Mathematics

On generalized hyperpolygons

Speaker: 
Laura Schaposnik
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work joint with Steven Rayan (arXiv:2001.06911). After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Time permitting, we shall conclude the talk by mentioning some partial results on current work on the construction of triple branes (in the sense of Kapustin-Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).

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Finding mirrors for Fano quiver flag zero loci

Speaker: 
Elana Kalashnikov
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

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.

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Exact Lagrangians in conical symplectic resolutions

Speaker: 
Filip Zivanovic
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

Conical symplectic resolutions are a vast family of holomorphic symplectic manifolds that appear in representation theory, algebraic and differential geometry, and also in theoretical physics. Their typical examples arise from the hyperkähler quotient construction (quiver and hypertoric varieties) but also from the representation theory of Lie algebras (resolutions of Slodowy varieties, slices in affine Grassmannians). In this talk, I will focus on their symplectic topology. In particular, we find families of non-isotopic exact Lagrangian submanifolds in them arising from different C*-actions. These Lagrangians have a very nice symplectic topology; in particular, we conjecture (work in progress) that all of their Floer-theoretic invariants are completely determined by their topology. At the end of the talk, I will discuss the special cases of Nakajima quiver varieties and resolutions of Slodowy varieties, where their count becomes feasible and interesting in its own.

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Birational geometry of quiver varieties

Speaker: 
Gwyn Bellamy
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

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).

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Euler numbers of Hilbert schemes of points on simple surface singularities

Speaker: 
Hiraku Nakajima
Date: 
Thu, Jun 25, 2020
Location: 
Zoom
Conference: 
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions
Abstract: 

We prove the conjecture by Gyenge, Némethi and Szendrői in arXiv:1512.06844, arXiv:1512.06848 giving a formula of the generating function of Euler numbers of Hilbert schemes of points Hilbn(C2/Γ) on a simple singularity C2/Γ, where Γ is a finite subgroup of SL(2). This is based on my preprint arXiv:2001.03834.

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Mathematical model, analysis and simulations of the COVID-19 pandemic with variable infection rate: Application to South Korea

Speaker: 
Meir Shillor
Date: 
Wed, Jun 24, 2020
Location: 
Zoom
Conference: 
CAIMS - PIMS Coronavirus Modelling Conference
CRG: 
Abstract: 

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

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Scenario tree and adaptive decision making on optimal type and timing for intervention and social-economic activity changes

Speaker: 
MIchael Chen
Kyeongah Nah
Date: 
Wed, Jun 24, 2020
Location: 
Zoom
Conference: 
CAIMS - PIMS Coronavirus Modelling Conference
CRG: 
Abstract: 

We assess Ontario’s reopening plans, taking into account the healthcare system capacity and uncertainties in contact rates during different reopening phases. Using stochastic programming and a disease transmission model, we find the optimal timing for each reopening phase that maximizes the relaxation of social contacts under uncertainties, while not overwhelming the health system capacity by an expected arrival time of a SARS-CoV-2 vaccine/drug.

* Written with Michael Chen and LIAM De-escalation Group

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Law of mass action and saturation in SIR model with applications to coronavirus

Speaker: 
Theodore Kolokolnikov
Date: 
Wed, Jun 24, 2020
Location: 
Zoom
Conference: 
CAIMS - PIMS Coronavirus Modelling Conference
CRG: 
Abstract: 

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.

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CAIMS - PIMS Coronavirus Modelling Conference - Panel

Speaker: 
Penelope Morel
Adrianne Jenner
Jane Heffernan
Wei Dai
Rohit Rao
Date: 
Wed, Jun 24, 2020
Location: 
Zoom
Conference: 
CAIMS - PIMS Coronavirus Modelling Conference
Abstract: 

A panel session was heard after the morning session of the third day of this conference. The panelists were the speakers from the 4 preceeding talks.

  1. The immune response to SARS-CoV-2: Friend or Foe? - Penelope Morel
  2. Modelling the systemic and tissue-level immune response to SARS-CoV-2 - Adrianne Jenner
  3. Models for immune system interaction and evolution - Jane Heffernan
  4. A Quantitative Systems Pharmacology Model of the Immune Response to SARS-COV-2 - Wei Dai, Rohit Rao
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A Quantitative Systems Pharmacology Model of the Immune Response to SARS-COV-2

Speaker: 
Wei Dai
Rohit Rao
Date: 
Wed, Jun 24, 2020
Location: 
Zoom
Conference: 
CAIMS - PIMS Coronavirus Modelling Conference
CRG: 
Abstract: 

Rapid development of a QSP model to support novel COVID-19 therapies. We intend to publish this model quickly to encourage community feedback. The simulated dynamics of immune response are modeled by describing viral activation of innate and adaptive immune processes involving both pro-inflammatory mediators regulating viral clearance and cell damage (e.g. neutrophils and cytotoxic lymphocytes) as well as counter-regulatory immune suppressive mediators (e.g. Treg cells and IL-10).

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