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
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
Robert Cornea, University of Waterloo
Stable Pairs on P2 via Spectral Correspondence
In this talk we will consider stable wild Vafa-Witten-Higgs bundles (or stable pairs for short) (E, ϕ) on P^2 where E is a rank two holomorphic vector bundle and ϕ : E -> E(d) is a holomorphic bundle map with d > 0. There is a way to construct stable pairs on called the spectral correspondence. This states that given a stable pair (E,ϕ) on P^2, there exists a surface Y and a 2:1 covering map pi: Y -> P^2 such that E is the push forward of a line bundle on Y and ϕ comes from the multiplication of a section on Y. So studying stable pairs (E,ϕ) on P^2 boils down to finding 2:1 covering maps Y -> P^2 and line bundles on Y. The study of constructing rank two vector bundles on P^2 via 2:1 coverings was studied by Schwarzenberger in 1960. We will demonstrate examples of stable pairs when d=1 and explain the cases briefly for d=2 and 3.
MC 5479