Georgia Institute of Technology College of Sciences

The Schinzel-Zassenhaus conjecture describes the narrowest collar width around the unit circle that contains a full set of conjugate algebraic integers of a given degree, at least one of which lies off the unit circle. I will explain what this conjecture precisely says and how it is proved. The method involved in this solution turns out to yield some other new results whose ideas I will describe, including to the closest interlacing of Frobenius eigenvalues for abelian varieties over finite fields, the closest separation of Salem numbers in a fixed interval, and the distribution of the short Kobayashi geodesics in the Siegel modular variety.

https://bluejeans.com/476147254/8544

Given a system of analytic functions and an approximate zero, we introduce inflation to transform this system into one with a regular quadratic zero. This leads to a method for isolating a cluster of zeros of the given system.

(This is joint work with Michael Burr.)

Associated to a smooth n-dimensional manifold are two infinite-dimensional groups: the group of homeomorphisms Homeo(M), and the group of diffeomorphisms, Diff(M). For manifolds of dimension greater than 4, the topology of these groups has been intensively studied since the 1950s. For instance, Milnor’s discovery of exotic 7-spheres immediately shows that there are distinct path components of the diffeomorphism group of the 6-sphere that are connected in its homeomorphism group.  The lowest dimension for such classical phenomena is 5. 

I will discuss recent joint work with Dave Auckly about these groups in dimension 4. For each n, we construct a simply connected 4-manifold Z and an infinite subgroup of the nth homotopy group of Diff(Z) that lies in the kernel of the natural map to the corresponding homotopy group of Homeo(Z). These elements are detected by (n+1)—parameter gauge theory. The construction uses a topological technique.  I’ll mention some other applications to embeddings of surfaces and 3-manifolds in 4-manifolds.
 

Zoom Link- https://brandeis.zoom.us/j/99772088777   (password- hyperbolic)

Here is alternative link where the password is embedded- https://brandeis.zoom.us/j/99772088777?pwd=WHpFQk1Fem5jZVRNRUwzVmpmck4xdz09 

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