research interests
My research interests lie in
Model Theory, a part of
Mathematical Logic, including connections to algebra and number theory.
Keywords: Mathematical logic, model theory, applications of model theory to algebra and number theory, Diophantine equations, asymptotic classes, measurable structures, pseudofinite structures, valued fields, henselianity, definable henselian valuations, Hilbert's Tenth Problem, arithmetic, homogeneous structures, infinite graphs
Model theory of valued fields, particularly those of positive characteristic.
My thesis included work on definable sets in thenselian fields, especially sets defined by existential formulas.
This is related to the problem of finding languages suitable for quantifier elimination in local fields of positive characteristic.
I am also very interested in definable henselian valuations.
This interest started in joint work with
Jochen Koenigsmann
in which we give an $\exists$$\emptyset$definition of $\mathbb{F}_{q}[[t]]$ in $\mathbb{F}_{q}((t))$, for $q$ a prime power.
Recently I have been working with Arno Fehm to obtain new results about definable henselian valuations and transfer principles for existential sentences. This leads to results about Hilbert's 10th Problem (i.e. existential decidability) in equicharacteristic henselian valued fields. As an example, we show that the existential theory of $\mathbb{F}_{q}((t))$ is decidable. In forthcoming joint work with Franziska Jahnke we investigate nearhenselian fields which are those thenselian fields which are not henselian, and AxKochen/Ershov principles in the language of fields.
Model theory of asymptotic classes, in the sense of Macpherson and Steinhorn. The original concept of a $1$dimensional asymptotic class (due to Macpherson–Steinhorn) was inspired by the asymptotic results (due to Chatzidakis–van den Dries–Macintyre) about sizes of definable sets across the class of finite fields.
In joint work with Dugald Macpherson, Charles Steinhorn, and Daniel Wolf we have been creating a framework (called Multidimensional asymptotic classes) which generalises the original concept to include many more examples and more exotic modeltheoretic behaviour. For example, an ultraproduct of a $1$dimensional asymptotic class is supersimple, but an ultraproduct of a multidimensional asymptotic class may not even be simple. In a multidimensional asymptotic class there is still an asymptotic description of the sizes of definable sets, but the functions describing the sizes of definable sets may be more exotic than the simpler monomial functions occurring in the earlier framework. We are currently preparing a paper which contains our work so far. In work with Charlotte Kestner we have been investigating the model theory of bilinear forms, by studying independence relations in the theory of an infinitedimensional vector space over an infinite field equipped with a nondegenerate bilinear form.
research articles
thesis

Sylvy Anscombe.
Definability in Henselian Fields.
DPhil Thesis, University of Oxford, 2013.
Supervised by Dr Jochen Koenigsmann.
published

Sylvy Anscombe and Jochen Koenigsmann.
An existential $\emptyset$definition of $\mathbb{F}_{q}[[t]]$ in $\mathbb{F}_{q}((t))$.
Journal of Symbolic Logic, 79:1336–1343, 2014.

Sylvy Anscombe and Arno Fehm.
The existential theory of equicharacteristic henselian valued fields.
Algebra & Number Theory, 103:665–683, 2016.

Sylvy Anscombe and FranzViktor Kuhlmann.
Notes on extremal and tame valued fields.
Journal of Symbolic Logic, 81:400–416, 2016.

Sylvy Anscombe and Arno Fehm.
Characterizing diophantine henselian valuation rings and valuation ideals.
Proceedings of the London Mathematical Society, 115:293–322, 2017.
submitted

Sylvy Anscombe and Franziska Jahnke.
Henselianity in the language of rings.
Manuscript, 2017.

Sylvy Anscombe.
Existentially generated subfields of large fields.
Manuscript, 2017.
 Sylvy Anscombe and Amery Gration.
The Mathematical Structure of NonRelativistic Classical Mechanics.
Manuscript, 2018. (new)
notes

Sylvy Anscombe.
Onedimensional $F$definable sets in $F((t))$.
Manuscript, 2015.
 Sylvy Anscombe.
Free homogeneous structures are generalised measurable.
Manuscript, 2016.
in preparation
 Sylvy Anscombe, Dugald Macpherson, Charles Steinhorn, and Daniel Wolf.
Multidimensional asymptotic classes and generalised measurable structures.
Manuscript, 2017.
 Sylvy Anscombe and Charlotte Kestner.
The model theory of bilinear forms.
Manuscript, 2016.
 Sylvy Anscombe.
Relative inseparable closure in field extensions.
Manuscript, 2016.
talks and meetings
slides, notes, abstracts, and posters
Here are various slides, notes, and abstracts of talks, as well as one or two posters.

