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Alex Pawelko, University of Waterloo

Riemannian Geometry of Knot Spaces

We will review the construction of knot spaces of manifolds, specifically over G2 and Spin(7) manifolds. We will then see an explicit construction of the Levi-Civita connection of the knot space, and see what this can tell us about the torsion of the induced special geometric structures on knot spaces of G2 and Spin(7) manifolds.

MC 5403

Sergey Grigorian, University of Texas Rio Grande Valley

Geometric structures determined by the 7-sphere

The 7-sphere is remarkable not only for its rich topological and algebraic properties but also for the special geometric structures it encodes. In this talk, we explore how the symmetries and stabilizer subgroups of Spin(7) acting on the 7-sphere, regarded as the set of unit octonions, give rise to G2​-structures on 7-manifolds, SU(3)-structures on 6-manifolds, and SU(2)-structures on 5-manifolds. We will trace how these structures arise naturally via the inclusions of Lie groups and are reflected in the geometry of sphere fibrations. This perspective highlights the role of the 7-sphere as a unifying object in special geometry in dimensions from 5 to 8.

MC 5417

Jon Cheah, University of Hong Kong

An advertisement of cluster algebras

Cluster algebras have had many surprising links with many areas of mathematics beyond their original purpose in studying total positivity. In this expository talk, we consider two discrete dynamical systems, namely Markov triples and Coxeter--Conway friezes. While the study of these examples predated that of cluster theory, we will see how the latter provides a conceptual explanation for the intergrality and positivity phenomena.

MC 5479

(snacks at 16:00)

Spiro Karigiannis, University of Waterloo

Schwarz Lemma for Smith maps

I will discuss a generalized Schwarz Lemma for Smith maps, proved recently by Broder-Iliashenko-Madnick, and explain how it generalizes the classical Schwarz Lemma from complex analysis.

MC 5403

Christine Eagles, University of Waterloo    

Algebraic Independence of solutions to multiple Lotka-Volterra systems          

A major problem in recent applications of the model theory of DCF_0 is determining when a given system of algebraic differential equations defines a strongly minimal set. A definable set S is strongly minimal if it is infinite and for any other definable set R (over any set of

parameters), either S\cap R or S\setminus R is finite. In joint work with Yutong Duan and Leo Jimenez, we classify exactly when the solution set to a Lotka-Volterra system is strongly minimal. In the strongly minimal case, we classify all algebraic relations between Lotka-Volterra systems and show that for any distinct solutions x_1,...,x_n (not in the algebraic closure of the base field F), trdeg(x_1, ..., x_m/F) = 2m.

MC 5403

Kaleb Ruscitti, University of Waterloo

Correspondence between logarithmic connections and framed parabolic bundles on the blow up of a nodal Riemann surface

In this seminar I will explain how a Mehta-Seshadri type correspondence between logarithmic connections and parabolic vector bundles works for a specific setting of interest. That is the blow up of the complex curve xy=t at the nodal point.

MC 5403

Joaco Prandi, University of Waterloo

When the weak separation condition implies the generalize finite type in R^d

Let S be an iterated function system with full support. Under some restrictions on the allowable rotations, we will show that S satisfies the weak separation condition if and only if it satisfies the generalized finite-type condition. To do this, we will extend the notion of net intervals from R to R^d. If time allows, we will also use net intervals to calculate the local dimension of a self-similar measure with the finite-type condition and full support.

QNC 1507 or Join on Zoom

Lionel Nguyen Van Thé, Université d'Aix-Marseille

Revisiting the canonical Ramsey theorem for finite vector spaces

The infinite Ramsey theorem (1931) asserts that for every integer m, if the collection of all m-subsets of natural numbers is finitely colored, then there exists an infinite subset whose m-subsets are all of the same color. This result does not hold anymore if the number of colors is not finite, but Erdös and Rado (1950) showed that there is still an infinite subset where the coloring takes a very particular form, called "canonical". Both of these results admit appropriate finite forms which hold in the context of vector spaces instead of sets by results of Graham-Leeb-Rothschild (1972) and Voigt (1984). The purpose of this talk will be to present a new approach to this latter result.

MC 5403

Guillaume Dumas, University of Maryland

Boundedness of weak quasi-cocycles for higher rank simple groups

If G is a second countable locally compact group, the Delorme-Guichardet theorem states that Kazhdan property (T) is equivalent to the fixed-point property for continuous affine isometric actions on Hilbert spaces—that is, every 1-cocycle with values in a Hilbert space is bounded. Many rigidity statements rely on property (T): for example, morphisms of G into R are trivial. However, it does not provide tools for studying quasi-homomorphisms, since these maps do not respect the group structure. In order to study this class of maps, Ozawa introduced wq-cocycles, which respect a cocycle identity up to a bounded error. A group is said to have property (TTT) if all wq-cocycles are bounded. In this talk, I will discuss the relationship between this property and other more analytical forms of “almost” property (T). I will also explain how to prove that a group possesses this property, with a focus on simple groups and their lattices.

QNC 1507 or Join on Zoom

Jack Jia, University of Waterloo

Langlands Correspondence : Local, Global, and Possibly Geometric.

The classical global Langlands Correspondence is one of the deepest phenomenon in mathematics: In GL_1, it is equivalent to class field theory; in GL_2, it subsumes the modularity conjecture, which leads to the proof of Fermat’s last theorem. (Disclaimer: I will not talk about these). In this talk, I will explain what the local and global L.C. means, and talk about the geometrization if time permits.

Join us for the Grad colloquium to meet your fellow graduate students. Refreshments will start at 5:00, and the talk will start at 5:30.

MC 5417