Events

Aleksandar Milivojevic, University of Waterloo

Realizing topological data by closed almost complex manifolds

I will talk about the topological obstructions to placing an almost complex structure on a smooth manifold. I will then discuss how the vanishing of these obstructions is in many cases sufficient to realize a given rational homotopy type (with a choice of cohomology classes) by an almost complex manifold (with those cohomology classes as its rational Chern classes).

MC 5403

Andy Zucker, University of Waterloo

Tameness, forcing, and the revised Newelski conjecture

The revised Newelski conjecture asserts that for any group definable in an NIP structure, the automorphism group of its definable universal minimal flow is Hausdorff in the so-called "tau-topology." Recently, the countable case of the conjecture was proven by Chernikov, Gannon, and Krupinski using a deep result of Glasner, which provides a structure theorem for minimal metrizable tame flows. With this result, they prove that the Ellis group of a minimal metrizable tame flow (the automorphism group of a related flow) has Hausdorff tau-topology, and the conjecture for groups definable in countable NIP structures follows. We prove the revised Newelski conjecture in full by showing that the Ellis group of any minimal tame flow has Hausdorff tau-topology. To do this, we introduce new set-theoretic methods in topological dynamics which allow us to apply forcing and absoluteness arguments. As a consequence, we obtain a partial version of Glasner's structure theorem for general minimal tame flows. Joint work with Gianluca Basso.

MC 5403

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

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

Cole Wyeth, University of Waterloo

Introduction to Algorithmic Complexity

The Kolmogorov complexity of an object is the size of the smallest "self-extracting archive" that could have generated it, which can be viewed as an algorithmic information content. For instance, an image of the Mandelbrot set (to finite resolution) may appear quite visually complex, but is actually rather algorithmically simple since it requires only a short rule and iteration number to generate it, while typical noise is algorithmically complex. In this introductory talk, I will introduce the plain and prefix versions of the Kolmogorov complexity along with some of their basic properties such as (in)computability level.

MC 5403

Alex Pawelko, University of Waterloo

Calibrated Geometry of a Strongly Nondegenerate Knot Space

We will discuss a modification of Lee-Leung's work of the Kaehler structure on the knot space that allows one to define an infinite-dimensional analogue of G2 manifolds, then explore their calibrated geometry.

MC 5403

Jashan Bal, University of Waterloo

Projectivity in topological dynamics

A compact space is defined to be projective if it satisfies a certain universal lifting property. Projective objects in the category of compact spaces were characterized as exactly the extremally disconnected compact spaces by Gleason (1958). Analogously, if we fix a topological group G, then one can consider projectivity in the category of G-flows or affine G-flows. We present some new results in this direction, including a characterization of amenability or extreme amenability for closed subgroups of a Polish group via a certain G-flow being projective in the category of affine G-flows or G-flows respectively. Lastly, we introduce a new property, called proximally irreducible, for a G-flow and use it to prove a new dynamical characterization of strong amenability for closed subgroups of a Polish group. In doing so, we answer a question of Zucker by characterizing when the universal minimal proximal flow for a Polish group is metrizable or has a comeager orbit.

QNC 1507 or Join on Zoom

Nic Banks, University of Waterloo

Classification results for intersective polynomials with no integral roots

In this bald-faced attempt to practice my thesis defence, we introduce strongly intersective polynomials - polynomials with no integer roots but with a root modulo every positive integer - of degree 5-10. We start by describing their relation to Hilbert's 10th Problem and an algorithm of James Ax. These are fascinating objects which make contact with many areas of math, including permutation group theory, splitting behaviour of prime ideals in number fields, and Frobenius elements from class field theory.

In particular, we explain the computation of a list of possible Galois groups of such polynomials. We also discuss constraints on the splitting behaviour of ramified primes; in the process, we argue that intersectivity can be thought of as a property of a Galois number field, together with its set of subfields of specified degrees.

MC 5479

Elan Roth, University of Pennsylvania

Formalizing Turing Reductions in Lean

Turing Reducibility and Turing Degrees simply characterize how difficult problems in mathematics are. We will begin by reviewing reductions and degrees, situating the degree structure as a partial order with a central spine of relativized halting problems. Then, we will explore Lean as a functional programming language and theorem prover looking at its capabilities and applications to modern mathematics. Finally, we will turn to the formal development of Turing reducibility, equivalence, and the induced degrees defined as a quotient. The goal is to demonstrate both how classical computability theory can be mechanized in Lean and how the resulting framework supports further formalization in logic and computable structure theory.

MC 5403

Lorenzo Sarnataro, University of Toronto

Index, Intersections, and Multiplicity of Min-Max Geodesics

The p-widths of a closed Riemannian surface are geometric invariants associated with the length functional. In a recent work, Chodosh and Mantoulidis showed that these invariants are realised as the weighted lengths of unions of closed immersed geodesics (possibly, with multiplicity). I will discuss joint work with Jared Marx-Kuo and Douglas Stryker, where we prove upper bounds for the Morse index and number of intersections of min-max geodesics achieving the p-width of a closed surface. A key tool in our analysis is a proof that for a generic set of metrics, the tangent cone at any vertex of any finite union of closed immersed geodesics consists of exactly two lines. We also construct examples to demonstrate that multiplicity one does not hold generically in this setting. Specifically, we construct an open set of metrics on S^2 for which the p-width is only achieved by p copies of a single closed geodesic.

MC 5417