4,320 Works

Lorenzo Fantini
I will introduce a non-archimedean version of the link of a singularity. This object will be a space of valuations, a close relative of non-archimedean analytic spaces (in the sense of Berkovich) over trivially valued fields. After describing the structure of these links, I will deduce information about the resolutions of surface singularities. If times allows, I will then characterize those normal surface singularities whose link satisfies a self-similarity property. The last part is a...

Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. First we define a real motivic integral which admits a change of variable formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for generically arc-analytic maps. Then we characterize in terms of the motivic measure, germs of arc-analytic homeomorphisms between real algebraic varieties which...

Relations between polynomial solutions, extensions, radical ideals and Lipschitz normal embeddings.

Maria Michalska
Take polynomials $f,g\in k[X]$, where $k$ is the field of complex or real numbers. Under certain assumptions we show equivalence of the following conditions: (i) $(f,g)$ is radical (ii) for every polynomial $h$ if there exists a pointwise solution of $$A\cdot f + B\cdot g =h$$ then there exists its polynomial solution (iii) every continuous function $$F=\left\{\begin{array}{ll} \alpha & on\ \{f=0\}\\ \beta & on\ \{g=0\} \end{array}\right.$$ with $\alpha,\beta\in{k}[X]$, is a restriction...

Stephen Tredwell

Topological invariants of Modular Tensor Categories

Peter Schauenburg

Classification Theory and the Construction of PAC Fields

Nick Ramsey
A field K is called pseudo-algebraically closed (PAC) if every absolutely irreducible variety defined over K has a K-rational point. These fields were introduced by Ax in his characterization of pseudo-finite fields and have since become an important object of model-theoretic study. A remarkable theorem of Chatzidakis proves that, in a precise sense, independent amalgamation in a PAC field is controlled by independent amalgamation in the absolute Galois group. We will describe how this theorem...

Artem Chernikov

Strongly minimal groups interpretable in o-minimal expansions of fields.

Assaf Hasson
We prove that if D=(G,+,\dots) is a strongly minimal non-locally modular group interpretable in an o-minimal expansion of a field and dim(G)=2 then D interprets an algebraically closed field K and D (as a structure) an algebraic group over K with all the induced K-structure. I will discuss some key aspects of the proof that may be of interest on their own right. Joint work with Y. Peterzile and P. Eleftheriou.

Anand Pillay

An answer to Kac's question on Coxeter exponents

Zhengwei Liu
In the ADE quiver theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix, captured by the Coxeter exponents. We formalize related notations and prove such a correspondence for a more general case: this includes the quiver of any module of any semisimple Lie algebra g at any level l. We answer a question posed by Victor Kac in 1994 and a recent comment by Terry...

Groupoids and Relative Internality

Leo Jimenez
We prove that in a stable theory, some 2-analysable types give rise to type definable groupoids, with some simplicial data attached to them, extending a well-know result linking groups to internal types. We then investigate how properties of these groupoids relate to properties of types. In particular, we expose some internality criteria.

Transfer of the Ramsey property by semi-retractions

Lynn Scow
In this talk we introduce a weaker form of bi-interpretability and see how it can be used to transfer the Ramsey property across classes in different first-order languages. This is a special case of a more general theorem about what we will call color-homogenizing embeddings.

On strongly minimal Steiner systems

John Baldwin
With Gianluca Paolini (in preparation), we constructed families of strongly minimal Steiner $( systems for every$k 3$. A quasigroup is a structure with a binary operation such that for each equation$xy=z$the values of two of the variables determines a unique value for the third. Here we show that the$2^{ Steiner $(2,3)$-systems are definably coordinatized by strongly minimal Steiner quasigroups and the Steiner $(2,4)$-systems are definably coordinatized by strongly minimal $SQS$-Skeins. Further...

Pseudofinite groups, arithmetic regularity, and additive combinatorics

Gabriel Conant
I will report on joint work with Pillay and Terry on arithmetic regularity (a group theoretic analogue of Szemeredi regularity for graphs) for sets of bounded VC-dimension in finite groups, which is proved using a local version generic compact domination for NIP formulas in pseudofinite groups. I will then present more recent work on nonabelian versions of certain "inverse theorems" from additive combinatorics, which are proved using pseudofinite model theory, and can be used to...

Retro-stability: The fine structure of classifiable theories

We give (equivalent) friendlier definitions of classifiable theories strengthen known results about how an independent triple of models can be completed to a model. As well, we characterize when the isomorphism type of a weight one extension $N/M$ is uniquely determined by the non-orthogonality class of the relevant regular type and discuss when $N$ is prime over $Ma$ for some finite $a\in N$. This is part of an ongoing project with Elisabeth Bouscaren, Bradd Hart,...

