Spiro Karigiannis
Organizational Meeting
We will meet to plan out the Differential Geometry Working Seminar for the Fall 2024 term.
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
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
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
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
There are at least two viewpoints on the modularity of elliptic curves over the rationals: it can be seen either as an analytic and representation-theoretic statement that the L-function of a curve is associated to a modular form, or as a geometric statement that the curve is a quotient of a modular curve. It is not clear that these remain equivalent for elliptic curves over number fields. For elliptic curves over real quadratic fields, analytic modularity is now known, and a form of geometric modularity was conjectured 40 years ago by Oda. Recent advances in the computation of rings of Hilbert modular forms have made it possible to verify the geometric modularity conjecture in special cases. In this talk I will describe my work in this direction, including some interesting auxiliary algebraic surfaces that arise in the course of the computations.
MC 5417
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
Alex Cowan, University of Waterloo
Statistics of modular forms with small rationality fields
We present (i) a new database of weight 2 holomorphic modular forms, and (ii) a new statistical methodology for assessing probabilistic heuristics using arithmetic data. With this methodology we discover examples of non-random behavior and strange behavior in our dataset and beyond. This is joint work with Kimball Martin.
MC 5479
Jiahui Huang, University of Waterloo
Various de Rham cohomologies in algebraic geometry
De Rham's theorem states that the de Rham cohomology of a smooth manifold is isomorphic to its singular cohomology. Various generalizations of the de Rham cohomology exist in algebraic geometry. In this talk we will take a look at algebraic de Rham cohomology for singular varieties, Chiral de Rham cohomology for smooth schemes, and derived de Rham cohomology for derived stacks.
MC 5403