Spring Lecture Series

UNDECIDABILITY

Principal Speaker - Bjorn Poonen, Massachusetts Institute of Technology - About the Speaker

Hilbert in 1900 asked for a method to decide, given a multivariable polynomial equation with integer coefficients, whether it has a solution in integers; this was the tenth in a list of 23 problems for the mathematicians of the coming century to resolve. It has been argued, based on Hilbert’s wording, that Hilbert believed that such a method existed and might someday be found. At that time, there was no precise notion of algorithm, but in the 1930s the work of Church and Turing provided rigorous models of computation. Moreover, Turing, inspired by work of Gödel, proved that certain problems, including the halting problem, were undecidable — no algorithm could give the correct answer on all instance of the problem. A few decades later, in 1970, Matiyasevich, building on work of Davis, Putnam, and Robinson, proved that an algorithm for solving all instances of Hilbert’s tenth problem did not exist. Both before and after that, other problems in mathematics were shown to be undecidable, via a growing web of clever reductions ultimately leading back to Turing’s halting problem.

The lecture series will survey problems from many branches of mathematics that are now known to be undecidable, as well as problems whose decidability status is still unknown.

Lecture 1: Undecidability in number theory.

We sketch key ideas leading to the negative answer to Hilbert’s tenth problem, including the notions of diophantine, listable, and computable sets of integers. We also survey progress towards many extensions of Hilbert’s tenth problem that were later considered.

Lecture 2: Undecidability in group theory, analysis, and topology.

Dehn asked in 1911 whether certain basic questions for finitely presented groups could be answered. For instance, he asked whether one could decide, given a word in the generators, whether it represents the identity element of the group. Novikov and Boone proved that this problem is undecidable. Later many general problems in analysis, such as deciding whether inequalities involving trigonometric functions hold for all real values of the inputs, were also proved undecidable. Likewise, basic questions in topology, such as whether two given manifolds are homeomorphic, were proved undecidable.

Lecture 3: Undecidability throughout mathematics.

We continue the theme by exhibiting examples of undecidable problems other fields of mathematics, including combinatorics, probability, dynamical systems, algebraic geometry, commutative algebra, and game theory. For some problems, such as deciding whether two finitely generated rings are isomorphic, it is still not known whether they are undecidable.

Lectures 4 and 5: Using elliptic curves towards undecidability.

The last two lectures will focus on the use of elliptic curves and arithmetic geometry towards proving the undecidability of Hilbert’s tenth problem over various rings, a line of work started by Denef. In particular, I will discuss recent breakthroughs in the field: Koymans and Pagano, building on these ideas and using also certain theorems in arithmetic combinatorics, proved that Hilbert’s tenth problem over the ring of integers of any number field is undecidable; a few months later, Alpöge, Bhargava, Ho, and Shnidman gave a second proof using a related but different method. The decidability status of the most famous extension of Hilbert’s tenth problem, for rational numbers, remains unknown.

Wednesday's Schedule of Talks

Ari Shnidman, Temple University - About the Speaker

Title: Solutions to diophantine equations and twist families

Abstract: There is an interesting tension in arithmetic geometry between the "purely geometric" and the "arithmetic". Twist families lie along the fault lines of this tension, as they are families of algebraic varieties that are geometrically isomorphic but arithmetically different. I'll discuss two recent results that exploit such families to say something about solubility of polynomial equations. The first is the recent resolution of Hilbert's 10th problem over finitely generated rings, and the second is related to Mazur's Program B. Building on recent work of Zywina, we show that all known rational points on all modular curves are explained by geometry in a precise sense.

Juan Pablo De Rasis, Ohio State University - About the Speaker

Title: First-order defining Campana and Darmon points in algebraic function fields in one variable over number fields

Abstract: Since Koenigsmann's famous development of a method to universally define integers in the rationals, several generalizations of his method have been attained to obtain universal definitions of rings of integers in number fields (Park), S-integers in global fields (Eisenträger and Morrison), and many other applications. A very recent generalization of Koenigsmann's methods by Becher, Daans, and Dittmann allowed to extend these results to the case of algebraic function fields in one variable by exploiting the use of the reciprocity theorem for quadratic Pfister forms. In this talk we adopt these generalized methods to produce first-order definitions of Campana points and Darmon points in the context of algebraic function fields in one variable over number fields.

Kirsten Eisenträger, Pennsylvania State University - About the Speaker

Title: Hilbert's Tenth Problem for finitely generated ℤ-algebras

Abstract: In this talk we show that Hilbert's Tenth Problem is undecidable for finitely generated ℤ-algebras
with infinitely many elements.

Friday's Schedule of Talks

Carlo Pagano, Concordia University - About the Speaker

Title: Additive combinatorics and descent

Abstract: In this talk I will overview a method introduced in joint work with Peter Koymans in 2024 that
enabled us to construct elliptic curves with positive but prescribed rank. This method enabled us to settle Hilbert 10th problem for all finitely generated rings and later in 2025 to show that every number field has an elliptic curve of rank 1. I will present this and other results obtained with this method, and, if time permits, ongoing work in progress with Zev Klagsbrun, addressing a question of Mazur-Rubin on reconstructing curves from their set of points as Galois set, in the case of a generic pair of elliptic curves with full 2-torsion.

Thomas Scanlon, University of California at Berkeley - About the Speaker

Title: The complexity of structures interpreted in ℂ(t)

Abstract: It has been proposed that we could show that the theory of field ℂ(t) of rational functions over
the complex numbers is undecidable by using the observation that the set CM of j-invariants of elliptic curves
with complex multiplication is definable in ℂ(t). However, it follows from an effective form of the André-Oort
conjecture due to Binyamini that the expansion of the field of complex numbers with a predicate for CM, in fact, decidable. The methods used to define CM may be used to produce other arithmetically complicated sets. We will discuss the logical complexity of these other expansions of the complex field and the relationship to conjectures in Diophantine geometry.