Events

Kevin Hare, University of Waterloo

Computational progress on the unfair 0-1 polynomial Conjecture

Let c(x) be a monic integer polynomial with coefficients 0 or 1. Write c(x)=a(x)b(x) where a(x) and b(x) are monic polynomials with non-negative real (not necessarily integer) coefficients. The unfair 0-1 polynomial conjecture states that a(x) and b(x) are necessarily integer polynomials with coefficients 0 or 1. We will discuss recent computationally progress towards this conjecture.

MC 5479

Kaleb D Ruscitti, University of Waterloo

Moduli of Line Bundles

As an example of a moduli problem that does not admit a fine moduli space, I have been studying the moduli space of line bundles. This admits a coarse moduli space: the quotient stack [pt/T], where T is a (algebraic) torus.

At first glance, [pt/T] seems very arcane, so I have been learning how one should understand this object. However it is an instructive simple case for motivating and working with moduli stacks. In this talk, I hope to present some different interpretations of [pt/T], so we can all be more comfortable with stacks.

MC 5479

Christine Eagles, University of Waterloo

Quantifier free internality and binding groups in ACFA

In ACFA, the definable closure of a set is not well understood. This presents an obstacle to understanding internality to the fixed field. Instead, we look at quantifier-free internality. In this talk we will follow Kamensky and Moosa (2024) by presenting quantifier-free internality and then stating a binding group theorem for rational types which are quantifier-free internal to the fixed field.

MC 5479

Gian Cordana Sanjaya, University of Waterloo

Squarefree discriminant of polynomials with prime coefficients

In 1991, Yamamura computed the density of monic polynomials of degree n which has discriminant not divisible by p^2 for any prime number p and positive integer n > 1. It is natural to conjecture that the density of monic polynomials of degree n with squarefree discriminant is the product of these local densities. This conjecture has been proved in 2022 by Bhargava, Shankar, and Wang in their paper, "Squarefree values of polynomial discriminants I".

In this talk, we consider a variant where the monic polynomials have prime coefficients. We compute the density of polynomials of degree n > 1 in this class which has squarefree discriminant, as an asymptotic density plus an explicit big-O error term. This is a joint work with Valentio Iverson and Xiaoheng Wang.

MC 5403

Kain Dineen, University of Waterloo

Gromov's non-squeezing theorem

I will discuss Gromov's non-squeezing theorem. We will prove the affine version of the theorem and discuss a potential generalization of it for maps preserving some power of the symplectic form. We will then discuss the general non-squeezing theorem and, as an application, prove the classical rigidity result that the symplectomorphism group of any symplectic manifold is (C^0)-closed in the diffeomorphism group.

MC 5479

Nicolas Chavarria Gomez, University of Waterloo

MCurve Excluding Fields IV

We continue reading through Will Johnson's and Vincent Ye's paper on the theory of existentially closed fields excluding a curve.

MC 5403

Roy Zhao, Tsinghua University

Unlikely Intersection Problems and The Pila-Zannier Method

The Zilber-Pink Conjecture or the Mordell-Lang Conjecture predict that the unlikely intersections, be it for dimension reasons or other geometrical reasons, between a variety and families of special subvarieties can be completely explained by only finitely many special subvarieties. In the past twenty years, Pila and Zannier introduced a new method to prove these types of problems by utilizing tools from o-minimality and functional transcendence. In this talk, we will give an overview of this method in some simple cases of the Andre-Oort Conjecture. Then, we will discuss our recent work and how it plays a key role in the Pila-Zannier method proof of the full Andre-Oort Conjecture.

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Jiahui Huang, University of Waterloo

Deformation of Complex Structures in Mirror Symmetry

In the spirit of relating the complex geometry of a Calabi-Yau manifold to the Kahler geometry of its mirror, this talk considers how their deformations relate to each other. We study deformations of complex structures via Kodaira-Spencer theory and Kahler structures via Gromov-Witten invariants. We will also look at how they relate to homological mirror symmetry.

MC 5479

Adam Jelinsky, University of Waterloo

The Completing Technique for sums of periodic complex valued functions

In Iwaniec and Kowalski's book on analytic number theory, they detail what they call the "completing technique" to evaluate bounds on incomplete sums of periodic functions Z^n->C by "completing" it by finding an equivalent complete sum over all Z/qZ. In this talk we will discuss how this completion technique can be used to prove the Polya-Vinogradov inequality, which gives a nearly tight bound on all sums of Dirichlet characters over the interval [N,N+M]. From this we will discuss other applications of this method, and give examples where this method fails to give a bound that is nontrivial.

MC 5403

Roberto Albesiano, University of Waterloo

From division to extension

The L^2 extension theorem of Ohsawa and Takegoshi, and the L^2 division theorem of Skoda are two fundamental results in complex analytic geometry. They are also intimately related: in fact, Ohsawa showed that a version of the latter can be proved as a corollary of the former. I will explain the main idea of Ohsawa and how, conversely, a version of the L^2 extension theorem can be obtained as an easy corollary of a Skoda-type L^2 division theorem with bounded generators.

MC 5479