Georgia Institute of Technology College of Sciences

Through the pioneering numerical computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed that nonlinear equations modeling wave propagation asymptotically decompose as a superposition of “traveling waves” and “radiation”. Since then, it has been a widely believed (and supported by extensive numerics) that “coherent structures” together with radiations describe the long-time asymptotic behavior of generic solutions to nonlinear dispersive equations. This belief has come to be known as the “soliton resolution conjecture”.  Roughly speaking it tells that, asymptotically in time, the evolution of generic solutions decouples as a sum of modulated solitary waves and a radiation term that disperses. This remarkable claim establishes a drastic “simplification” to the complex, long-time dynamics of general solutions. It remains an open problem to rigorously show such a description for most dispersive equations.  After an informal introduction to dispersive equations, I will survey some of my recent results towards understanding the long-time behavior of dispersive waves and the soliton resolution using techniques from both partial differential equations and inverse scattering transforms.

Two years after the beginning of the pandemic, we are still working to understand the mechanisms of immunopathology in COVID-19. Immune responses following SARS-CoV-2 infections are heterogeneous, and biomarkers of this variability remain to be elucidated. In collaboration with experimentalists and clinicians, we have deployed various mathematical and computational approaches to understand longitudinal immunological data from patients, and to generate new hypotheses about the factors determining COVID-19 severity and disease dynamics.
To answer foundational questions about immunopathology and heterogeneity in COVID-19, we have developed a multi-scale, mechanistic mathematical model of the immune response to SARS-CoV-2 that includes several innate and adaptive immune cells and their communication via signalling networks. By generating a population of virtual patients, we identified dysregulated rates of monocyte-to-macrophage differentiation that distinguishes disease severity in these in silico patients. Further, our results suggest that maximal IL-6 concentrations can be used as a predictive biomarker of CD8+ T cell lymphopenia. Using the same cohort of virtual patients, we have also studied the influence of variant on immunopathology by combining our model with data of intra-host viral evolution. We predicted that the combined effects of mutations affecting the spike proteins and interferon evasion on the severity of COVID-19 are mostly determined by the innate host immune response. Our approaches can be used to study the factors regulated immunopathology during SARS-CoV-2 infections, and represent a quantitative framework for the study of COVID-19 and other viral diseases.

Recording link: https://bluejeans.com/s/6CmKwHWWc2O

Meeting link: https://bluejeans.com/961048334/8189

A long-standing problem in the field of graph coloring is the Erdős–Faber–Lovász conjecture (posed in 1972), which states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$, or equivalently, that a nearly disjoint union of $n$ complete graphs on at most $n$ vertices has chromatic number at most $n$.  In joint work with Dong Yeap Kang, Daniela Kühn, Abhishek Methuku, and Deryk Osthus, we proved this conjecture for every sufficiently large $n$.  Recently, we also solved a related problem of Erdős from 1977 on the chromatic index of hypergraphs of small codegree.  In this talk, I will survey the history behind these results and discuss some aspects of the proofs.