webnotice

Nicolas Banks, University of Waterloo

Classification Results for Intersective Polynomials With No Integral Roots

In this thesis defence, we introduce strongly intersective polynomials - polynomials with no integer roots but with a root modulo every positive integer - of degree 5-10. 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 5417 or Join on Teams

Aleksa Vujičić, University of Waterloo

PhD Thesis Defence

In 1972, Baggett showed that a locally compact group G is compact if and only if its dual space of irreducible representations G^ is discrete.

Curiously however, there are non-discrete groups whose duals are compact, and such a group was identified in the same paper.

In a similar vein, one can define the Fell group Op* ⋊ Qp, where Op denotes the p-adic integers, and Qp the p-adic numbers).

Baggett shows that this is a noncompact whose dual is not countable.

In this talk, we shall discuss the dual space structure of this and other related groups.

It is well known that p-adics are an instance of a local field, that is a non-discrete locally compact field.

In the corresponding thesis, we generalise the results of Baggett to what we call the local Fell groups, the local field equivalents of the Fell group.

We also work in this local context for all forthcoming results, though we state it in terms of p-adics to simplify matters.

This talk is divided into two parts.

In the first, we analyse the Fourier and Fourier-Stieltjes algebras of these local Fell groups, which are of the form A ⋊ K for A abelian and K compact.

These local Fell groups fall into a particular class of groups induced by actions for which the stabilisers are 'minimal', and we call such groups cheap groups.

For groups of this form, we show that B(G) = B_∞(G) ⊕ A(K) ∘ q_K, where B_∞(G) is the Fourier space generated by purely infinite representations.

We also show that in groups with countable open orbits (such as the local Fell groups) this simplifies further to B(G) = A(G) ⊕ A(K) ∘ q_K.

In an attempt to generalise this to higher dimensional analogues, for which the above does not hold true, we examine the structure of B_∞(G).

In particular, we obtain a result for dimension two in terms of the projective space, and we show that this is in some sense the 'best' decomposition that can be made.

In the second portion, we study the amenability of the central Fourier algebra ZA(G) = A(G) ∩ L1(G) for G = Op ⋊ Op*.

We show that ZA(G) contains as a quotient the Fourier algebra of a hypergroup, which is induced by the action of Op* ↷ Op.

In general, if H is a hypergroup induced by an action K ↷ A, then there is a corresponding dual hypergroup H^ by the dual action.

When this is the case, we show that this H satisfies A(H) = L1(H^), mimicking the classical result for groups.

We also show that if H^ has orbits which 'grow sufficiently large', then via a result of Alaghmandan, the algebra L1(H^) is not amenable.

In particular, this shows that ZA(G) is also not amenable, reaffirming a conjecture of Alaghmandan and Spronk.

MC 2009

Chantal David, Concordia University

Non vanishing for cubic Hecke L-functions

I will discuss recent work with Alexander Dunn, Alexandre de Faveri and Joshua Stucky, in which we prove that a positive proportion of Hecke L-functions associated to the cubic residue symbol modulo squarefree Eisenstein integers do not vanish at the central point. Our principal new contribution is the asymptotic evaluation of the mollified second moment with power saving error term. No such asymptotic formula was previously known for a cubic family (even over function elds). Our new approach makes crucial use of Pattersons evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Browns cubic large sieve, and a Lindelof-on-average upper bound for the second moment of cubic Dirichlet series that we establish.

MC 5417

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

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

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

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

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

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