| Date | Speaker | Title | Abstract |
| 09/19/2007 | Prof M. Loss | Lieb-Thirring Inequalities | Lieb-Thirring inequalities are bounds on sums of eigenvalues of -\delta -V in terms of the potential V. More generally one can try to bound sums of powers of eigenvalues in terms of the potential. In this talk I review this topic, mention some applications, indicate some of the proofs and discuss some open problems. |
| 09/26/2007 | Prof C. Heil | Music, Time-Frequency Shifts, and Linear Independence | Fourier series provide a way of writing almost any signal as a superposition of pure tones, or musical notes. But this representation is not local, and does not reflect the way that music is actually generated by instruments playing individual notes at different times. We will discuss Fourier series, and then present time-frequency representations, which are a type of local Fourier representation of signals. This gives us a mathematical model for representing music. While the model is crude for music, it is in fact a powerful mathematical representation that has appeared widely throughout mathematics (e.g., partial differential equations), physics (e.g., quantum mechanics), and engineering (e.g., time-varying filtering). We ask one very basic question: are the notes in this representation linearly independent? This seemingly trivial question leads to surprising mathematical difficulties. |
| 10/3/2007 | Prof S. Basu | Postponed to 10/24 | TBA |
| 10/10/2007 | Prof J. Etnyre | Symplectic Geometry --- the language of classical mechanics | I will begin this talk by discussing a classical question concerning periodic motions of particles in classical physics. In trying to better understand this question we will develop the notion of a symplectic structure. This is a fundamental geometric object that provides the "right way" to think about classical mechanics, and many many other things too. I will then indicate some recent results (just a few months old) relating to our initial naive questions about periodic motions. |
| 10/17/2007 | Prof M.G. Westdickenberg | Rare Events and Action Minimization | When a system is perturbed by random noise, unexpected things can happen. Even a small noise term will, given enough time, lead to behavior that would never be seen in the deterministic setting. Large deviation theory is a mathematical theory that answers some questions about stochastically-driven rare events in the limit of small noise. At its heart is the large deviations action functional, through which questions about a stochastic equation reduce to solving a deterministic calculus of variations problem. As a particular example, we will consider the action functional associated to the Allen-Cahn PDE, where an interesting sharp-interface limit emerges. |
| 10/24/2007 | Prof S. Basu | Algorithmic and topological complexity of semi-algebraic sets. | Semi-algebraic sets are subsets of ${\mathbb R}^n$ defined in terms of a finite number of real polynomial equalities and inequalities. They have important applications in many areas of science and engineering. Moreover, they satisfy important finiteness properties. A semi-algebraic set can have only finitely many connected components, finite Betti numbers etc. It is in fact possible to bound the Betti numbers, as well as the number of topological types, of semi-algebraic sets in terms of the number and the degrees of the polynomials defining them. In this talk I will give a survey of recent results on obtaining tight bounds on the Betti numbers, as well as on the number of homotopy types, of semi-algebraic sets in terms of the parameters mentioned above. I will also describe how some of these results generalize to the more abstract setting of definable sets in any o-minimal structure. Finally, time permitting, I will describe recent progress in developing efficient algorithms for computing the Betti numbers of semi-algebraic sets. Certain parts of this work is joint with R. Pollack, M-F. Roy and (separately) with N. Vorobjov. |
| 11/7/2007 | Prof. R. Pan | Singularity formation in hyperbolic conservation laws | Hyperbolic conservation laws states basic physical laws for conserved quantities that propagate with finite speed. I will discuss how nonlinearity coupled with hyperbolicity leads to singularity formation. I will also discuss the development on the methods toward proofs of singularity formations. Examples include: Invicid Burgers' equations, compressible Euler equations, and relativistic Euler equations. |
| 11/21/2007 | Thanks Giving | TBA | TBA |
| 11/28/2007 | J. Etnyre, W. Trotter, M.G. Westdickenberg | How to apply to an Academic position in Math. | Prof. Trotter will lead a discussion about how the hiring process works for academic positions in math. |