Events

Benoit Charbonneau, University of Waterloo

Invariant connections and Wang’s theorem

In this working seminar, we will study the classification result for invariant connections on principal bundles on homogeneous spaces proved by Hsien-Chung Wang in 1958 and learn, to paraphrase Gonçalo Oliveira, some useful facts on invariant connections.

MC 4058

Matthew Satriano, University of Waterloo

An introduction to toric stacks

Toric stacks are a tractable subclass of stacks due to their combinatorial structure. They can serve as an introduction to stacks in the same way that toric varieties can be an introduction to schemes. We will show how one can gain insight into the geometry of toric stacks with simple pictures of fans and marked points.

MC 5403

Beining Mu, University of Waterloo

Sacks' Splitting Theorem

In this talk, I will present Sacks’ Splitting Theorem, which states that every nonzero computably enumerable degree can be split into the join of two strictly lower computably enumerable degrees, as an example of finite injury priority argument. I will discuss two different proofs of the theorem, one of which is the classical way of how people think about finite injury arguments, while the other is a modern way of presenting a priority argument where a priority tree is involved.

MC 5403

Olivier Lalonde, University of Waterloo

Quantum chromatic numbers, orthogonal representations, and the Hadamard’conjecture

Cameron, Montanaro, Newman, Severini and Winter gave a construction which shows that, for \(n \in \{2,4,8\}\) any graph G which admits a real \(n\)-dimensional orthogonal representation satisfies \(\chi_q(G) \leq n\).This result can be recast as the statement that \(\chi_q(S^{n-1}_\mathbb{R}) = n\)  for these values of \(n\), where \(S^{n-1}_\mathbb{F}\) stands for the orthogonality graph of the unit sphere in \(\mathbb{F}^n\). We investigate possible extensions of their construction. We first show that their hypothesis that the orthogonal representation be real-valued is required by proving that \(\chi_q(S^{n-1}_\mathbb{C}) > n\) for all \(n \geq 3\). We also exhibit a finite subgraph \(G_{19}\) of \(S^{2}_\mathbb{C}\)  and show that \(k+4 = \chi_q^{(1)}(G_{19} \vee K_k) > \xi_{\mathbb{C}}(G_{19} \vee K_k) = k+3\) for all \(k\), so that the joins \(G_{19} \vee K_k\) form a family of finitary witnesses of the aforementioned separation for the special case of rank-one colorings. As a byproduct, we show that \(\xi_\mathbb{R}(G_{19}) = 4\), thereby separating the real and complex orthogonal ranks. For the case of the real sphere, we show that \(\chi_q(S^{n-1}_\mathbb{R}) > n\) whenever \(n \neq 2\) and \(n\) is not a multiple of 4. On the other hand, we show that \(\chi_q(S^{n-1}_\mathbb{R}) = n\) does does hold whenever a Hadamard matrix of order \(n\) exists. Hence, assuming the Hadamard conjecture, it follows that the CMNSW construction can be extended to real \(n\)-dimensional orthogonal representations if and only if \(n=2\) or \(n\) is a multiple of 4. Our method of proof involves showing the equivalence between the existence of such a construction and the ability to find a maximal code space for Clifford-algebraic errors given a clean ancilla, and we believe that the representation-theoretic techniques we use for tackling the latter problem could be of independent interest. It also follows from this equivalence that \(\chi^{(1)}_q(S^{n-1}_\mathbb{R}) = n\) if and only if \(n \in \{2,4,8\}\), thereby settling a conjecture of Zeng and Zhang.

QNC 1201

Julius Frizzell, University of Waterloo

Generic Measures

We will begin to discuss generic measures and their applications to ergodic theory in proving Roth’s theorem.

MC 5417

Tommaso Pacini, University of Torino

Anisotropic calibrations, adiabatic limits, and mirror symmetry

Calibrations, adiabatic limits and Fueter maps play an important role in the theory of man ifolds with special holonomy and in the corresponding gauge theory. The goal of this seminar is to show how they can be fitted into a very general framework, defined via distributions and the concept of “anisotropic calibrations”. This framework (i) applies in a uniform way across special holonomy, (ii) provides an identification between certain Fueter maps and calibrated submanifolds, (iii) introduces new degrees of freedom which may be useful towards genericity arguments, (iv) provides techniques for both explicit and abstract existence results for Fueter maps. This is joint work with Kotaro Kawai (BIMSA, China). The seminar will be largely non-technical. Details can be found in the arXiv paper with the same title.

MC 5403

Moritz Weber, Saarland University

Quantum Mathematics, quantum symmetries and quantum information

Since the early days of the foundation of quantum mechanics, 100 years ago, it was clear that a new kind of mathematics was needed in order to capture the new physics. At that time, John von Neumann formulated his principles of quantum mechanics and one of the main features was noncommutativity - the fact, that two observables A and B need not to commute. This was the starting point of a systematic study of noncommuting operators which quickly emancipated from "just a physics tool" to an own branch in mathematics as such. More and more often, it is called quantum mathematics nowadays and it comprises C*-algebras (aka quantum
topology), von Neumann algebras (aka quantum measure theory), Connes’s noncommutative geometry (aka quantum differential geometry), quantum groups and many more. I will give a brief survey on quantum mathematics, and I will then focus on an introduction to quantum symmetries and their link to quantum information theory.

MC 5501