### 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 t-henselian 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 near-henselian fields which are those t-henselian fields which are not henselian, and Ax-Kochen/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 model-theoretic 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 infinite-dimensional vector space over an infinite field equipped with a nondegenerate bilinear form.

• ### research articles

##### thesis
1. Sylvy Anscombe. Definability in Henselian Fields. DPhil Thesis, University of Oxford, 2013. Supervised by Dr Jochen Koenigsmann.
##### published
1. 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.
2. Sylvy Anscombe and Arno Fehm. The existential theory of equicharacteristic henselian valued fields. Algebra & Number Theory, 10-3:665–683, 2016.
3. Sylvy Anscombe and Franz-Viktor Kuhlmann. Notes on extremal and tame valued fields. Journal of Symbolic Logic, 81:400–416, 2016.
4. Sylvy Anscombe and Arno Fehm. Characterizing diophantine henselian valuation rings and valuation ideals. Proceedings of the London Mathematical Society, 115:293–322, 2017.
##### submitted
1. Sylvy Anscombe and Franziska Jahnke. Henselianity in the language of rings. Manuscript, 2017.
2. Sylvy Anscombe. Existentially generated subfields of large fields. Manuscript, 2017.
3. Sylvy Anscombe and Amery Gration. The Mathematical Structure of Non-Relativistic Classical Mechanics. Manuscript, 2018. (new)
##### notes
1. Sylvy Anscombe. One-dimensional $F$-definable sets in $F((t))$. Manuscript, 2015.
2. Sylvy Anscombe. Free homogeneous structures are generalised measurable. Manuscript, 2016.
##### in preparation
1. Sylvy Anscombe, Dugald Macpherson, Charles Steinhorn, and Daniel Wolf. Multidimensional asymptotic classes and generalised measurable structures. Manuscript, 2017.
2. Sylvy Anscombe and Charlotte Kestner. The model theory of bilinear forms. Manuscript, 2016.
3. 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.
1. Questions of definability in fields of formal power series. Talk, Leeds, 2010.
2. Existential definability in t-henselian fields. Poster, Leeds, 2011.
3. Aspects of definability in henselian fields of positive characteristic. Talk, Konstanz, 2011.
4. $F$-definability in $F((t))$. Talk, Oxford, 2012.
5. Definable henselian valuations. Poster, Edinburgh, 2014.
6. Free homogeneous structures are generalised measurable. Talk, Durham, 2015.
7. Characterizing Diophantine henselian valuation rings and valuation ideals. Talk, AMS Rutgers, Nov 2015.
8. Dimension and dividing in generalised measurable structures. Abstract, Workshop on Finite and Pseudofinite Structures, Leeds, Jul 2016.
9. Generalized measurable structures with the Tree Property. Abstract, Logic Colloquium, Leeds, Aug 2016.
10. Generalized measurable structures with the Tree Property. Slides, Logic Colloquium, Leeds, Aug 2016.
11. Measure and dimension in model theory. Abstract, British Logic Colloquium, Edinburgh, Sep 2016.
12. Measure and dimension in model theory. Slides, British Logic Colloquium, Edinburgh, Sep 2016.
13. Viewing free homogeneous structures and bilinear forms as 'generalised measurable'. Abstract, Logic Seminar, Manchester, Oct 2016.
14. Research presentation. Slides, Galway, Oct 2017.
##### meetings, conferences, workshops
Here are some links to meetings that I have attended or will attend.

### online tools

• my articles on the arXiv
• my page on Google Scholar
• my page on ORCiD
• my page on Mathematics Genealogy Project

• 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