Mathematics

Quantum cryptography is known for enabling functionalities that are unattainable using classical information alone. Recently, Secure Software Leasing (SSL) has emerged as one of these areas of interest. Given a circuit ? from a circuit class, SSL produces an encoding of ? that enables a recipient to evaluate ? and also enables the originator of the software to later verify that the software has been returned, meaning that the recipient has relinquished the possibility to further use the software. Such a functionality is unachievable using classical information alone, since it is impossible to prevent a user from keeping a copy of the software. Recent results have shown the achievability of SSL using quantum information for compute-and-compare functions (a generalization of point functions). However, these prior works all make use of setup or computational assumptions. We show that SSL is achievable for compute-and-compare circuits without any assumptions.
We proceed by studying quantum copy-protection, which is a notion related to SSL, but where the encoding procedure inherently prevents a would-be quantum software pirate from splitting a single copy of an encoding for ? into two parts each allowing a user to evaluate ?. Using quantum message authentication codes, we show that point functions can be copy-protected without any assumptions against one honest and one malicious evaluator. We then show that a generic honest-malicious copy-protection scheme implies SSL. By prior work, this yields SSL for compute-and-compare functions.

This is joint work with Anne Broadbent, Stacey Jeffery, Supartha Podder, and Aarthi Sundaram.

The entanglement of free fermions on Hamming graphs will be discussed. This will be used to showcase how tools of algebraic combinatorics such as the Terwilliger algebra are well suited for this analysis. The usefulness of a Heun operator generalization will also be stressed and extensions to other association schemes will be mentioned.

I will describe a way to compute anomalies in general (2+1)D fermionic topological phases. First, a mathematical characterization of symmetry fractionalization for (2+1)D fermionic topological phases is presented, and then this data will be used to define a (3+1)D state sum for a topologically invariant path integral that depends on a generalized spin structure and G bundle on a 4-manifold. This path integral is a cobordism invariant and describes a (3+1)D fermion symmetry-protected topological state (SPT). The special case of time-reversal symmetry with ?2=−1? gives a ℤ16 invariant of the 4D Pin+ smooth bordism group, and gives an example of a state sum that can distinguish exotic smooth structure.

Please note, the last 3 minutes of the talk are missing from the video

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 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.