Events

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

Anton Bernshteyn, UCLA

Minimal subdynamics and its applications

Topological dynamics is the study of continuous group actions. The classical and best understood cases are when the acting group is either $\mathbb{Z}$ or $\mathbb{R}$, while the general theory remains far less clear. In particular, the tools to approach the dynamics of non-Abelian discrete groups have only begun to emerge in recent years, in part thanks to the arrival of powerful techniques from combinatorics and descriptive set theory. In this talk, I will illustrate this confluence of ideas by discussing the following very basic problem: If $\Delta$ is a subgroup of $\Gamma$, does $\Gamma$ have a free continuous action on a compact Hausdorff space without nontrivial closed $\Delta$-invariant subsets? This talk is based on joint work with Joshua Frisch.

MC 5501

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

Cole Wyeth, University of Waterloo

Introduction to Prefix-free Kolmogorov Complexity

I will continue last week's introduction to algorithmic complexity by upgrading from the plain Kolmogorov complexity to the prefix-free Kolmogorov complexity, which offers a more effective explanation of effective explanations. This alternative formalization of algorithmic complexity can be motivated in terms of probabilistic programs with "random seeds." I will explain why the probabilistic  approach might be considered heretical (by Kolmogorov), and prove some slightly more sophisticated properties of the prefix-free Kolmogorov complexity. As time permits, I will also define its conditional version, which has been used to construct the information distance and its practical counterpart, the normalized compression distance, which was applied to bioninformatics by my advisor Professor Ming Li.

MC 5403

Jacques Van Wyk, University of Waterloo

The Clutching Construction

The clutching construction is a technique in differential topology to construct fibre bundles over spheres. I will explain how the clutching construction works, and how it can be used to define symplectic fibre bundles over spheres.

MC 5403

Owen Sharpe, University of Waterloo

Vaughan's Identity, Exponential Sums over Primes, and Exponential Sums with Additive Coefficients

Vaughan's Identity is a technique for summing sequences of the form f(n) Lambda(n), where Lambda(n) is the von Mangoldt function. We apply it to f(n) = e(alpha n) to obtain asymptotics for the sums of sequences of the form e(alpha p). Finally, we show an application to summing sequences of the form f(n) e(alpha n), where f is an additive function satisfying f(p) = 1 for all prime p.

MC 5479

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

Karl Dilcher, Dalhousie University

Heronian triangles, Gauss primes, and some linear recurrences

We will see that certain sequences of Heronian triangles, that is, triangles with sides of integer length and with integer area, occur in an unexpected way in the study of some specific factorials. In particular, we will consider the multiplicative order of ((p-1)/4)! modulo a prime p = 1 (mod 4). The question of when this order can be a power of 2 leads to the concept of a "Gauss prime". Apart from explaining these various connections, I will derive some divisibility properties of the sequences in question.

Time allowing, I will also discuss factorials ((p-1)/3)! modulo primes p = 1 (mod 6), and generalizations of such factorials. Quite recently, a close relationship between "exceptional primes" in this setting and Iwasawa theory was established by M. Stokes in his Ph.D. thesis.

(Joint work with John Cosgrave.)

MC 4064