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
Elisabeth Werner, Case Western Reserve University
Affine invariants in convex geometry
In analogy to the classical surface area, a notion of affine surface area (invariant under affine transformations) has been defined. The isoperimetric inequality states that the usual surface area is minimized for a ball. Affine isoperimetric inequality states that affine surface area is maximized for ellipsoids. Due to this inequality and its many other remarkable properties, the affine surface area finds applications in many areas of mathematics and applied mathematics. This has led to intense research in recent years and numerous new directions have been developed. We will discuss some of them and we will show how affine surface area is related to a geometric object, that is interesting in its own right, the floating body.
MC 5501