MSRI-UP 2014: Arithmetic Aspects of Elementary Functions
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Nancy Rodriguez, Stanford University
The competition between dispersal and aggregation in biological and ecological aggregation
In recent years much research has been devoted to trying to understand the phenomenon of aggregation in biological and ecological systems. In this talk I will discuss the derivation of a general class of models for the competition between aggregation and dispersal of species (one very well-known model is the Patlak-Keller-Segel model for chemotaxis). I also will address one of the most fundamental questions in PDE theory, the existence and uniqueness of solutions for the aggregation model. Using ODE examples, I will motivate the interest in studying the issue of well-posedness.
Trachette Jackson, University of Michigan
Mathematical Models of Tumor- Induced Blood Vessel Formation
Cancer is the collective name given to an entire class of diseases characterized by rapid, uncontrolled cell growth. There are over 200 different types of cancer, each classified by the type of cell that is initially affected. Normally, our bodies form new cells only as we need them. However when cells acquire mutations that disrupt the tightly controlled processes of cell division and death, a self-sustaining wave of cellular multiplication can occur and result in the formation of a tumor. To ensure its continued growth, a tumor must acquire a continuous supply of nutrients and the ability to export metabolic waste. It does this by recruiting new blood vessels from the nearby existing vasculature, a process known as tumor-induced angiogenesis. Angiogenesis provides the necessary blood supply for the growth of solid tumors beyond a few millimeters in diameter. A recent advancement in cancer treatment has been combining traditional chemotherapeutic agents with drugs that interfere with a tumor's ability to stimulate blood vessel formation. In this talk, we explore mathematical models of tumor-induced blood vessel formation and discuss some related treatment strategies.
Richard Laugesen, University of Illinois at Urbana-Champaign
Two pictures and a quotation: Dido, drums, and isospectrality
Why does your cat curl into a ball on a cold night? What did the cat teach Lord Rayleigh? And what might you be missing if you close your eyes at the orchestra?
Lisa R. Goldberg, University of California, Berkeley
What is Bayes' Rule and Why Does It Matter?
Bayes' Rule is a cornerstone of modern statistics that was found independently at the end of the nineteenth century by Pierre-Simon Laplace and by Thomas Bayes. A simple equation relating two conditional probabilities, the "rule" has a short proof that masks its depth and subtlety. We will discuss the application of Bayes' Rule to the Monty Hall problem and other statistical puzzles.