Introduction to the course

Title: Self-similar fractals and dimension

Title: The gambler's ruin problem and an interesting asymptotic extension

Abstract: In 1656, Pascal posited to Fermat something akin to the following: "Players A and B start with 'a' and 'b' points, respectively. Flips of a fair coin determine how points are transferred between them. The game ends when one player acquires all a+b points (and the other player thus has none). Given a and b, how likely is Player A to win?" In this talk I will answer Pascal's question by visualizing the game as a 1D random walk with absorbing boundary conditions. I will then discuss an extension of the problem involving the limit of many random walkers, which has numerous applications to physics, chemistry, and biology.

STOCHASTIC ADAPTIVE CHEMOTHERAPY CONTROL OF COMPETITIVE RELEASE IN TUMORS

Abstract:  Adaptive chemotherapy seeks to manage chemoresistance by delaying the competitive release of a resistant sub-population, and to manage cancer by maintaining a tolerable tumor size rather than seeking a cure. Models typically follow interactions between infinite populations of sensitive (S) and resistant (R) cell types to derive a chemotherapy dosing strategy C(t) that maintains the balance of the competing sub-populations. Our models generalize to include healthy (H) cells, and finite population sizes. With finite population size, stochastic fluctuations lead to escape of resistant cell populations that are predicted to be controlled in the deterministic case. We test adaptive schedules from the deterministic models on a finite-cell (N = 10,000 – 50,000) stochastic frequency-dependent Moran process model. We quantify the stochastic fluctuations and variance (using principal component coordinates) associated with the evolutionary cycle for multiple rounds of adaptive chemotherapy, and show that the accumulated stochastic error over multiple rounds follows power-law scaling. This accumulates variability and can lead to stochastic escape which occurs more quickly with a smaller total number of cells. Moreover, we compare these adaptive schedules to standard approaches, such as low-dose metronomic (LDM) and maximum tolerated dose (MTD) schedules, finding that adaptive therapy provides more durable control than MTD even when we include the effects of finite population size. Although low-dimensional, this simplified model elucidates how well applying adaptive chemotherapy schedules for multiple rounds performs in a stochastic environment. Increasing stochastic error over rounds can erode the effectiveness of adaptive therapy.

Applying for the NSF GRFP

Abstract:  This is a panel about applying for the National Science Foundation (NSF) Graduate Research Fellowship Program (GRFP).  This is a program that graduating seniors as well as first and second year graduate students, may apply for.  It provides funding for 3 years of financial support to US Citizens, US Nationals and permanent residents who intend or are pursuing a Masters or Ph.D. in various fields including Mathematics.  This panel will provide useful information for anyone who is thinking about graduate school in Mathematics (whether or not they are eligible to apply for the NSF GRFP this year).

Title: What should you do if you miss a dose of medication?

Abstract:  Medication adherence is a major problem for patients with chronic diseases that require long term pharmacotherapy. What should patients do if they miss a dose of medication? How can physicians design drug regimens to mitigate nonadherence? Why are some medications effective despite lapses in adherence? In this talk, I will describe recent efforts to address these questions using mathematical modeling.

Gradient Dynamics in a Valley: Super Slow Motion of Interfaces

Abstract:Paul Fife and Charles Conley studied a dynamical system modeling the spread of a genetic trait through a population over many generations. In doing so, they developed an abstract theorem, using a topological tool, with applications going far beyond the biological model. Consider a curve (manifold) M of stationary solutions to a dynamical system and suppose that each point of M is exponentially stable in directions perpendicular to M. They showed that small changes in the dynamical system retained some semblance of M, namely, traveling waves in the perturbed dynamical systems. I will have more to say about invariant manifolds in Thursday’s talk. Now for something completely different. Not really, but I can’t resist Monty Python lines sometimes (ask your parents, sorry, grandparents about MP). Diverging slightly from the above, in this talk, I will go on to talk about places like Alta, where there are mountains and valleys and how something like gravity tended to propel me into the valleys from where I would walk out by going along the valley to what I hoped would be a route to the chairlift, so I could do it again. The dynamics I described is a gradient dynamical system, where I would go down in the steepest way possible until getting into a valley, where motion became rather slow. I will describe a class of dynamical systems, being the gradient of an “energy” J, that has a peculiar property for points near M, a set of points forming a curve (manifold): If the distance of an initial point u0 from M is less than some number g, and if its energy J(u0) is low, then that distance is less than b, for some number less than g! The solution then stays within distance b from M. One such dynamical system is a vector version of the system studied by Fife and Conley and also having small diffusion. I’ll indicate how the statement on boldface applies and produces incredibly slow motion of interfaces in the spatial profile of a solution. This is joint work with Giorgio Fusco (U. L’Aquila) and Georgia Karali (U. Crete).

Panel: Population Health Sciences PhD, Emphasis in Biostatistics

We train the next generation of biostatisticians with the methodological and collaborative skills needed to design and analyze studies aimed at addressing important population health problems. Our graduates are well positioned for meaningful careers in academia, industry, or government.

Learn more about our PhD Program at bit.ly/PHSPhD

Panel: Healthcare Data Science 2023 Internship Program

Cryptography, Freedom, Democracy
How Basic Science Affects Everyone

To most people, research in basic science seems irrelevant, and
consequently, citizens, legislators, government funding agencies, and
corporations are disinclined to support it.

Nevertheless, basic science can have deep impacts on our lives.  This
talk examines two developments in basic science in the Twentieth
Century. The first of them, Albert Einstein's work in 1905, changed
the field of physics, and the course of history.  The second, the
invention of public-key cryptography in 1975, has important
consequences for secure communications.

Many of mankind's discoveries have potential for both good and bad.
The talk concludes with a discussion of some recent uses of technology
that pose the very serious risk of our complete loss of privacy,
freedom, and democracy.

Title: Power Grid Operation and Planning: Convergence of Engineering, Economics, and Machine Learning

Abstract:  Power grid fuels our everyday life and has successfully contributed to the economic surplus of all countries across the globe. Today, the power grid is undergoing a massive change with the integration of renewable energy resources, and faces a growing number of natural disasters and cyber attacks that threaten the reliable delivery of power to the communities. This talk will discuss how new mathematical optimization and machine learning models are changing the way power grids are operated to adapt to the challenges the grid faces. 

Topic: Yoneda Lemma

Abstract:  I'll give a quick introduction to Category theory and state Yoneda Lemma. Then If time permits, I'll give a couple of quick application.

Title: Area without Numbers

 Abstract:  Ancient Greek civilization didn't have the greatest set of numbers. Therefore the way they approached geometry was a little different from what we're used to. This talk will explore the notion of quadrature -- producing a square with an area equal to that of a given figure.

Title: What is Representation Theory? 

 Abstract:  Representation theory is a fundamental field of mathematics and has been shown to be useful in nearly every other area. The main idea behind this is to transform a hard problem about symmetry to a simpler problem about linear algebra. In this talk we will look at why such a thing is possible and how understanding the representation theory of a group can tell us about its structure in the case of finite groups.