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Spiro Karigiannis, University of Waterloo

Organizational Meeting

The Differential Geometry Working Seminar is an opportunity for all participants (students, postdocs, and faculty) to learn new things, to teach each other new things, and to get more practice in giving talks. It is very informal and confusion is common/encouraged. That's how we learn.

As usual, we'll start the term by attempting to decide on as much of the schedule as we can for the coming term. We'll have one speaker per week (most weeks) on Thursdays at 2:30pm.

MC 5403

Aareyan Manzoor, University of Waterloo

There is a non-Connes embeddable equivalence relation

Connes embeddability of a group is a finite dimensional approximation property. It turns out this property depends only on the so-called group von Neumann algebra. The property can be extended to all von Neumann algebras. The fact that there is a von Neumann algebra without this property was proved in 2020 using the quantum complexity result MIP*=RE. It is still open for group von Neumann algebras. I will discuss the best-known partial result, which is that there is a group action without this property. In particular, this implies the negation to the Aldous-Lyons conjecture, a big problem in probability theory

QNC 1507 or Join on Zoom

Steve Gonek, University of Rochester

The Density of the Riemann Zeta Function on the Critical Line

K. Ramachandra asked whether the curve f(t)=\zeta(1/2+it), t \in \R, is dense in the complex plane. We show that if the Riemann hypothesis, a zero-spacing hypothesis, and a plausible assumption about the uniform distribution modulo one of the normalized ordinates of the zeros of the zeta function hold, then the answer is yes.

Join on Zoom

Mattias Jonsson, University of Michigan

Pure Math Colloquium: Algebraic, analytic, and non-Archimedean geometry:

Algebraic geometry is (in part) concerned with solutions to polynomial equations with complex coefficients. It can be studied using complex analytic geometry, taking into account the standard absolute value on the complex numbers. There is a parallel world of non-Archimedean geometry, using instead the trivial absolute value. I will explain some relationships between the three types of geometry.

MC 5501

Beatrice-Helen Vritsiou, University of Alberta

On the Hadwiger-Boltyanski illumination conjecture for convex bodies with many symmetries

Let us think of a convex body in R^n (convex, compact set, with non-empty interior) as an opaque object, and let us place point light sources around it, wherever we want, to illuminate its entire surface. What is the minimum number of light sources that we need? The Hadwiger-Boltyanski illumination conjecture from 1960 states that we need at most as many light sources as for the n-dimensional hypercube, and more generally, as for n-dimensional parallelotopes. For the latter their illumination number is exactly 2^n, and they are conjectured to be the only equality cases.

The conjecture is still open in dimension 3 and above, and has only been fully settled for certain classes of convex bodies (e.g. zonoids, bodies of constant width, etc.). In this talk I will briefly discuss some of its history, and then focus on recent progress towards verifying the conjecture for all 1-symmetric convex bodies and certain cases of 1-unconditional bodies.

MC 5501

Noah Slavitch, University of Waterloo

Generic Absoluteness in Set Theory

We give an overview of the study of generic absoluteness for V in set theory, including a discussion of projective absoluteness and Sealing.

MC 5403

Xuemiao Chen, University of Waterloo

Space of lines-II

I will continue to talk about some related constructions on the space of oriented lines in the three dimensional Euclidean space.

MC 5403

Alex Pawelko, University of Waterloo

Gerbes of Coassociative Submanifolds and the first Chern class

We will define (bundle) gerbes, a generalization of principal S^1-bundles, and define connections on gerbes, whose corresponding forms over a trivialization are 2-forms and whose curvatures are 3-forms. Then, we will build a gerbe from coassociative submanifolds of a G_2 manifold, and study its analogue of the first Chern class, an integral cohomology class in degree 3.

MC 5403

Anton Iliashenko, Beijing Institute of Mathematical Sciences and Applications

Hyperbolicity and Schwarz Lemmas in Calibrated Geometry

We will define two new notions of hyperbolicity for a general Riemannian manifold equipped with a calibration, which generalize the notions of Kobayashi and Brody hyperbolicity from complex geometry. For this we introduce a decreasing Finsler pseudo-metric that specializes to the Kobayashi-Royden pseudo-metric in the Kahler case; and derive the generalization of the classical Schwarz Lemma but for Smith immersions. We will talk about how these notions of hyperbolicity relate to one another and will see some examples. This is joint work with Kyle Broder and Jesse Madnick.

MC 5417

Samantha Nadia Pater, Cuiwen Zhu and Hanwu Zhou

The Hasse Principle for Diagonal Forms via the Circle Method

The Hasse principle predicts that a Diophantine equation should have a rational solution whenever it has solutions in reals and p-adics for all primes p. For diagonal forms, this principle can be analyzed via the Hardy–Littlewood circle method. In this talk, we examine how the major and minor arc contributions are handled to establish asymptotic formulas for the number of integral solutions. Moreover, we would present a sketch of Jorg Brudern and Trevor D. Wooley's proof of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables.

MC 5417