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
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
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