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Hong Wang, NYU Courant

Kakeya sets in R^3

A Kakeya set is a compact subset of R^n that contains a unit line segment pointing in every direction.  Kakeya set conjecture asserts that every Kakeya set has Minkowski and  Hausdorff dimension n. We prove this conjecture in R^3 as a consequence of a more general statement about union of tubes. This is joint work with Josh Zahl.

M3 1006

Julius Frizzell, University of Waterloo

The Bernoulli Discrepancy

The Bernoulli numbers were originally defined by Jacob Bernoulli with the goal of finding a general formula tocompute the sum of the first n consecutive m-th powers. In modern math, they have many different applications,most notably in analytic number theory, where they are used to analytically continue the Riemann Zeta functionvia Euler-Maclaurin summation. However, modern mathematicians use a slightly different definition to the oneoriginally given by Bernoulli (which differs in only one term). In this talk, we will discuss the differencesbetween the two definitions, looking at just some of the many examples given by Peter Luschny in his ”BernoulliManifesto”.

MC 5501

(Refreshments will start at 4:30)

Faisal Romshoo, University of Waterloo

Deformations of calibrations, II

We will continue where we left off last time, completing the proof of when the obstructions for the calibrationsvanish. If time permits, we will go through the proof of the fact that if an orbit is metrical, elliptic andtopological, then the corresponding moduli space is a smooth manifold.

MC 5403

Xiao Zhong, University of Waterloo

Topics in Arithmetic Dynamics

This thesis studies several problems in arithmetic dynamics, focusing on preimages of invariant subvarieties,common zeros of iterates of rational functions, and periodic curves for polynomial endomorphisms. Weinvestigate stabilization phenomena for rational points in backward orbits and develop dynamical cancellationresults for semigroups of polynomials. We also prove a finiteness theorem for common zeros of iterates ofcompositionally independent rational functions, answering a question of Hsia and Tucker. Finally, we studypolynomial endomorphisms of the projective plane with many periodic curves, showing that families containinga Zariski dense set of periodic curves must be invariant under an iterate, and we classify maps admittinginfinitely many periodic curves of bounded degree.

MC 5479

Fateme Peimany, University of Waterloo

Model Theory Working Seminar

We continue to study the structure of groups definable in CCM.

MC 5479

Jérémy Champagne, University of Waterloo

Weyl's Equidistribution Theorem in function fields and multivariable generalizations

This thesis is concerned with finding a suitable function field analogue to the classical equidistribution theorem of Weyl. More specifically, we are interested in the distribution of polynomial values f(x) as x runs over the ring Fq[T], and where the coefficients of f(X) are taken from the field of formal power series Fq((1/T)). Classically, results of this type were all subject to the constraint deg f <p where p:=char(Fq). In 2013, Lê, Liu and Wooley were able to break this characteristic barrier using modern developments regarding Vinogradov's Mean Value Theorem.

The first set of results in this thesis consists in a resolution of the main conjecture made by Lê-Liu-Wooley, which establishes the largest possible class of equidistributed polynomial sequences f(x) that can be determined by irrationality conditions on the coefficients of f(X). This is done by introducing a new transformation f(X) -> f^τ(X) which preserves the size of Weyl sums, and is such that f^τ(X) does not involve any powers divisible by p.

The second sets of results is concerned with a multivariate generalization of the method of Lê-Liu-Wooley. As such, we use a multivariate version of Vinogradov's Mean Value Theorem together with the Large Sieve Inequality to obtain suitable minor arc estimates for Weyl sums in d variables. We then use these minor arc estimates to study the distribution of polynomial values f(x_1,...,x_d)$ as (x_1,...,x_d) runs over Fq[T]^d, and we also consider the case where each of x_1,...,x_d is required to be monic.

MC 6029

William Dan, University of Waterloo

Solovay Reducibility

Having discussed the relationship between Solovay reducibility and the newly introduced reducibilities, K-reducibility and C-reducibility, we turn back to study its relationship with previously discussed reducibilities, Turing reducibility and wtt-reducibility. Then, if time permits, we will completely finish sections 9.1 and 9.2 by discussing a final characterization of Solovay reducibility and going beyond random left-c.e. reals to look at random left-d.c.e. reals.

MC 5403

Liam Orovec, University of Waterloo

Greedy and Lazy expansions for Pisot and Salem Numbers

We call a sequence (a_i) a beta-representation for x provided that the sum over all positive integers ofa_i*beta^{-i} is equal to x. We call the lexicographically largest of of these sequences the greedy expansion for xunder base beta and the smallest we call the lazy expansion. A real number is called a Parry number if its greedyexpansion for 1 is eventually periodic or finite. Similarly we label those real numbers whose lazy expansion for 1is eventually periodic as lazy Parry numbers. Given a PV number with minimal polynomial M(x), we know thatfor sufficiently large value of m that the polynomial T_m^{\pm}(x) has a Salem root. We give criteria that relatesthe greedy and lazy expansions for 1 under these Salem numbers to the greedy and lazy expansions for 1 underthe PV number. We consider Salem numbers of degree 4, the minimal such degree. We will prove these arealways lazy Parry numbers and give explicit constructions for their lazy expansions for 1. We compare theseexpansions to the results of Boyd, that proved that they are also Parry numbers.

MC 5501

Fateme Peimany, University of Waterloo

Strongly minimal groups in CCM

We continue our study of the structure of groups definable in CCM, now in our second session on this topic, withthe goal of proving that every strongly minimal group is either a complex torus or a (commutative) linearalgebraic group.

MC 5479

Spencer Kelly, University of Waterloo

Constructing a Slice Theorem in Infinite Dimensions

The slice theorem is a powerful tool for understanding proper group actions on manifolds; however it does nothold on infinite dimensional manifolds, nor does there exist a general infinite dimensional extension of it.However, on specific infinite dimensional manifolds, working on a case-by-case basis, we have been able toconstruct analogues of the slice theorem. In this talk, we will investigate one of these cases, namely the space ofconnections on a bundle over a compact Riemannian manifold, acted on by the gauge group.

MC 5403