I will discuss recent results in the theory of symmetry-enriched topological phases, with a focus on the (2+1) case. I will review the classification of symmetry-enriched topological order and present general formula to compute relative 't Hooft anomaly for bosonic topological phases. I will also discuss partial results for fermionic topological phases and open questions.
Topological orders have a mathematical axiomatization in terms of their higher fusion categories of extended operators; the characterizing property of these higher fusion categories is that they are satisfy a nondegeneracy condition. After overviewing some of the higher category theory that goes into this axiomatization, I will describe what we do and don't know about the classification of topological orders in various dimensions.
The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit QED, I will present a hyperbolic generalization of Bloch theory, based on ideas from Riemann surface theory and algebraic geometry. The theory is formulated despite the non-Euclidean nature of the problem and concomitant absence of commutative translation symmetries. The general theory will be illustrated by examples of explicit computations of hyperbolic Bloch wavefunctions and bandstructures.
The COVID-19 pandemic has passed its initial peak in most countries in the world, making it ripe to assess whether the basic reproduction number (R0) is different across countries and what demographic, social, and environmental factors other than interventions characterize vulnerability to the virus. In this talk, I will show the association (linear and non-linear) between COVID-19 R0 across countries and 17 demographic, social and environmental variables obtained using a generalized additive model. Moreover, I will present a mathematical model of COVID-19 that we designed and used to explore the effects of adopting various vaccination and relaxation strategies on the COVID-19 epidemiological long-term projections in Ontario. Our findings are able to provide public health bodies with important insights on the effect of adopting various mitigation strategies, thereby guiding them in the decision-making process.
Born in N. Battleford, Saskatchewan (1945), member of the Gordon First Nation in Saskatchewan and a first generation Chinese Canadian, the Honorable Dr. Lillian Eva Quan Dyck is well-known for her extensive work in the senate on Missing & Murdered Aboriginal Women. She was the first female First Nations senator and first Canadian born Chinese senator. Prior to being summoned to the senate by the Rt. Hon. Paul Martin in 2005, she was a Full Professor in the Department of Psychiatry and Associate Dean, College of Graduate Studies & Research at the University of Saskatchewan.
She earned a BA, MSc in Biochemistry and Ph.D. in Biological Psychiatry, all from the University of Saskatchewan. She was conferred a Doctor of Letters, Honoris Causa by Cape Breton University in 2007. She has also been recognized in a number of ways, such as: A National Aboriginal Achievement Award for Science & Technology in 1999 and most recently the YWCA Saskatoon Women of Distinction Lifetime Achievement Award in 2019. She has been presented three eagle feathers by the Indigenous community.
Extremal length is a conformal invariant that plays an important
role in Teichmueller theory. For each essential closed curve on a Riemann
surface, it furnishes a function on the Teichmueller space. The extremal
length systole of a Riemann surface is defined as the infimum of extremal
lengths of all essential closed curves. Its hyperbolic analogue is the
hyperbolic systole: the infimum of hyperbolic lengths of all essential
closed curves. While the latter has been studied profusely, the extremal
length systole remains widely unexplored. For example, it is known that in
genus 2, the hyperbolic systole has a unique global maximum: the Bolza
surface. In this talk we introduce the extremal length systole and show
that in genus two it attains a strict local maximum at the Bolza surface,
where it takes the value square root of 2. This is joint work with Maxime
Fortier Bourque and Franco Vargas Pallete.
PIMS Workshop on New Trends in Localized Patterns in PDES
Abstract:
Narrow escape (NE) problems are concerned with the calculation of the mean first passage time (MFPT) for a Brownian particle to escape a domain whose boundary contains N small windows (traps). NE problems arise in escape kinetics modeling in chemistry and cell biology, including receptor trafficking in synaptic membranes and RNA transport through nuclear pores. The related Narrow capture (NC) problems are characterized by the presence of small traps within the domain volume; such traps may be fully absorbing, or have absorbing and reflecting boundary parts. The MFPT of Brownian particles traveling in domains with traps is commonly modeled using a linear Poisson problem with Dirichlet-Neumann boundary conditions. We provide an overview of recent analytical and numerical work pertaining to the understanding and solution of different variants of NE and NC problems in three dimensions. The discussion includes asymptotic MFPT expressions in in the limit of small trap sizes, the cases of spherical and non-spherical domains, same and different trap sizes, the dilute trap fraction limit and MFPT scaling laws for N 1 traps, and the global optimization of trap positions to seek globally and locally optimal MFPT-minimizing trap arrangements. We also present recent comparisons of asymptotic and numerical solutions of NE problems to results of full numerical Brownian motion simulations, in the usual case of constant diffusivity, as well as considering more realistic anisotropic diffusion near the domain boundary.
PIMS Workshop on New Trends in Localized Patterns in PDES
Abstract:
We propose an extension of the well-known Klausmeier model of vegetation to two plant species that consume water at different rates. Rather than competing directly, the plants compete through their intake of water, which is a shared resource between them. In semi-arid regions, the Klausmeier model produces vegetation spot patterns. We are interested in how the competition for water affects co-existence and stability of patches of different plant species. We consider two plant types: a thirsty species and a frugal species, that only differ by the amount of water they consume, while being identical in all other aspects. We find that there is a finite range of precipitation rate for which two species can co-exist. Outside of that range, (when the rate is either sufficiently low or high), the frugal species outcompetes the thirsty species. As the precipitation rate is decreased, there is sequence of stability thresholds such that thirsty plant patches are the first to die off, while the frugal spots remain resilient for longer. The pattern consisting of only frugal spots is the most resilient. The next-most-resilient pattern consists of all-thirsty patches, with the mixed pattern being less resilient than either of the homogeneous patterns. We also examine numerically what happens for very large precipitation rate. We find that for sufficiently high rate, the frugal plant takes over the entire range, outcompeting the thirsty plant.
PIMS Workshop on New Trends in Localized Patterns in PDES
Abstract:
The singularly perturbed Gierer-Meinhardt system has been a prototypical reaction diffusion system for the analysis of localized multi spike solutions. Motivated by recent interest in bulk-surface coupled systems, in this talk we address the structure and linear stability of multi spike solutions in the presence of inhomogeneous boundary conditions. Such inhomogeneities are shown to lead to the formation of both stable symmetric and asymmetric boundary bound spike solutions in one-dimensional domains and analogous solutions in higher dimensions.
PIMS Workshop on New Trends in Localized Patterns in PDES
Abstract:
The stability and dynamic properties of spike-type solutions to the Gierer- Meinhart equations are well understood. We examine the effect of adding noise to the equations on the spike-dynamics. We derive a stochastic ordinary differential equation for the motion of a single spike as well as the distribution of spike location from the associated Fokker-Plank equation. With sufficiently large amplitude noise, it is possible for the spike to reach the boundary of the domain and become effectively trapped for some time. In this case, we calculate the expected time to capture.