Introductory presentations for DOmath 2022.

I'll discuss rigorous nonlinear stability results for traveling waves in a class of reaction-diffusion systems that arise in chemical reaction models. The class includes systems in which there is no diffusion in some equations. The results are detailed enough to show, for example, that the results of adding some heat or adding some reactant to a combustion front are different.

Network science is a rapidly growing interdisciplinary field with methods and applications drawn from across the natural, social, and information sciences. Perhaps surprisingly, very few approaches use techniques from the rich literature of structural graph theory. In this talk, we discuss some first steps towards integrating what have been predominantly theoretical results into tools for scalable network analysis. Tree-like structures arise extensively in network science - for example, hierarchical structures in biology, hyperbolic routing in the internet, and core-periphery behavior in social networks. As such, this talk focuses on ways to use tree decompositions, key combinatorial objects used in graph minor theory, in tandem with k-cores and Gromov hyperbolicity to provide structural characterization of and improve inference on complex networks. We also discuss new algorithms using tree decompositions to enable scalable solution of certain graph optimization problems in a high performance computing environment.

It is a fair assumption that many of us in the math department enjoyed math puzzles in our youth and this helped to bring us to where we are. I know I did (and do!). I recently had to solve a classic style of problem: find the nth term of a sequence of integers. I tried everything I knew but only had a pile of scratched out notes to show for it. And then I was told about generating functions. Although not a total panacea for all things sequential, generating functions provide a staightforward blueprint for deriving nth-term formulas and more. I will present a few basic examples and some notes on the excellent book I used as a reference, but the majority of the talk will discuss my particular problem and its solution via generating functions. The main goal will be to impress upon younger grad students the power of this method where other more familiar methods fail.

Mathematics is being used in many ways to improve the analysis and interpretation of DNA and other molecules that can affect our health. I will describe how math was used to identify genes associated with tumor progression, and to develop methods to identify and quantify genetic variations without expensive and time-consuming sequencing. resulting in a rapid, economical test for transplant compatibility, a cancer therapy, and numerous clinical diagnostic assays. I will also discuss some surprising mathematical connections discovered in the course of this work.

At the turn of this century it was realized that social and communication networks were best modeled by graphs that were "small worlds" and/or had power law degree distributions. I will discuss two examples. The first is a situation where physicist's mean field arguments give the wrong answer about the spread of epidemic. The second, inspired by a gypsy moth outbreak in the late 1980s in NY leads to chaotic behavior. I will concentrate on what is true rather than why, so the talk should be accessible to a wide audience.

Let S be a smooth projective surface over a field k. If S contains sufficiently many curves of nonpositive self-intersection, we show that that there are two such curves C_1, C_2 and positive integers a, b so that aC_1 + bC_2 has positive self-intersection. I will then describe some applications of this result to the geometry and topology of complex projective surfaces uniformized by the unit ball or the product of two Poincare disks. This is joint with Ted Chinburg.

We will discuss the proof of the conjecture due to M. Khovanov relating the algebraic and topological categorification of the chromatic polynomial. We show that there exists a spectral sequence relating the chromatic graph homology and the homology of a graph configuration space and discuss higher differentials.

The widespread presence of synchronized neuronal oscillations within the brain suggests that a mechanism must exist that is capable of decoding such activity. Two realistic designs for such a decoder include: 1) a read-out neuron with a high spike threshold, or 2) a phase-delayed inhibition network motif. Despite requiring a more elaborate network architecture, phase-delayed inhibition has been observed in multiple systems, suggesting that it may provide inherent advantages over simply imposing a high spike threshold. We use a computational and mathematical approach to investigate the efficacy of the phase-delayed inhibition motif in detecting synchronized oscillations, showing that phase-delayed inhibition is capable of detecting synchrony far more robustly than a high spike threshold detector. Furthermore, we show that in a system with noisy encoders where stimuli are encoded through synchrony, phase-delayed inhibition enables the creation of a decoder that can respond both reliably and specifically to a stimulus, while a high spike threshold does not.

If a partial differential equation has coefficients that vary with respect to the independent (spatial) variables, how do the fluctuations in the coefficients effect the solution? In particular, if these fluctuations have a statistical structure, can anything thing be said about the statistical behavior of the solutions? I'll consider these questions in the context of a scalar reaction diffusion equation. Without the variable coefficients, the equation admits stable traveling wave solutions. It turns out that the stability of wave-like solutions persists in a heterogeneous environment, and this fact can be used to derive a central limit theorem for the wave when the environment has a certain statistical structure.

I'll try to explain two interesting mathematical issues: First, how can one prove stability of the wave-like solution in this general setting, since spectral techniques don't seem applicable? Second, how can one use the structure of the problem to say something about how randomness in the environment effects the solution?

Schedule (approximate times): 1:00-2:00: An, Deng, McPhail-Snyder research groups 2:00-2:20: break 2:20-3:00: Robles, Stokols research groups