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Gregory G. Smith, Queen's University

Sums of Squares and Projective Geometry

A multivariate real polynomial is nonnegative if its value at any real point is greater than or equal to zero.  These special polynomials play a central role in many branches of mathematics including algebraic geometry, optimization theory, and dynamical systems.  However, it is very difficult, in general, to decide whether a given polynomial is nonnegative.  In this talk, we will review some classical methods for certifying that a polynomial is nonnegative.  We will then present novel certificates in some important cases. This talk is based on joint work with Grigoriy Blekherman, Rainer Sinn, and Mauricio Velasco.

MC 5501

Peter Crooks, Utah State University

Lie-theoretic constructions in the Moore-Tachikawa category

I will briefly review the Moore-Tachikawa conjecture, as well as the representation theory underlying its formulation. This will lead to an outline of recent, affirmative evidence for the conjecture. I will also detail a systematic association of topological quantum field theories to Lie-theoretic data. Distinguished roles will be played by the partial Grothendieck-Springer resolutions and their Poisson-geometric relatives. This represents joint work with Maxence Mayrand. 

MC5417

Aleksandar Milivojevic, University of Waterloo

Formality in rational homotopy theory

I will introduce the notion of formality of a manifold and will discuss some topological implications of this property, together with a computable obstruction to formality called the triple Massey product. I will then survey a conjecture relating formality and the existence of special holonomy metrics.

MC 5479

Kaleb D Ruscitti, University of Waterloo

ntroducing the Log Canonical Threshold of a Singularity

Given a variety X, an ideal sheaf a, and a point p in X, the log canonical threshold of a at p is a birational invariant which generalizes the order of a. It appears in asymptotic expansions of certain intergrals, in the minimal model program for log-canonical pairs, and in many other algebraic geometry contexts.   In this seminar, I will give an introduction to this invariant, following the IMPANGA Lecture Notes on Log Canonical Thresholds by Mircea Mustață.

MC 5403

Joey Lakerdas-Gayle, University of Waterloo

Fundamentals of Computability Theory 1

This semester in the Computablility Theory Learning Seminar, we will be learning general Computability Theory following Robert Soare's textbook. This week, we will prove some of the fundamental theorems about Turing machines in Chapter 1 and 2.

MC 5403

Robert Haslhofer, University of Toronto

Mean curvature flow through singularities

A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow first arose as a model of evolving interfaces in material science and has been extensively studied over the last 40 years. In this talk, I will give an introduction and overview for a general mathematical audience. To gain some intuition we will first consider the one-dimensional case of evolving curves. We will then discuss Huisken's classical result that the flow of convex surfaces always converges to a round point. On the other hand, if the initial surface is not convex we will see that the flow typically encounters singularities. Getting a hold of these singularities is crucial for most striking applications in geometry, topology and physics. In particular, we will see that flow through conical singularities is nonunique, but flow through neck singularities is unique. Finally, I will report on recent work with various collaborators on the classification of noncollapsed singularities in R^4.

MC 5501

John Yin, Ohio State University

A Chebotarev Density Theorem over Local Fields

I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajović. This is joint work with Asvin G and Yifan Wei.

MC 5479

Brent Nelson, Michigan State University

Uniqueness of almost periodic states on hyperfinite factors

Murray and von Neumann initiated the study of "rings of operators" in the 1930's. These rings, now known as von Neumann algebras, are unital *-algebras of operators acting on a Hilbert space that are closed under the topology of pointwise convergence. Elementary examples include square complex matrices and essentially bounded measurable functions, but the smallest honest examples come from infinite tensor products of matrix algebras. These latter examples are factors—they have trivial center—and are hyperfinite—they contain a dense union of finite dimensional subalgebras. Highly celebrated work of Alain Connes from 1976 and Uffe Haagerup from 1987 showed that these infinite tensor products are in fact the unique hyperfinite factors. Haagerup eventually provided several proofs of this uniqueness, and one from 1989 included as a corollary a uniqueness result for so-called periodic states. This result only holds for some infinite tensor products of matrix algebras and is known to fail for certain other examples, but in recent joint work with Mike Hartglass we show that it can be extended to the remaining examples when periodicity is generalized to almost periodicity. In this talk, I will discuss these results beginning with an introduction to von Neumann algebras that assumes no prior knowledge of the field.

MC 5501

Spiro Karigiannis

Organizational Meeting

We will meet to plan out the Differential Geometry Working Seminar for the Fall 2024 term.

MC 5479

Andy Zucker

Topological groups with tractable minimal dynamics

For Polish groups, there are several interesting dividing lines in how complicated their minimal flows can be. While metrizability of the universal minimal flow is the most obvious, a theorem of Ben Yaacov, Melleray, and Tsankov suggests the broader class of Polish groups whose universal minimal flows have a comeager orbit. In joint work with Gianluca Basso, we find natural extensions of these classes to general topological groups, obtaining the classes of topological groups with ``concrete minimal dynamics'' or ``tractable minimal dynamics,'' respectively. Both classes admit a wide variety of non-trivial characterizations. In particular, the class of groups with tractable minimal dynamics is the largest class of topological groups admitting any form of KPT correspondence, allowing us to show that this class is absolute between models of set theory.

MC 5479