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Christine Eagles, University of Waterloo

The Zilber dichotomy in DCF_m II

We continue to read Omar Le\'on S\'anchez' paper on the Zilber dichotomy in partial differentially closed fields

MC 5403

Gerrik Wong, University of Waterloo

Tidy Subgroups and Ergodicity

We will continue talking about applications of tidy subgroups to ergodic automorphisms on totally disconnected locally compact groups.

MC 5403

Jérémy Champagne, University of Waterloo

Equidistribution and the probability of coprimality of some integer tuples

" What is the probability of two random integers being coprime? "

This question, sometimes called " Chebyshev’s Problem », is very natural and happens to have a very straightforward answer. Using only elementary methods, one can easily show that the natural density of pairs (m,n) with gcd(m,n)=1 is exactly 1/zeta(2)=6/pi^2=60.8..%.

Knowing this, one might seek certain g:N->N for which the density of n’s with gcd(n, g(n))=1 is also 1/zeta(2), which give a certain sense of randomness to the function g. Many functions with that property can be found in the literature, and we have a special interest for those of the form g(n)=[f(n)] where f is a real valued function with some equidistributive properties modulo one; for example, Watson showed in 1953 that g(n)=[αn] has this property whenever α is irrational. In this talk, we use a method of Spilker to obtain a more general framework on what properties f(n) must have, and also what conditions can replace coprimality of integer pairs.

MC 5403

Jesse Huang, University of Waterloo

Enumerative Mirror Symmetry

Continuing on with the introduction to mirror map and Yukawa coupling, we will discuss Gromov-Witten invariants and quantum cohomology which give rise to the statement of enumerative mirror symmetry. The statement extends to certain non-Calabi-Yau toric varieties, whose mirror information can be extracted from compactificatification of SYZ discussed on Monday.

MC 5479

Andrew Hanlon, Dartmouth College

Mirror Symmetry Seminar: Compactifying 2D mirror symmetry for the algebraic torus

Join on Zoom

Jacques van Wyk, University of Waterloo

The Mathematics of Tuning an Instrument; or, Why a Piano Is Always out of Tune

Have you ever wondered why a musical scale is seemingly arbitrarily split into twelve notes? Why twelve? And, how are these notes related? As we will see, there is no one answer to this question—there are multiple systems to define the twelve-note scale, and each one has its own advantages and disadvantages. I will be bringing my guitar and my trumpet to demonstrate how this ambiguity affects the way each instrument is tuned and played, and how, with some instruments like the piano, compromises are made that affect music in subtle ways.

MC 5501

Snacks will be served after.

Paul Cusson, University of Waterloo

Holomorphic vector bundles over an elliptic curve

We'll go over the classification of holomorphic vector bundles over an elliptic curve, with a focus on the rank 1 and 2 cases. For the case of line bundles, we'll show that the space of degree 0 line bundles is isomorphic to the elliptic curve itself. The classification of rank 2 bundles rests on the existence of two special indecomposable 2-bundles of degree 0 and 1, which we will describe in detail. The general case for higher ranks would then follow essentially inductively

MC 5479

William Verreault, University of Toronto

On the minimal length of addition chains

An addition chain is a sequence of increasing numbers, starting with 1 and ending with n, such that each number is the sum of two previous ones in the sequence. A challenging problem is, given a positive integer n, to find the minimal length of an addition chain leading to n. I will present bounds on the distribution function of this minimal length, which are sharp up to a small constant. This is joint work with Jean-Marie De Koninck and Nicolas Doyon.

MC 2034

AJ Fong, University of Waterloo

Finite automorphism groups of fans (with some adjectives)

A fan (with some aformentioned adjectives) is a subdivision of n-space into polyhedral cones from the origin subject to some conditions. I will make this precise and describe a classification of the finite automorphism groups when n=2.

MC 5403

Rahim Moosa, University of Waterloo

Zilber dichotomy in DCF_m

We will start reading Omar Leon Sanchez' recent paper by that name.

MC 5403