Questions of definability in fields of formal power series.
Talk, Leeds, 2010.

Existential definability in thenselian fields.
Poster, Leeds, 2011.

Aspects of definability in henselian fields of positive characteristic.
Talk, Konstanz, 2011.

$F$definability in $F((t))$.
Talk, Oxford, 2012.

Definable henselian valuations.
Poster, Edinburgh, 2014.

Free homogeneous structures are generalised measurable.
Talk, Durham, 2015.

Characterizing Diophantine henselian valuation rings and valuation ideals.
Talk, AMS Rutgers, Nov 2015.

Dimension and dividing in generalised measurable structures.
Abstract, Workshop on Finite and Pseudofinite Structures, Leeds, Jul 2016.

Generalized measurable structures with the Tree Property.
Abstract, Logic Colloquium, Leeds, Aug 2016.

Generalized measurable structures with the Tree Property.
Slides, Logic Colloquium, Leeds, Aug 2016.

Measure and dimension in model theory.
Abstract, British Logic Colloquium, Edinburgh, Sep 2016.

Measure and dimension in model theory.
Slides, British Logic Colloquium, Edinburgh, Sep 2016.

Viewing free homogeneous structures and bilinear forms as 'generalised measurable'.
Abstract, Logic Seminar, Manchester, Oct 2016.

Research presentation.
Slides, Galway, Oct 2017.
meetings, conferences, workshops
Here are some links to meetings that I have attended or will attend.
 Special Session on Advances in Valuation Theory, Rutgers, 14th  15th November 2015
 ALaNT 4, Telč, 13th  17th June 2016
 SEEMOD, Oxford, 5th  6th July 2016
 Geometry, Number Theory, and Logic, Oxford, 6th July 2016
 Workshop on finite and pseudofinite structures, Leeds, 27th  29th July 2016
 Logic Colloquium, Leeds, 31st July  6th August 2016
 British Logic Colloquium, Edinburgh, 6th  8th September 2016
 visit to the Nesin Mathematics Village, Şirince, 10th  17th September 2016
 Mathematisches Forschungsinstitut Oberwolfach, 23rd  29th October 2016
 LYMoTS in Manchester, 30th May 2017
 Model Theory in Wroclaw, Wroclaw, 30th June  2nd July 2017
 Model Theory in Bedlewo, Bedlewo, 2nd  8th July 2017
 New Directions in the Model Theory of Valued Fields, Preston, 4th August 2017
 Homogeneous structures, permutation groups, and connections to set theory, Leeds, 10th  12th September 2017
 Diophantine problems, Manchester, 11th  15th September 2017
 LYMoTS, Preston, 21st October 2017
 Model Theory, Combinatorics and Valued Fields, Institut Henri Poincaré, Paris, 18th February  7th April 2018
 From permutation groups to model theory: a workshop inspired by the interests of Dugald Macpherson, on the occasion of his 60th birthday, International Centre for Mathematical Sciences, Edinburgh, 17th  20th September 2018
online tools
my articles on the arXiv
my page on academia.edu
my page on Google Scholar
my page on ORCiD
my page on Mathematics Genealogy Project
useful links
Model Theory group, UCLan
Mathematics group, UCLan
Jeremiah Horrocks Institute, University of Central Lancashire
University of Central Lancashire
Logic@Leeds, University of Leeds
School of Mathematics, University of Leeds
Mathematical Logic research group, University of Oxford
The Mathematical Institute, University of Oxford
The London Mathematical Society
The British Logic Colloquium
The Association for Symbolic Logic
arXiv
MathSciNet
AMS Collaboration Distance
Mathematics Genealogy Project