Georgia Institute of Technology College of Sciences

Normal 0 false false false EN-US X-NONE X-NONE   Over the last several decades, cancer has become a global pandemic of epic proportions. Unfortunately, treatment strategies resulting from the traditional approach to cancer have met with only limited success. This calls for a paradigm shift in our understanding and treating cancer.    In this talk, we present an entirely mechanistic, comprehensive mathematical model of cancer progression in an individual patient accounting for primary tumor growth, shedding of metastases, their dormancy and growth at secondary sites. Parameters of the model were estimated from the age and volume of the primary tumor at diagnosis and volumes of detectable bone metastases collected from one breast cancer and 12 prostate cancer patients. This allowed us to estimate, for each patient, the age at cancer onset and inception of all detected metastasis, the expected metastasis latency time and the rates of growth of the primary tumor and metastases before and after the start of treatment. We found that for all patients: (1) inception of the first metastasis occurred very early when the primary tumor was undetectable; (2) inception of all or most of the surveyed metastases occurred before the start of treatment; (3) the rate of metastasis shedding was essentially constant in time regardless of the size of the primary tumor, and so it was only marginally affected by treatment; and most importantly, (4) surgery, chemotherapy and possibly radiation bring about a dramatic increase in the rate of growth of metastases. Although these findings go against the conventional paradigm of cancer, they confirm several hypotheses that were debated by oncologists for many decades. Some of the phenomena supported by our conclusions, such as the existence of dormant cancer cells and surgery-induced acceleration of metastatic growth, were first observed in clinical investigations and animal experiments more than a century ago and later confirmed in numerous modern studies. 

There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, methods in linear and nonlinear partial differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes. Examples will consider both "bottom up" and "top down" approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory.

The construction of the Berkovich space associated to a rigid analytic variety can be understood in a general topological framework as a type of local compactification or uniform completion, and more generally in terms of filters on a lattice. I will discuss this viewpoint, as well as connections to Huber's theory of adic spaces, and draw parallels with the usual metric completion of $\mathbb{Q}$.

The primary objective of this thesis is to make a quantitative study of complex biological networks. Our fundamental motivation is to obtain the statistical dependency between modules by injecting external noise. To accomplish this, a deep study of stochastic dynamical systems would be essential. The first part is about the stochastic dynamical system theory. The classical estimation of invariant measures of Fokker-Planck equations is improved by the level set method. Further, we develop a discrete Fokker-Planck-type equation to study the discrete stochastic dynamical systems. In the second part, we quantify systematic measures including degeneracy, complexity and robustness. We also provide a series of results on their properties and the connection between them. Then we apply our theory to the JAK-STAT signaling pathway network.