Scientific

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler--Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. I will report on a joint work with Lingfei Yi, where we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems. This confirms a central conjecture of the geometric Langlands program for hypergeometrics. The key tool we use is the notion of rigid automorphic data due to Zhiwei Yun. This talk is based on the preprint arXiv:2006.10870.

One interesting feature of the classification of smooth Fano varieties up to dimension three is that they can all be described as certain subvarieties in GIT quotients; in particular, they are all either toric complete intersections (subvarieties of toric varieties) or quiver flag zero loci (subvarieties of quiver flag varieties). Fano varieties are expected to mirror certain Laurent polynomials; given such a Fano toric complete intersection, one can produce a Laurent polynomial via the Landau-Ginzburg model. In this talk, I’ll discuss finding mirrors of four dimensional Fano quiver flag zero loci via finding degenerations of the ambient quiver flag varieties. These degenerations generalise the Gelfand-Cetlin degeneration, which in the Grassmannian case has an important role in the cluster structure of its coordinate ring.

In this talk I will report on joint work in progress with A. Craw and T. Schedler on the birational geometry of quiver varieties. We give an explicit local description of the birational transformations that occur under variation of GIT for quiver varieties. The main consequence of this local picture is that one can show that all Q-factorial terminalizations of quiver varieties (excluding the (2,2) case) can be obtained by VGIT. I will try to explain what our results mean in two concrete classes of examples. Namely, for framed affine Dynkin quivers (corresponding to wreath product quotient singularities) and star shape quivers (corresponding to hyperpolygon spaces).

The talk describes a substantial extension of the Middle East Respiratory Syndrome (MERS) model constructed, analyzed and simulated in Al-Asuoad et. al. BIOMATH 5 (2016)1, Al-Asuoad, Oakland University Dissertation (2017), and Al-Asuoad and Shillor, BIOMATH 7(1)(2018)2 to the case of the current COVID-19 Respiratory Syndrome pandemic that is sweeping the globe. It is caused by the new SARS-CoV-2 coronavirus that has been identified in December 2019 and since then outbreaks have been reported in all parts of the world. To help predict the dynamics and possible controls of the pandemic we developed a mathematical model for the pandemic. The model has a compartmental structure similar but more complex to the SARS and MERS models. It is a coupled system of nonlinear ordinary differential equations (ODEs) and a differential inclusion for the contact rate parameter. The talk will describe the model in detail, mention some of its analysis, and describe our computer simulations of the pandemic in South Korea. The main modeling novelties are in taking into account the shelter-in-place directives, the rates at which the populations obey them and the observed changes in the infectiveness of ‘contact number’ of the SARS-CoV-2 virus. The model predictions are fitted to some of the data from the outbreak in South Korea. Since the DFE (in South Korea) is found to be asymptotically stable, the pandemic will eventually die out (as long as some control measures remain in place). And, indeed, the model simulations show that the COVID-19 will in the near future be contained. However, the containment time and the severity of the outbreak depend crucially on the contact coefficients and the isolation or shelter-in-place rate constant. The simulations show that when randomness is added to the model coefficients the model captures the pandemic dynamics very well. Finally, the model highlights the importance of isolation of infected individuals and may be used to assess other control measures. It is general and will be used to analyze outbreaks in other parts of the world.

*with Aycil Cesmelioglu and Anna M. Spagnuolo

1 http://dx.doi.org /10.11145/j.biomath.2016.12.141
2 http://dx.doi.org/10.11145/j.biomath.2018.02.277

It is common in SIR models to assume that the infection rate is proportional to the product S*I of susceptible and infected individuals. This form is motivated by the law of mass action from chemistry. While this assumption works at the onset of the outbreak, it needs to be modified at higher rates such as seen currently in much of the world (as of June 2020). We propose a physics-based model which leads to a simple saturation formula based on first principles incorporating the spread radius and population density. We then apply this modified SIR model to coronavirus and show that it fits much better than the ``classical'' law of mass action.