webnotice

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

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.

Join on Zoom

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

Ben Webster, University of Waterloo

Intro to 3-d mirror symmetry

This will be an overview talk, aiming to get people hyped up for the 3-d mirror symmetry seminar.

MC 2017

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

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

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

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

Joey Lakerdas-Gayle, University of Waterloo

Compactness and connectives in continuous logic

We will look at the compactness theorem and systems of connectives following "Model Theory for Metric Structures" by Ben Yaacov, Berenstein, Henson, and Usvyatsov.

MC 5403

Carlo Pagano, Concordia University

Hilbert 10 via additive combinatorics

In 1900 Hilbert proposed a list of problems that have been very influential throughout the last century. In 1970 Matiyasevich, building on earlier work of Davis—Putnam—Robinson, proved that Hilbert's 10th problem is undecidable for Z. The problem of extending this result to any ring that is finitely generated over Z (eg ring of integers in number fields) has attracted significant attention since 1970 and, thanks to the efforts of many mathematicians, the task has been reduced to an arithmetic problem about elliptic curves. This problem so far had been solved only conditional on the BSD conjecture (one of the Millenium problems) by Mazur—Rubin.

In joint work with Peter Koymans we have combined additive combinatorics (Green—Tao’s celebrated theorem) with 2-descent (an old technique dating back to Fermat) to solve this problem about elliptic curves unconditionally. This shows that Hilbert 10 is undecidable over any finitely generated infinite commutative ring.

In this colloquium I will provide a gentle introduction to this undecidability result, giving a glimpse of how mathematical logic, number theory and additive combinatorics meet into one story.

MC 5501