### The Model theory of groups

Not stable since xRy has the order property. A field F is pseudo-finite if every sentence in the field language true of F holds in some finite field. Axiomatized in the graph language by sentences stating that if A and B are finite disjoint sets of vertices, with A K n-1 -free, then there is a vertex connected to everything in A and nothing in B. This theory can be interpreted in a real closed field, and thus is NIP. Shown to be o-minimal by Wilkie Totally categorical of Morley rank 2.

## Activities

Complete theory of the universal and existentially closed countable graph omitting a 'bowtie' sum of two triangles sharing a single vertex. Complete theory of the ring of integers a completion of Peano Arithmetic. Has dp-rank n. The unique universal and ultrahomogeneous separable metric space with distances bounded by 1. There are few options for what language to use for this structure. Alternatively, one consider the Urysohn sphere as a metric structure in continuous logic. Hrushovski constructed an example of a strongly minimal theory, which is not locally modular and does not interpret an infinite group.

This disproved Zilber's conjecture that a strongly minimal theory must either be locally modular or interpret an infinite field. Given integers m and n , let K m n be the class of graphs obtained from the complete graph on n vertices by replacing each edge with a path containing at most m new vertices.

A graph G is superflat if for all m there is some n such that G omits K m n.

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G is ultraflat if there is some n such that for all m , G omits K m n. Define the following strictly stable superflat graph. Any superflat graph is stable; any ultraflat graph and so any planar graph is superstable.

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The graph above interprets infinitely refining equivalence relations , and so is strictly stable. A variety defined over a field K is absolutely irreducible if it is not the union of two algebraic sets defined over some algebraically closed extension of K A field K is pseudo-algebraically closed PAC if every absolutely irreducible variety defined over K has a K -rational point.

The theory of a perfect bounded PAC field is supersimple.

Shown to be NSOP 1 in this was first claimed in , but errors were found in the proof. See also , ,. A variety defined over a field K is absolutely irreducible if it is not the union of two algebraic sets defined over some algebraically closed extension of K. A field K is pseudo-algebraically closed PAC if every absolutely irreducible variety defined over K has a K -rational point. Shown to be NSOP 1 in. Defined in the 2-sorted language of vector spaces over an algebraically closed field. See also ,.

A field K , of characteristic 0, is bounded if, for every n , K has only finitely many extensions of degree n. A field K is pseudo real closed if, for every absolutely irreducible variety V defined over K , if V has a K r -rational point for every real closure K r of K , then V has a K -rational point. We assume here that the pseudo real closed field K is not real closed or algebraically closed, which ensures the independence property, and also that K has at least one order, which ensures the strict order property a pseudo real closed field with no orders is pseudo algebraically closed.

Therefore the theory is strictly stable. Geometry and Topology.

Finite and Residually Finite Groups. The conference programme will include invited lectures, a limited number of contributed talks and a poster session. Artin groups, CAT 0 geometry and related topics. Event listing ID:. AIM Workshop: Random walks beyond hyperbolic groups.

## Group theory

This workshop, sponsored by AIM and the NSF, will be devoted to extending results on random walks known for Lie groups or hyperbolic groups, to the more general class of groups which have actions on non-proper Gromov hyperbolic spaces. Workshop — Geometric Structures in Group Theory.

It belongs to the G2-series that are about strong and beautiful mathematics, especially those involving group actions on combinatorial objects. The main goal of G2G2-Summer School is to bring together experts and students to exchange ideas and to enrich their mathematical horizon. We organize two short courses and four colloquium talks to let participants see order and simplicity from possibly new perspectives and share insights with experts. Arithmetic Aspects of Algebraic Groups. The investigation of arithmetic groups has been an active and important area of mathematical research ever since it arose in the work of Gauss, Klein, Poincare, and other famous mathematicians of the 18th and 19th centuries.

## Group theory - Wikipedia

New points of view have recently led to progress on classical problems, opened new directions of inquiry, and revealed unexpected connections with other areas of mathematics. The workshop will bring together experts in the area, researchers in related fields, and young mathematicians who wish to learn about the most recent advances and the most promising directions for the future of the field.

Number Theory, Arithmetic. Introductory preschool of the trimester.

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Model theory and combinatorics , 29 January - 2 February Model theory of valued fields , 5 - 9 March Model theory and applications , 26 - 30 March This document was last modified on. Presentation Main conferences Preschool Model theory, combinatorics and valued fields Workshop Model theory and combinatorics Workshop Model theory of valued fields Conference Model theory and applications.

Download the poster: pdf or jpg. Aims of the programme Model theory is a branch of mathematical logic which deals with the relationship between formal logical languages e.