Rahim Moosa, University of Waterloo
Zilber dichotomy in DCF_m
We will start reading Omar Leon Sanchez' recent paper by that name.
MC 5403
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.
Andrew Hanlon, Dartmouth College
Mirror Symmetry Seminar: Compactifying 2D mirror symmetry for the algebraic torus
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
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
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
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
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
Habiba Kadiri, University of Lethbridge
An explicit version of Chebotarev’s Density Theorem.
This talk will first provide a (non-exhaustive) survey of explicit results on zero-free regions and zero densities of the Riemann zeta function and their relationship to error terms in the prime number theorem. This will be extended to Dirichlet L functions and Dedekind zeta functions, where new challenges arise with potential exceptional zeros. We will explore estimates for the error terms for prime counting functions across various contexts, with a specific attention to number fields. Chebotarev’s density theorem states that prime ideals are equidistributed among the conjugacy classes of the Galois group of any normal extension of number fields. An effective version of this theorem was first established by Lagarias and Odlyzko in 1977. In this article, we present an explicit refinement of their result. Key aspects of our approach include using the following: smoothing functions, recently established zero-free regions and zero-counting formula for zeros of the Dedekind zeta function, and sharp bounds for Bessel-type integrals. This is joint wok with Sourabh Das and Nathan Ng.
MC 2034
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