Dook'o'oosliid



Erik Walsberg Erik Walsberg
Visiting Assistant Professor
Department of Mathematics
University of California, Irvine

Email: ewalsber at uci dot edu
Office: 410 Rowland



Large fields and the Etale-open topology:

Galois groups of large fields with simple theory, with Anand Pillay

Etale-open topology and the stable field conjecture, with Will Johnson, Minh Chieu Tran, and Vincent Ye

Here's a talk that I gave in the Notre Dame logic seminar on this stuff, and here's a talk I gave in the online Chulalongkorn University math seminar. The ND talk was written for model theorists and the CU talk was written for a general audience.

The slides from my second talk in the Notre Dame seminar.

Topological approach to the theory of large fields? Some notes on large fields. Very much a work in progress.

The Interpolative Fusion Program:

This is a way of decomposing theories into generically interacting simpler theories. It turns out that many interesting theories decompose in this way, for example the theory of random graphs decomposes as a fusion of two theories each interpretable in the theory of equality and the theory of differentially closed fields of characteristic zero decomposes as a fusion of two theories each interpretable in the theory of algebraically closed field of characteristic zero. Theorems about specific theories often turn out to be special cases of general results on the fusion.

Interpolative Fusions, with Alex Kruckman and Minh Chieu Tran. Journal of Mathematical Logic, accepted.
(There is a much longer paper on arxiv, this is about half of the arxiv paper, the rest of the arxiv paper will be submitted with the NSOP1 results at some point.)

Model theory over the reals:
There is a rich collection of logically tame first order structures over the real line. Definable sets in these structures are geometrically tame objects, this has in particular lead to the development of o-minimal geometry. The goal of this project is to show that logical tameness implies geometric tameness, to show that definable sets in first order structures that satisfy minimal tameness properties are geometrically tame objects. We have found a suprising connection with the monadic second order theory of one sucessor (S1S). For example an expansion of the ordered real additive group that defines a fractal set interprets S1S. This is sharp as S1S interprets expansions that define fractal sets.


Interpreting the Monadic Second Order Theory of One Successor in Expansions of the Real Line
,
with Philipp Hieronymi, Israel Journal of Mathematics, accepted.

How to Avoid a Compact Set, with Antongiulio Fornasiero and Philipp Hieronymi, Advances in Mathematics, Volume 317 / Sept. 2017 / pp. 758-785.

Wild Theories with O-minimal Open Core, with Philipp Hieronymi and Travis Nell, Annals of Pure and Applied Logic, Volume 169 / Issue 2 / Feb. 2018 / pp. 146-163.

On Continuous Functions Definable in Expansions of the Ordered Real Additive Group, with Philipp Hieronymi

Expansions of the Real Field by discrete subgroups of Gl_n(C), with Philipp Hieronymi and Samantha Xu, accepted by Proceedings of the AMS

Fractals and the Monadic Second Order Theory of One Successor, with Philipp Hieronymi

Continuous Regular Functions, with Alexi Block Gorman, Philipp Hieronymi, Elliot Kaplan, Ruoyu Meng, Zihi Wang, Ziqin Xiong, and Hongru Yang. Logical Methods in Computer Science, Volume 16, Issue 1
(This work was done in the research project Automata and Differentiable Functions at the Illinois Geometry Lab in Spring 2018).

Coarse Dimension and Definable Sets in Expansions of the Ordered Real Vector Space. Accepted by Illinois Journal of Mathematics.

NIP:

Dp-minimal Valued Fields, with Franziska Jahnke and Pierre Simon,Journal of Symbolic Logic / Volume 82 / Issue 01 / March 2017 / pp. 151-165.

Tame Topology over Dp-minimal Structures, with Pierre Simon, Notre Dame Journal of Formal Logic, accepted.

A Family of Dp-minimal Expansions of the Additive Group of Integers, with Minh Chieu Tran

Externally definable quotients and NIP expansions of the real ordered additive group

An NIP structure which does not interpret an infinite group but whose Shelah completion interprets an infinite field
(I say that a NIP structure is "Shelah complete" if every externally definable set is definable and call the Shelah expansion of a NIP structure the "Shelah completion".
The Shelah completion is Shelah complete by a theorem of Shelah.)

Dp and other minimalities with Pierre Simon

Dp-minimal expansions of (Z,+) via dense pairs via Mordell-Lang

Nippy proofs of p-adic results of Delon and Yao

A P-adic structure which does not interpret an infinite field but whose Shelah completion does


O-minimal Structures:

The Marker-Steinhorn Theorem via definable linear orders, Notre Dame Journal of Formal Logic, accepted.

Hausdorff dimension of definable metric spaces in o-minimal expansions of the real field., with Jana Maříková, Fundamenta Mathematicae, accepted.

Directed sets and topological spaces definable in o-minimal structures. with Pablo Andujar Guerrero and Margaret Thomas. Accepted (mod a few revisions) by Journal of the London Mathematical Society.

My thesis (I removed some typos and corrected some small mistakes, so this is not the exact version of my thesis that was submitted to UCLA)


Other:

Isometric Embeddings of Snowflakes into Finite-Dimensional Banach Spaces, with Enrico Le Donne and Tapio Rajala, Proceedings of the AMS, Volume 16 / Number 2 / 2018 / pp. 685-693.