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Ila Varma, University of Toronto

Counting Number Fields by P^1 height

When do two irreducible polynomials with integer coefficients define the same number field? One can define an action of GL_2 × GL_1 on the space of polynomials of degree n so that for any two polynomials f and g in the same orbit, the roots of f may be expressed as rational linear transformations of the roots of g; thus, they generate the same field. In this article, we show that almost all polynomials of degree n with size at most X can only define the same number field as another polynomial of degree n with size at most X if they lie in the same orbit for this group action. (Here we measure the size of polynomials by the greatest absolute value of their coefficients.)

This improves on work of Bhargava, Shankar, and Wang, who proved a similar statement for a positive proportion of polynomials. Using this result, we prove that the number of degree n fields such that the smallest polynomial defining the field has size at most X is asymptotic to a constant times X^{n+1} as long as n \ge 3. For n = 2, we obtain a precise asymptotic of the form 27/(pi^2) * X^2

This is joint work with Arango-Pineros, Gundlach, Lemke Oliver, McGown, Sawin, Serrano Lopez, and Shankar.

MC 5479

Henrique Sà Earp, Unicamp

Updates on flows of geometric structures

Aiming at a public with interests among Riemannian and complex geometry, Lie groups and parabolic PDEs, Iwill recall some results towards a general analytical theory for flows of H-structures, obtained with Fadel,Loubeau and Moreno. Then, as a concrete example, I will present some recent progress on SU(2)-flows on 4-manifolds, initiating a classification programme of ‘parabolic’ flows, based on a representation-theoretic methodoriginally due to Bryant in the context of G2 geometry, obtained with Fowdar.

MC 5501

Fateme Peimany, University of Waterloo

Definable Groups in Differentially Closed CCM-structures (DCCM)

How do definable groups behave when a compact complex manifold is equipped with a generic derivation? Inthis talk, I will discuss an ongoing project to understand definable groups within the theory of DifferentiallyClosed CCM-structures (DCCM). The presentation centers on a foundational model-theoretic analogy: just asadding a derivation to algebraic geometry bridges ACF and DCF, a parallel expansion maps CCM to DCCM inthe analytic setting. By reviewing established results for definable groups in ACF, DCF, and CCM, we willexplore how these classifications translate into the differential-analytic framework of DCCM.

MC 5417

Jules Ribolzi, University of Waterloo

A Chevalley structure theorem for meromorphic groups

We continue to study definable groups in the standard model of CCM. We will look at the proof of a Chevalleystructure type theorem for meromorphic groups (that is, meromorphic groups are regular in the sense of Fujiki)from the article of Pillay and Scanlon.

MC 5479

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