Events

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

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

Faisal Romshoo, University of Waterloo

Constructing examples of Smith maps

I will start off by giving some background on Smith maps, which are special k-harmonic maps between two Riemannian manifolds. Smith maps have deep connections with both calibrated geometry and conformal geometry. I will then discuss my current work, where I am trying to construct explicit examples of Smith immersions.

MC 5403

AJ Fong, University of Waterloo

Galois actions on torus-invariant curves of toric surfaces

We are interested in studying the actions of the absolute Galois group on curves on varieties. Toric surfaces form an instructive class of examples since they are geometrically well-understood, and it is natural to study torus-invariant curves here. In this talk I will describe upper bounds on the Galois actions, and ongoing work regarding the inverse problem.

MC 5479

Barbara Csima, University of Waterloo

Measuring complexity of structures via their Scott Sentences

Scott’s Isomorphism Theorem shows that each countable structure can be uniquely defined, up to isomorphism, by a sentence of infinitary logic, now known as the Scott Sentence of the structure. The complexity of a structure’s Scott Sentence can then be viewed as a measure of complexity of the structure. In this talk, we will discuss the relationship of Scott complexity with other measures of complexity, as well as discuss the Scott Complexity of certain structures.

MC 5403

Kateryna Tatarko, University of Waterloo

Isoperimetric problem: from classical to reverse

The well-known classical isoperimetric problem states that the Euclidean ball has the largest volume among all convex bodies in R^n of a fixed surface area. We will discuss the question of reversing this result for the special class of convex bodies which are intersections of (finitely or infinitely many) balls of radius 1/lambda for some lambda>0. In particular, we will discuss the problem of determining which bodies in this class minimize the volume for a prescribed surface area and completely resolve it in R^3.

MC 5501

Alex Cowan, University of Waterloo

Murmurations from functional equations

Unexpected and striking oscillations in the average a_p values of sets of elliptic curves, dubbed murmurations, were recently discovered using techniques from data science. Since then, similar patterns have been discovered for many other types of arithmetic objects. In this talk we present a new approach for studying murmurations, revolving around mean values of L-functions in the critical strip and guided by random matrix theory. With our approach, we prove murmurations in many cases conditional on standard conjectures, and unconditionally for all GL_1 automorphic representations. To handle the case of elliptic curves knowledge is needed of the distribution of conductors of elliptic curves when ordered by height, which is of independent interest.

MC 5417

Cole Wyeth, University of Waterloo

ntroduction to Algorithmic Probability and the Coding Theorem

Building on the prefix-free Kolmogorov complexity discussed at our last meeting, I will introduce the basic objects of algorithmic probability. In particular, with a theory of effective explanations in hand, it is natural to ask which strings are more probable a priori? After all, it is harder to predict the data before you have seen it! The distributions generated by probabilistic Turing machines can be fully characterized as the (normalized) "lower semicomputable semimeasures," which naturally leads to the so-called "discrete universal distribution" m by simply mixing them all together. I will sketch a proof of Leonid A. Levin's coding theorem, which tells us that -lg m(x) = K(x) up to constants, meaning that all of our work was, in the most satisfying possible sense, for nothing: we can take only the shortest algorithmic explanation without losing anything. However, this is all just a warm-up: we will find that the situation is much more intricate when we turn to the prediction of infinite sequences, which I hope to gesture at, time permitting. 

MC 5403

Eduardo de Lorenzo Poza, KU Leuven

Algebraic Geometry Seminar:Singularities via arc spaces and Floer homology

Given an isolated hypersurface singularity, the arc-Floer conjecture relates the cohomology of the associated contact loci with the Floer homology of the monodromy iterates. In this talk we will explain the origin of this conjecture and what is known about it, and we will explore the key ingredients of the proof of the conjecture in the cases of plane curve singularities and semihomogeneous singularities. This is joint work with Javier de la Bodega and Jiahui Huang.

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