NTP_2 groups with f-generics and PRC fields

Samaria Montenegro
This is a joint work with Alf Onshuus and Pierre Simon. In this talk we focus on groups with f-generic types definable in NTP2 theories. In particular we study the case of bounded PRC fields. PRC fields were introduced by Prestel and Basarav as a generalization of real closed fields and pseudo algebraically closed fields, where we admit having several orders. We know that the complete theory of a bounded PRC field is NTP2 and...

An Â« Ahlbrandt-Ziegler Reconstruction Â» for theories which are not necessarily countably categorical

Itaï Ben Yaacov
It is by now almost folklore that if T is a countably categorical theory, and M its unique countable model, then the topological group G(T) = Aut(M) is a complete invariant for the bi-interpretability class of T . This gained renewed interest recently, given the correspondences between dynamical properties of G(T) and classification-theoretic properties of T . From a model-theoretic point of view, the obvious drawback is the restriction to countably categorical theories. As a...

Equivalence query learning and the negation of the finite cover property

Hunter Chase
There are multiple connections between model-theoretic notions of complexity and machine learning. NIP formulas correspond to PAC-learning by way of VC-dimension, and stable formulas correspond to online learning by way of Littlestone dimension, also known as Shelah's 2-rank. We explore a similar connection between formulas without the finite cover property and equivalence query learning. In equivalence query learning, a learner attempts to identify a certain set from a set system by making hypotheses and receiving...

A gentle invitation to the Landau-Ginzburg/conformal field theory correspondence

Ana Ros Camacho
The Landau-Ginzburg/conformal field theory correspondence is a physics result dating from the late 80s-early 90s which in particular predicts some relation between categories of representations of vertex operator algebras and categories of matrix factorizations. Up to date, we have several examples available yet it lacks a precise mathematical statement. In this talk, we will review the actual state-of-art in this topic and discuss future directions of research.

Localized Lascar group

Jan Dobrowolski
The notion of the localized Lascar-Galois group $Gal_L(p)$ of a type $p$ appeared recently in the context of model-theoretic homology groups, and was also used by Krupinski, Newelski, and Simon in the context of topological dynamics. After a brief introduction of the context, we will discuss some basic properties of localized Lascar-Galois groups. Then, we will focus on the question about how far $Gal_L(tp(acl(a)))$ can be from $Gal_L(tp(a))$. This is a joint work with B....

Amenability and definability

Krzysztof Krupinski
I will discuss some aspects of my recent paper (still in preparation) with Udi Hrushovski and Anand Pillay. Most of the main results are of the form "a version of amenability implies a version of G-compactness". Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some extension results that we obtain for measure-like functions (which we call...

Interpolative Fusions

Alex Kruckman
Fix languages L_1 and L_2 with intersection L_\cap and union L_\cup. An L_\cup structure M is interpolative if whenever X_1 is an L_1-definable set and X_2 is an L_2-definable set, X_1 and X_2 intersect in M unless they are separated by L_\cap-definable sets. When T_1 is an L_1 theory and T_2 is an L_2 theory, we say that a theory T_\cup^* is the interpolative fusion of T_1 and T_2 if it axiomatizes the class of...

NSOP_1 theories

Byunghan Kim
Let $T$ be an NSOP$_1$ theory. Recently I. Kaplan and N. Ramsey proved that in $T$, the so-called Kim-independence ($\phi(x,a_0)$ Kim-divides over $A$ if there is a Morley sequence $a_i$ such that $\{\phi(x,a_i)\}_i$ is inconsistent) satisfies nice properties over models such as extension, symmetry, and type-amalgamation. In a joint work with J. Dobrowolski and N. Ramey we continue to show that in $T$ with nonforking existence, Kim-independence also satisfies the properties over any sets, in...

Applications of Gauging and Anyon Condensation

Eric Rowell
I will discuss some specific applications of gauging/anyon condensation. This will include some approaches to classification of various types of braided categories (metaplectic, super-modular) and rank-finiteness for braided fusion categories. I will also present some speculations on other possible applications, for example to verify the property F conjecture, and to find a sensible structure theorem for braided fusion categories. This will be based on several joint projects (some completed, some on-going).

o-minimal flows on nilmanifolds

Kobi Peterzil
Let G be a real algebraic unipotent group and let Lambda be a lattice in G, with p:G->G/Lambda the quotient map. Given a definable subset X of G, in some o-minimal expansion of the reals, we describe the closure of p(X) in G/Lambda in terms definable families of cosets of real algebraic subgroups of G of positive dimension. The family is extracted from X independently of Lambda.

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