Wednesdays at 4:15 PM in 380X.
Consider the sample paths of a solution X(t) to an SDE, as they cross between two disjoint subsets in the state space. Those portions of the path which bridge the two states are called transition paths. Understanding the behavior of transition paths is important for applications in chemistry, for example, where they represent reactive trajectories between reactant and product states. I will identify the probability law of these paths in terms of an auxiliary SDE with singular drift, and I will explain how empirical sampling of the original process X(t) is related to solutions of this auxiliary SDE. Using these ideas, one can derive various representation formulas for statistics of the transition paths, such as mean crossover time and reactive current. http://arxiv.org/abs/1303.1744
I will describe a recently discovered equivalence between the first two objects mention in the title, as well as an interesting but not widely known connection between pursuit curves and high frequency vibrations.
Many objects in solid state physics have properties that vary dramatically depending on the arithmetics of certain physically relevant and measurable parameters. Quasiperiodic operators is one field where such phenomena are rampant. We will discuss how this can be interpreted in a non-contradictory way and, in particular, will present some recent continuity/discontinuity results.
I will discuss some recent results on developing new factorizations for matrices obtained from discretizing differential and integral operators. A common ingredient of these new factorizations is the interpolative decomposition for numerically low-rank matrices. As we shall see, these factorizations offer efficient algorithms for applying and inverting these operators. This is a joint work with Kenneth Ho.
The development of drug resistance is a major challenge in the treatment of cancer. In this talk we will overview some of the aspects of drug resistance that have been studied by the mathematical community. We will focus on two examples: 1) Modeling the dynamics of cancer stem cells and their role in developing drug resistance. When combined with clinical and experimental data, our mathematical analysis provides new insights on how to approach treating Chronic Myelogenous Leukemia. 2) Studying the role of cell density and mutations on the dynamics of drug resistance in solid tumors. This is another example in which the mathematical analysis leads to insights on the design of new treatment protocols. This is a joint work with C. Tomasetti, J. Greene, O. Lavi, and M. Gottesman.
We will describe the hot spots conjecture of J. Rauch, known results and counterexamples. After that I will consider arguably the simplest unknown case: acute triangles. We will discuss recent progress sparked by a new method due to Miyamoto.
Universal properties related to the KPZ equation is an extremely active research field right now. It is connected to many different areas in mathematics and mathematical physics: probability theory and statistical mechanics, PDEs, random matrices, integrable systems, and many others. While it was a big recent progress, mainly in deriving exact formulas for correlation functions, the problem of universality is still largely open. We shall discuss some recent results in this direction, as well as possible approaches to establishing of the full universality.
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