# SCIENTIFIC PROGRAMS AND ACTIVTIES

October 24, 2014

## Geometry and Model Theory Seminar 2010-11 at the Fields Institute

Organizers: Ed Bierstone, Patrick Speissegger

### Overview

The idea of the seminar is to bring together people from the group in geometry and singularities at the University of Toronto (including Ed Bierstone, Askold Khovanskii, Grisha Mihalkin and Pierre Milman) and the model theory group at McMaster University (Bradd Hart, Deirdre Haskell, Patrick Speissegger and Matt Valeriote).

As we discovered during the programs in Algebraic Model Theory Program and the Singularity Theory and Geometry Program at the Fields Institute in 1996-97, geometers and model theorists have many common interests. The goal of this seminar is to further explore interactions between the areas. It served as the main seminar for the program on O-minimal structures and real analytic geometry, which focussed on such interactions arising around Hilbert's 16th problem.

The seminar meets once a month at the Fields Institute, Room 230, on a Thursday announced below, for one talk 2-3pm and a second talk 3:30-4:30pm. Please subscribe to the Fields mail list to be informed of upcoming seminars.

### SEMINARS 2010-11

June 30, 2011 Chris Miller
Oscillatory trajectories, Minkowski dimension and definability.

In recent joint work with A. Fornasiero and P. Hieronymi, we showed that an expansion of the real field (in the sense of model
theory) avoids defining the set of all integers if and only if every compact definable subset of the line has either nonempty interior or Minkowski dimension zero. I will give an outline of the proof, and discuss consequences for expansions of the real field by oscillatory trajectories of planar vector fields.

May 19, 2011

Patrick Speissegger, McMaster
11-12 am

Non-oscillatory vs. interlaced trajectories
A recent result of Rolin, Sanz and Schaefke describes analytic vector fields in arbitrary dimension whose trajectories are pairwise interlaced but also o-minimal. They also give an example of a vector field in dimension 5 that has a non-oscillatory trajectory that is not o-minimal. With Le Gal and Sanz, we have been investigating whether any lower dimensional examples of the latter kind might exist. I'll give an update of what we have found so far.

Gal Binyamini, Weizmann Institute
2-3 pm

Multiplicity and order of contact for regular and singular foliations
I will begin by considering the order of contact between trajectories of singular polynomial vector fields and algebraic hypersurfaces. The goal is to obtain upper bounds depending solely on the degrees of the field and the hypersurface, where possible. I will describe an algebraic approach to this problem. I will then consider a generalization of this algebraic approach to the study of contact between leafs of higher-dimensional foliations and algebraic surfaces of complementary dimension. If time permits, I will also discuss an application of this approach to the study of non-isolated intersection multiplicities.

Vincent Grandjean, Fields Institute
3:30-4:30 pm

Geodesics and Exponential map at singular points of generic 2-cuspidal surfaces
Assume a Riemannian manifold (M,g) is given. Let X be a locally closed subset of M, that is singular at some of its point.The smooth part of X comes equipped with a Riemannian metric induced from the ambient one. We would like to understand how do geodesics on the regular part of X behave in a neighbourhood of a singular point. It turns out that very little is known (or even explored) about very elementary singularities (conical, edges or corners). One of the purpose of specifying these simple singular "manifolds" was to study the propagation of singularities for the wave equation on such a singular "manifold" (Melrose, Vasy, Wunsch,...).

In this joint work with Daniel Grieser (Oldenburg, Germany), we want to address a naive and elementary question: Given an isolated surface singularity X in (M,g), can a neighbourhood of the singular point be foliated by geodesics reaching the singular point ? Then could we define an exponential-like mapping at a such point, or if you like get some polar-like coordinates where the half-lines would be replaced by a geodesic ray? And what would be the regularity of such an exponential mapping? Conical singularities of any dimension (Melrose & Wunsch) satisify this property and it can even be done analytically if the setting is real analytic (Lebeau). With D. Grieser, we have exhibited simple classes of examples of non-conical real analytic surfaces with an isolated singularity, and cuspidal-like, in the Euclidean 3-space, such that the geodesics reaching the singular point behave differently according to the considered class.

We obsviously have in mind singular real algebraic sets, or germs or subanalytic sets, as model of the singularities we are interested in dealing with. If times allow I will make some connections with Hardt conjecture about the subanaliticity (or definability in a some larger o-minimal structure) of the inner distance on semi-algebraic sets, and try to explain how the current work could allow to provide counter-examples.

### PAST SEMINARS

March 24, 2011

Dmitry Novikov (Weizmann Institute of Science)
Finiteness theorems for zeros of pseudo-abelian integrals

The limit cycles appearing in perturbations of Darboux integrable planar polynomial vector fields are closely connected to the zeros of so-called pseudo-abelian integrals, a generalization of abelian integrals. Therefore, one is interested in upper bounds for the number of zeros of such integrals. Even for an individual system this question is far less trivial than its analogue for abelian integrals. The natural conjecture, generalizing the Varchenko-Khovanskii theorem, claims existence of an upper bound depending on the degrees only. I'll describe the progress toward solving this conjecture.

Nov. 18, 2010

Gareth O. Jones (University of Manchester)
Counting rational points on certain Pfaffian surfaces

I'll briefly discuss a result, due to Pila and Wilkie, on the density of rational points on sets definable in o-minimal expansions of the real field. I'll then state Wilkie's conjecture on how this result can be improved in certain cases, and discuss progress towards this conjecture.

Janusz Adamus (University of Western Ontario)
Tameness of complex dimensions in real analytic sets

Given a real-analytic set $R$ in a complex ambient space, a natural question to ask is how much of the complex structure is inherited (locally) by $R$. Two important ways of measuring this influence at a point $p \in R$ are concerned with the "outer" and "inner" (complex) dimensions of the germ $R_p$. Namely, the minimal dimension of a complex germ containing $R_p$, and the maximal dimension of a complex germ contained in $R_p$. We will consider the problem of semianalytic filtration of $R$ by these complex dimensions.
This is a joint work with Serge Randriambololona and Rasul Shafikov.

Oct. 21, 2010

Jana Mariková (McMaster University)
Beyond T-convexity

We let R be an o-minimal field, V a convex subring, and k_ind the corresponding residue field with structure induced from R.

In 1995, T-convex subrings of o-minimal fields were identified by van den Dries and Lewenberg as a good analogue of convex subrings of real closed fields. One of the nice properties of the T-convex case is that k_ind turns out to be o-minimal. On the other hand, standard valuations don't necessarily come from T-convex subrings. We consider here the class of all (R,V) such that k_ind is o-minimal. It includes both the T-convex case, as well as standard valuations, and we show that it is in fact first order axiomatizable.

Alex J. Wilkie (University of Manchester)
Complex analytic functions in an o-minimal context.

I present an approach to Zilber's conjecture (that the complex exponential field is quasi-minimal) based on an analytic continuation conjecture for complex functions definable in o-minimal structures. As an application of the ideas involved here, I prove that the smallest ring R of analytic germs (at 0, say, in the complex plane) containing the polynomials and closed under taking logarithms and raising to real powers has the following property: every f in R has an analytic continuation along all but finitely many rays emanating from 0. (If we replace "finitely many" by "countably many" the result is obvious.) This is, of course, a purely mathematical result, but I do not know how to prove it without going via definability in certain 0-minimal structures.

Sept. 23, 2010

Vincent Grandjean (Fields Institute)
Gradient trajectories lying on isolated surface singularities do not oscillate at their limit point.
Abstract

Tobias Kaiser (Universität Regensburg)
Tame measures
Abstract