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Carlo Pagano, Concordia University

Hilbert 10 via additive combinatorics

In 1900 Hilbert proposed a list of problems that have been very influential throughout the last century. In 1970 Matiyasevich, building on earlier work of Davis—Putnam—Robinson, proved that Hilbert's 10th problem is undecidable for Z. The problem of extending this result to any ring that is finitely generated over Z (eg ring of integers in number fields) has attracted significant attention since 1970 and, thanks to the efforts of many mathematicians, the task has been reduced to an arithmetic problem about elliptic curves. This problem so far had been solved only conditional on the BSD conjecture (one of the Millenium problems) by Mazur—Rubin.

In joint work with Peter Koymans we have combined additive combinatorics (Green—Tao’s celebrated theorem) with 2-descent (an old technique dating back to Fermat) to solve this problem about elliptic curves unconditionally. This shows that Hilbert 10 is undecidable over any finitely generated infinite commutative ring.

In this colloquium I will provide a gentle introduction to this undecidability result, giving a glimpse of how mathematical logic, number theory and additive combinatorics meet into one story.

MC 5501

Gian Cordana Sanjaya, University of Waterloo

Squarefree discriminant of polynomials with prime coefficients

In 1991, Yamamura computed the density of monic polynomials of degree n which has discriminant not divisible by p^2 for any prime number p and positive integer n > 1. It is natural to conjecture that the density of monic polynomials of degree n with squarefree discriminant is the product of these local densities. This conjecture has been proved in 2022 by Bhargava, Shankar, and Wang in their paper, "Squarefree values of polynomial discriminants I".

In this talk, we consider a variant where the monic polynomials have prime coefficients. We compute the density of polynomials of degree n > 1 in this class which has squarefree discriminant, as an asymptotic density plus an explicit big-O error term. This is a joint work with Valentio Iverson and Xiaoheng Wang.

MC 5403

Christine Eagles, University of Waterloo

Quantifier free internality and binding groups in ACFA

In ACFA, the definable closure of a set is not well understood. This presents an obstacle to understanding internality to the fixed field. Instead, we look at quantifier-free internality. In this talk we will follow Kamensky and Moosa (2024) by presenting quantifier-free internality and then stating a binding group theorem for rational types which are quantifier-free internal to the fixed field.

MC 5479

Akash Singha Roy, University of Georgia

Residue-class distribution and mean values of multiplicative functions

The distribution of values of arithmetic functions in residue classes has been a problem of great interest in elementary and analytic number theory. The analogous question commonly studied for multiplicative functions is the distribution of their values in coprime residue classes. In work studying this problem for large classes of multiplicative functions, Narkiewicz obtained criteria deciding when a family of such functions is jointly uniformly distributed among the coprime residue classes to a fixed modulus. In the first part of this talk, we shall extend Narkiewicz's criteria to moduli that are allowed to vary in a wide range. Our results are essentially the best possible analogues of the Siegel-Walfisz theorem in this setting. One of the primary themes behind our arguments is the quantitative detection of a certain "mixing" (or ergodicity) phenomenon in multiplicative groups via methods belonging to the "anatomy of integers", but we also rely heavily on more classical analytic arguments, tools from arithmetic and algebraic geometry, and from linear algebra over rings.

In the second part of this talk, we shall gain a finer understanding of these distributions, such as the second-order behavior. This shall rely on extending some of the most powerful known estimates on mean values of multiplicative functions (precisely, the Landau-Selberg-Delange method) to a result that is much more uniform in certain important parameters. We will see several applications of this extended result in other interesting settings as well.

This talk is partially based on joint work with Prof. Paul Pollack.

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Kaleb Ruscitti, University of WaterlooA category theory joke

A category theory joke

In this talk I will tell one joke. To ensure that all participants find the joke funny, I will spend the first 50 minutes explaining the background material (applied category theory) required for the joke.

MC 5501

Rahim Moosa, University of Waterloo

Curve excluding fields III

We continue to read the paper by Johnson and Ye.

MC 5403

Jesse Huang, University of Waterloo

Mirror symmetry for complex tori

We discuss various forms of mirror symmetry using the example of a complex torus and its compactifications.

MC 5479

Jesse Huang, University of Waterloo

Organizational Meeting

This is an organizational meeting for the mirror symmetry learning seminar. We will skim through the reading list and topics to cover and assign talks. All are welcome!

MC 2017

Amanda Maria Petcu, University of Waterloo

Cohomogeneity one solitons of the hypersymplectic flow

Given a manifold X^4 x T^3 where X^4 is hypersymplectic, one can give a flow of hypersymplectic structures that evolve according to the equation dt(w) = d(Q d^*(Q^{-1} w)), where w is the triple that gives the hypersymplectic structure and Q is a 3x3 symmetric matrix that relates the symplectic forms w_i to one another. We will let X^4 be R^4 with a cohomogeneity one action and explain what it means to be a soliton for the hypersymplectic flow and examine a (potentially hyperkahler) metric that comes from this set-up.

MC 5479

Owen Sharpe, University of Waterloo

The Selberg-Delange Method

For complex w and z, the expression w^z is ambiguous, requiring a choice of branch of log(w). In particular, there is no way to make w^z an entire function of w; a branch cut will always be present. In turn, this makes it difficult to perform contour integration and calculate residues with functions of the form f(w)^z, which are fundamental operations in number theory. We describe Selberg's method for performing such computations and some of its applications, such as those by Selberg and Delange. Incidentally, we will also discuss Hankel's formula for the Gamma function and Perron's formula for partial sums of Dirichlet series.

MC 5403