorithm for factoring polynomials. Hidden in this description was a subalgorithm which later became known as the LLL-algorithm. Their method for factoring polynomials was never practical, but the LLL-algorithm is used to more efficiently solve gcd multiplier problems, class group problems, and many others. I will describe the LLL-algorithm, as well as its applications to factoring polynomials and to the extended Euclidean algorithm.
First, I will define ergodicity, explain what it means for a function to be ergodic on a space and give some examples on simple spaces. Then, I will discuss moduli spaces and what it means means for a function to be ergodic on a moduli space. Finally, I will state a theorem and give what proof I can in the remaining time.