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Lawrence Mouillé, Syracuse University

"Positive intermediate Ricci curvature with maximal symmetry rank"

The Grove-Searle Maximal Symmetry Rank Theorem (MSRT) is a foundational result in the study of manifolds with positive sectional curvature and large isometry groups. It provides a classification of closed, positively curved manifolds that admit isometric actions by tori of large rank. In this talk, I will present progress towards extending the MSRT to positive intermediate Ricci curvature, a condition that interpolates between positive sectional curvature and positive Ricci curvature. Grove and Searle were able to employ concavity of distance functions to establish their MSRT, but this feature is not available for positive intermediate Ricci curvature. I will discuss how we can overcome this barrier using a strengthening of Wilking's Connectedness Lemma. A portion of this talk is from joint work with Lee Kennard.

MC 5417

Organisational Meeting

We will discuss the format of the seminar and determine the first set of speakers. If you would like to speak or otherwise participate in the meeting and are unable to attend, please contact AJ Fong.

MC 5417

An organizational meeting for the Differential Geometry Working Seminar will be held at 2:30pm on January 9, 2024.

M3 4206

Niclas Technau, Max Plank Institute

"Counting Rational Points Near Manifolds"

Choose your favourite, compact manifold M. How many rational points, with denominator of bounded size, are near M? We report on joint work with Damaris Schindler and Rajula Srivastava addressing this question. Our new method reveals an intriguing interplay between number theory, harmonic analysis, and homogeneous dynamics.

MC 5501

Jesse Peterson, Vanderbilt University

"Amenability and von Neumann algebras"

Amenability for groups is a notion that was first introduced by von Neumann in 1929 in order to provide a conceptual explanation for the Banach-Tarski paradox. The notion has since been exported to many different areas of mathematics and continues to hold a distinguished position in fields such as group theory, ergodic theory, and operator algebras. For von Neumann algebras the notion plays a fundamental role, with the classification of amenable von Neumann algebras by Connes and Haagerup being considered a touchstone of the area. In this talk, I will give a survey of amenability and von Neumann algebras, emphasizing my own contributions related to von Neumann algebras associated with lattices in Lie groups.

MC 5501

Lucas Mason-Brown, University of Oxford

"Unitary representations of semisimple Lie groups and conical symplectic singularities"

One of the most fundamental unsolved problems in representation theory is to classify the set of irreducible unitary representations of a semisimple Lie group. In this talk, I will define a class of such representations coming from filtered quantizations of certain graded Poisson varieties. The representations I construct are expected to form the "building blocks" of all unitary representations.

MC 5501

We will be streaming the last two lectures of Alain Connes’ Lecture Series titled “From rings of operators to noncommutative geometry” given at the Fields Institute. The lecture series details the origin and impact of non-commutative geometry to various areas in mathematics and will end with recent advances in the program of the operator theoretic approach to the Riemann Hypothesis. Everyone is welcome to join us for a joint viewing experience. For more information and the Zoom link for the first part of the Lecture Series see: http://www.fields.utoronto.ca/activities/23-24/Alain-Connes

MC 5479

Kaleb Ruscitti, Department of Pure Mathematics, University of Waterloo

"Homomorphic Encryption (or: my summer as a Fed)"

In this talk, I will describe homomorphic encryption, which are encryption schemes that allow one to evaluate polynomial functions on the encrypted data. I will introduce the basics of encryption and then the general theory of homomorphic encryption, and then discuss some of the applications to online privacy that I looked at during my summer research internship.

MC 5501

Arul Shankar, University of Toronto

"Secondary terms in the first moment of the 2-Selmer groups of elliptic curves"

Ranks of elliptic curves are often studied via their 2-Selmer groups. It is known that the average size of the 2-Selmer group of elliptic curves is 3, when the family of all elliptic curves is ordered by (naive) height. On the computational side, Balakrishnan, Ho, Kaplan, Spicer, Stein, and Weigand collect and analyze data on ranks, 2-Selmer groups, and other arithmetic invariants of elliptic curves, when ordered by height. Interestingly, they find a persistently smaller average size of the 2-Selmer group in the data. Thus it is natural to ask whether there exists a second order main term in the counting function of the 2-Selmer groups of elliptic curves. In this talk, I will discuss joint work with Takashi Taniguchi, in which we prove the existence of such a secondary term.

MC 5501

Andy Zucker, Department of Pure Mathematics, University of Waterloo

"Ultracoproducts of G-flows"

Given a topological group G, a G-flow is a continuous action of G on a compact Hausdorff space X. This talk will discuss a notion of ultracoproduct for G-flows, which arise from considering ultraproducts of commutative G-C*-algebras by Gelfand duality. We apply the construction to develop an understanding of the properties of various classes of subflows of a flow, i.e. minimal, topologically transitive, etc. For groups which are locally Roelcke precompact, ultracoproducts of G-flows lead to a well-behaved notion of weak containment for a wide class of G-flows, and in particular for all G-flows when G is locally compact. In joint work with Gianluca Basso, we apply ultracoproducts of G-flows to achieve a new characterization of those Polish groups G with the property that every minimal flow has a comeager orbit.

MC 5479