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Sean Monahan, Department of Pure Mathematics, University of Waterloo

"Approximating rational points on horospherical varieties"

I will discuss a recent paper that I wrote with Matt Satriano, the title of which is conveniently the title of this talk (arXiv 2308.11847). David McKinnon has a conjecture known as the “curve of best approximation (COBA)” conjecture, which says that among the sequences which approximate a given rational point on a projective variety, the ones that do their job the best must lie on a rational curve inside the variety. To probe this conjecture, Matt and I explored the case where the variety is horospherical so that we could use the tractable combinatorial theory which these varieties possess. Unfortunately, the stars did not align since this talk will not take place in MC 5417, which is roughly the centre point between the offices of Matt, David, and myself.

QNC 1502

Benjamin Anderson-Sackaney, University of Saskatchewan

"Amenability of Fusion Modules and Coideals"

The coideals of a quantum group offer a quantum analogue of a subgroup of a group. For certain classes of coideals there is an obvious quantum analogue of a quasi-regular representation. For a larger class of coideals recently introduced by De Commer and Dzokou Talla, namely, the so-called $g$-integral coideals, we will introduce a notion of a $g$-quasi-regular representation. We will then define a notion of $g$-coamenability that generalizes the notion of a coamenable inclusion of groups. We will also introduce a notion of amenability of a fusion module equipped with a dimension function that is compatible with a dimension function on the given fusion algebra. This notion gives a characterization of $g$-coamenability at the tensor categorical level.

This seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Ila Varma, University of Toronto

"Counting number fields and predicting asymptotic"

A guiding question in number theory, specifically in arithmetic statistics, is: Fix a degree n and a Galois group G in S_n. How does the count of number fields of degree n whose normal closure has Galois group G grow as their discriminants tend to infinity? In this talk, we will discuss the history of this question and take a closer look at the story in the case that n = 4, i.e. the counts of quartic fields.

MC 5501

Christine Eagles, Department of Pure Mathematics, University of Waterloo

"NIP"

We continue to read through Pierre Simon's "A Guide to NIP theories". 

MC 5403

Michael Albanese, Department of Pure Mathematics, University of Waterloo

"Blowups and examples!"

We will discuss blowups and prove that the blowups of S^2 bundles over S^2 are diffeomorphic. We will use the remaining time to discuss other examples.

MC 5403

Yash Singh, Department of Pure Mathematics, University of Waterloo

"Lines and surfaces in P^3"

We use our understanding of Grassmannians so far to study the geometry of lines and surfaces in P^3.

This seminar will be held both online and in person:

• MC 5417
• Zoom link: https://uwaterloo.zoom.us/j/93801243240?pwd=blpUMVJ1dExUYVlEMkJVWjlscUNqZz09

Jing Zhang, University of Toronto

"Compactness and incompactness in higher dimensional combinatorics"

We describe an organizing framework to study higher dimensional infinitary combinatorics based on \v{C}ech cohomology, originating from works by Barry Mitchell, Barbara Osofsky and others. A central combinatorial notion is $n$-dimensional coherence sequences, generalizing the 1-dimensional ones studied extensively by Todorcevic using the method of minimal walks. We will discuss ZFC results suggesting $\aleph_n$ is not "compact for $(n+1)$-dimensional combinatorics" and consistency results that any regular cardinal greater or equal to $\aleph_{\omega+1}$ can be "compact for $n$-dimensional combinatorics for all $n$". The talk will be purely combinatorial. Joint work with Jeffrey Bergfalk and Chris Lambie-Hanson.

MC 5479

Francisco Villacis, Department of Pure Mathematics, University of Waterloo

"An Overview of the Gelfand-Cetlin System"

The Gelfand-Cetlin system is an integrable system first introduced by V. Guillemin and S. Sternberg in 1983 in order to study the quantization of complex flag varieties. This integrable system shares many properties with moment maps coming from torus actions, such as having a polytope as image and the fibres above the interior points of the polytope are Lagrangian tori. There is one big difference between toric moment maps and the Gelfand-Cetlin system: the latter allows for Lagrangian fibres which are not tori. This phenomenon makes this system an important object to study in the context of mirror symmetry.  In this talk, I will give a brief overview of the Gelfand-Cetlin system and discuss the classification by Y. Cho, Y. Kim and Y-G. Oh of the Lagrangian fibres.

MC 5417

Joey Lakerdas-Gayle, Department of Pure Mathematics, University of Waterloo

"Effective Descriptive Set Theory 4"

We will continue to introduce effective descriptive set theory following Andrew Marks's notes.

MC 5479

Frederik Benirschke, University of Chicago

"Isometric embeddings and totally geodesic submanifolds of Teichmüller spaces"

Classical results by Royden, Earle, and Kra imply that the biholomorphism group of Teichmüller space, the isometry group of the Teichmüller metric, and the mapping class group of the underlying surface are all isomorphic. In other words, every isometry of Teichmüller space is induced by a homeomorphism of the underlying surface.

In this talk, we present a generalization, obtained in joint work with Carlos Serván, where we relax isometries to isometric embeddings. The main result is that isometric embeddings of Teichmüller spaces are coverings constructions, except for some low-dimensional special cases. In other words: Isometric embeddings are induced by branched coverings of the underlying surfaces.

Time permitting, we explain how our techniques can be used to rule out the existence of certain totally geodesic submanifolds.

QNC 2501