Events

Patrick Speissegger, McMaster University

We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands (i) the expansion of the real field by all multisummable (in the positive real direction) power series, and (ii) the reduct of Ran∗ generated by all convergent generalized power series with natural support. In particular, its expansion by the exponential function defines both the Gamma function on (0, ∞) and the Zeta function on (1, ∞).  (Joint work with Jean-Philippe Rolin and Tamara Servi)

MC 5403

Adele Padgett, McMaster University

I will explain an open problem in the model theory of ordered structures and outline a possible strategy for its resolution. The problem is whether there are o-minimal fields that are “transexponential”, i.e., which define functions that eventually grow faster than any tower of exponentials. In recent work, I gave evidence that the real field expanded by a particular transexponential function, call it E, could be o-minimal. After some background, I will describe how a criterion of Lion which grew out of Wilkie’s proof that the real exponential field is o-minimal could be used in the transexponential case.

MC 5479

Gianluca Basso, Université Lyon 1

Duchesne, Monod and Wesolek described how to associate to each permutation group of countable degree a group acting on a certain one-dimensional tree-like continuum. This is called its kaleidoscopic group. We reframe the construction in terms of countable structures and determine which dynamical properties are preserved when passing to the kaleidoscopic group. This requires a novel structural Ramsey theorem and produces a new class of examples exhibiting a poorly understood phenomenon, namely non-metrizable universal minimal flows with a comeager orbit. 

This is joint work with Todor Tsankov.

MC 5479

Elliot Kaplan, McMaster University

Eventual polynomial growth is a common theme in combinatorics and commutative algebra. The quintessential example of this phenomenon is the Hilbert polynomial, which eventually coincides with the linear dimension of the graded pieces of a finitely generated module over a polynomial ring. A later result of Kolchin shows that the transcendence degree of certain field extensions of a differential field is eventually polynomial. More recently, Khovanskii showed that for finite subsets A and B of a commutative semigroup, the size of the sumset A+tB is eventually polynomial in t. I will present a common generalization of these three results in terms of finitary matroids (also called pregeometries). Time permitting, I’ll discuss other instances of eventual polynomial growth (like the Betti numbers of a simplicial complex) and how these polynomials can be used to bound model-theoretic ranks (like thorn-rank). This is joint work with Antongiulio Fornasiero.

MC 5479

Ronnie Nagloo, University of Illinois at Chicago

In this talk, we revisit the problem of describing the 'finer' structure of geometrically trivial strongly minimal sets in DCF0. In particular, I will explain how recent work joint with David Blázquez-Sanz, Guy Casale and James Freitag on Fuchsian groups (discrete subgroups of SL2(R)) and automorphic functions has lead to intriguing questions around the $\omega$-categoricity conjecture of Daniel Lascar. This conjecture was disproved in its full generality by James Freitag and Tom Scanlon using the modular group SL2(Z) and its automorphic uniformizer (the j-function). I will explain how their counter-example fits into the larger context of arithmetic Fuchsian groups and has allowed us to 'propose' refinements to the original conjecture.

MC 5479

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

Scott's Isomorphism Theorem states that every countable structure can be described up to isomorphism (among countable structures) by a single sentence of $L_{\omega_1 \omega}$ known as its Scott sentence. Here, $L_{\omega_1 \omega}$ is the extension of finitary first-order logic obtained by allowing countably infinite conjunctions and disjunctions. Each structure has a least complexity Scott sentence in the hierarchy of $\Pi_{\alpha}, \Sigma_{\alpha},$ and $d$-$\Sigma_\alpha$ formulas known as its Scott complexity. In this talk, we compute the Scott complexity of several kinds of linear orders and groups.

MC 5479

Amador Martin-Pizarro, University of Freiburg

In 1992 Lascar proved that the group of field automorphisms of the complex numbers which fix pointwise the algebraic closure of the rationals is simple, assuming the continuum hypothesis. His proof used strongly the topological features of the group of automorphisms of a countable structure, as a Polish group. 

In 1997 Lascar gave a different proof of the above, without assuming  the continuum hypothesis. The new proof needed just the stability of the theory of the field of complex numbers (and particularly stationarity of types as a way to merge two elementary maps) as well as the fact that the field of complex numbers is saturated in its own cardinality. In a recent preprint with T. Blossier, Z. Chatzidakis and C. Hardouin, we have adapted a proof of Lascar to show that certain groups of automorphisms of various theories of fields with operators are simple. It particularly applies to the theory of difference closed fields, which is simple and hence has possibly no  models which are saturated in their uncountable cardinality.

MC 5479

Natasha Dobrinen, University of Notre Dame

We develop infinite-dimensional Ramsey theory for Fraïssé limits of finitely constrained binary relational FAP classes.  In our spaces, Souslin-measurable colorings are Ramsey, and for spaces of so-called strong diaries we obtain analogues of the Ellentuck Theorem.  Our results are optimal and recover exact big Ramsey degrees via uniform clopen colorings.  A key step in the proof is the development of the new notion of an A.3(2)-ideal and a proof that Todorcevic's Abstract Ramsey Theorem still holds when Axiom A.3(2) is replaced by the weaker assumption of an A.3(2)-ideal.  This is joint work with Andy Zucker.

MC 5479

Alexi Block Gorman, McMaster University

Buchi automata are the natural extension of finite automata to a model of computation that accepts infinite-length inputs. We say a subset X of the reals is k-regular if there is a Buchi automaton that accepts (one of) the base-k representations of every element of X, and rejects the base-k representations of each element in its complement. Such sets include the classical two-thirds Cantor set. In general, call a subset of the reals a Cantor set if it is nonempty, compact, has no isolated points, and no interior.  Let V_k be a ternary predicate on Euclidean 3-space such that V_k(x,u,d) holds if and only if u is an integer power of k, and d is the coefficient of the term u in some base-k expansion of x. For a fixed k, all of the k-regular subsets of Euclidean space are definable in the expansion of the reals as an ordered additive group by the predicate V_k. In this talk, we will discuss current and ongoing progress toward a characterization of when we can define the V_k relation, and hence all k-regular subsets of Euclidean space, from an arbitrary k-regular Cantor set.

MC 5479

David Meretzky, University of Notre Dame

A field is bounded if it has finitely many extensions of each finite degree. A theorem of Serre says that if a field is bounded then the Galois cohomology of finite linear algebraic groups over that field is finite and can be understood in terms of finite group cohomology. In this talk, we will introduce a boundedness condition for differential fields. This condition will yield an analogous theorem to Serre's in the differential setting, giving some control over Kolchin's differential Galois cohomology for certain classes of differential algebraic groups related to the Picard-Vessiot theory. We will discuss the relationship between Kolchin's cohomology theory and the Picard-Vessiot theory. This is joint work in progress with Anand Pillay.

MC 5479