Events

Ragini Singhal, University of Münster

Solutions and singularities of the Ricci-harmonic flow and Ricci-like flows of G2-structures

We find explicit solutions and singularities of the Ricci-harmonic flow of $G_2$-structures on 7-dimensionalcontact Calabi-Yau manifolds and the 7-dimensional Heisenberg group. We prove that the natural co-closed$G_2$-structure on a contact Calabi-Yau manifold as the initial condition leads to an ancient solution of the Ricci-harmonic flow with a finite time Type I singularity. These are the first examples of Type I singularities of the Ricci-harmonic flow. We also obtain similar (but different) results for the Ricci-like flows of $G_2$-structures studied by Gianniotis--Zacharopoulos in arXiv:2505.06872 (J. Geom. Anal. 36.2 (2026)) and of the negative gradient flow of an energy functional of $G_2$-structures studied by Weiss--Witt. The talk is based on a joint work with Shubham Dwivedi (Hamburg).

MC 5417

Joey Lakerdas-Gayle, University of Waterloo

Priority Arguments on Trees

We will introduce terminology for priority trees following Steffen Lempp's notes and compare the classicalpriority argument for Sacks Cone Avoidance Theorem with a proof that uses a priority tree.

MC 5403

Adina Goldberg, Mittag-Leffler Institute

Quantum games and quantum graphs via the double category of quantum relations

Binary relations are ubiquitous. Classical nonlocal games (Bell scenarios) can be thought of as binary relations between question pairs and answer pairs. Classical undirected graphs can be thought of as symmetric binary relations on a vertex set. There is a notion of quantum relation: a quantum analogue of binary relation. Just as relations can be transformed by functions, quantum relations can be transformed by quantum functions. The interactions between quantum relations and quantum functions are made precise by the double category of quantum sets, quantum functions, and quantum relations. We define this new double category building on the work of Weaver, Kornell, and Musto-Reutter-Verdon. We illustrate how the category suggests, distinguishes, and motivates quantizations of some constructions/results from classical graph theory and nonlocal games. Basicterms related to C*-algebras, bimodules, and category theory will come up, but the talk can also be followed by analogy to the classical setting.

QNC 1201

Julius Frizzell, University of Waterloo

Mutliple recurrence for weakly-mixing transformations (Part II)

We will continue to discuss weakly-mixing transformations and work towards proving a special case of the multiple recurrence theorem.

MC 5417

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