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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

Casey Blacker, Augusta University

Geometric and algebraic reduction of multisymplectic manifolds

A symplectic Hamiltonian manifold consists of a Lie group action on a symplectic manifold together with an associated moment map. In special cases, the moment map distinguishes a smooth submanifold to which the Lie group action restricts, and the quotient inherits the structure of a symplectic manifold. In every case, it is possible to construct a reduced Poisson algebra that plays the role of the space of smooth functions on the reduced space.

In this talk, we will discuss an adaptation of these ideas to the multisymplectic setting. Specifically, we will exhibit a geometric reduction scheme for multisymplectic manifolds in the presence of a Hamiltonian action, an algebraic reduction procedure for the associated L-infinity algebras of classical observables, and a comparison of these two constructions. This is joint work with Antonio Miti and Leonid Ryvkin.

MC 5417

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

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

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

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

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

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

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

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