Sep 8 - Sep 11, 2025 at University of Warwick, UK
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The space of real numbers, although arguably a familiar space for people thinking mathematically, plays host to scary and intuition-defying objects such as the Cantor set and the Weierstrass function. O-minimal geometry provides a framework for studying sets and functions with tame topological and geometric properties, generalizing semi-algebraic and subanalytic geometry, while retaining key finiteness conditions. Over the years, o-minimality has had interactions with a wide range of areas such as diophantine geometry, Hodge theory, theoretical computer science, combinatorics, dynamical systems, physics, and machine learning, where it has been essential to go beyond just algebraic sets while at the same time having control over complexity and structure.
This conference will explore recent advances in o-minimal geometry and its interactions with other areas. Talks will cover both foundational aspects and emerging directions, highlighting new perspectives and open problems. Please register below and join us!
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Alignments of definable groups and explicit bounds in general Elekes-Szabó
An influential theorem of Elekes and Szabó indicates that the intersections of a given algebraic variety with large finite grids of points may have maximal size only for varieties that are closely connected to algebraic groups. Techniques from model theory --- variants of Hrushovski's group configuration and of Zilber's trichotomy principle --- are very useful in recognizing these groups, and led to far reaching generalizations of Elekes-Szabó in the last decade. In this talk, focusing on the o-minimal case, we provide a generalization of the earlier result from Chernikov-Peterzil-Starchenko to arbitrary co-dimension, in particular obtaining explicit bounds in a theorem of Bays-Breuillard over the complex numbers.
A non-model-complete pfaffian chain
I will discuss some recent work with van Hille, Kirby and Speissegger in which give an example of a pfaffian chain such that the theory of the corresponding expansion of the real ordered field is not model complete.
Classification of types in o-minimal expansions of ordered abelian groups and real closed fields
We give a classification of 1-variable types in extensions of o-minimal expansions of ordered abelian groups and real closed fields. This is achieved by a valuation theoretic analysis of types, leading to the trichotomy: (i) immediate transcendental (ii) value transcendental (iii) residue transcendental. As application, we give necessary and sufficient conditions for a power bounded o-minimal expansion of a real closed field (in a language of arbitrary cardinality) to be $\kappa$-saturated. The conditions are in terms of the value group, residue field, and $\kappa$- bounded pseudo-Cauchy sequences of the natural valuation on the real closed field. A further application is a characterization of recursively saturated models. This provides a construction method for saturated and recursively saturated models, using fields of generalized power series. This is based on joint work with P. D'Aquino and K. Lange.
Real and p-adic Nash groups
A real Nash function is a real-valued analytic function on an open semialgebraic subset of $\mathbb{R}^n$ whose graph is semialgebraic. The category of Nash manifolds has been widely studied. A Nash group is a Nash manifold with Nash group structure. Any group definable in the real field can be definably equipped with the structure of a Nash group. The category of Nash groups is (strictly) in between that of real algebraic groups and that of real Lie groups. The question is how to describe Nash groups in terms of real algebraic groups. Same definitions and questions with the p-adic field in place of the reals. I discuss conjectures and old and recent work on them.
The homeomorphism type of semi-algebraic sets is elementary
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Beyond o-minimality: neostability and tame topology
The NIP structures ($\mathbb{R}$, $+$, $\cdot$, $<$, $2^\mathbb{Z}$) and ($\mathbb{R}$, $+$, $<$, $\mathbb{Q}$) are not o-minimal, yet they have motivated generalizations of o-minimality such as d-minimality and having o-minimal open core. In this talk, we explore the interplay between neostability notions --- particularly NTP$_2$ and NIP --- and certain tame topological properties. Our main result is that NTP$_2$ expansions of ($\mathbb{R}$, $+$, $<$) and of $(\mathbb{Q}_p, +, \cdot)$ have constructible open core.
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Bézout's bounds for rational and lacunary complex algebraic plane curves
I will explain which Bézout's bounds one can obtain in the complex case for rational plane curves and lacunary algebraic curves. More precisely, I will give lower and upper fewnomial bounds on the number of intersection points in a ball of the complex plane, between a rational curve $P$ and a lacunary algebraic curve $Q=0$. These bounds depend only on the initial terms of $P$ and on the support of $Q$. This question is related to deep questions in algebraic complexity, such as the Valiant version of P vs NP. This is a joint work with Sébatien Tavenas.
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Vandermonde Varieties and the Topology of Symmetric Semialgebraic Sets
This talk concerns the role of Vandermonde varieties in the study of the topology of symmetric semialgebraic subsets of $R^n$ (where $R$ is some real closed field). In type A, Basu and Riener have leveraged symmetry relative to the action of the symmetric group $S_n$ on $R^n$ in the study of the cohomology of semialgebraic sets, proving length restrictions on which partitions appear in the isotypic decompositions of the cohomology spaces. We seek to extend these results to the remaining classical reflection groups (types B=C and D). Vandermonde varieties, which in type A are defined by the first several generators of the ring of $S_n$-invariant polynomials, play a key role in these arguments. We discuss analogous results in the remaining types. We also observe how equivariance in a construction for compact replacement (originally due to Gabrielov and Vorobjov) strengthens the restriction theorem and consequent algorithmic results. This is joint work with Dr. Saugata Basu (Purdue University).
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Fields with(out) Generic Derivations
We investigate the existence of generic derivations in expansions of fields. Specifically, we provide examples of field expansions that admit a generic derivation and study their model-theoretic properties. Furthermore, we show that exponential fields, in the absence of compatibility conditions between the derivation and the exponentiation, do not admit a generic derivation.
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Towards Model-Theoretic Learnability Results
In recent years, the interaction between Model Theory and Statistical Learning Theory has increasingly received attention. Notably, Laskowski [1] established a fundamental connection between NIP and the Vapnik--Chervonenkis (VC) dimension, while the Fundamental Theorem of Statistical Learning links VC dimension to probably approximately correct (PAC) learning. When analyzing the PAC learnability of hypothesis spaces definable over tame ordered fields, measurability requirements must be taken into account.
In this talk, we will explore the measurability of definable sets and functions in such contexts, with a view toward model-theoretic applications of the Fundamental Theorem of Statistical Learning. Since ordered fields are naturally endowed with the order topology, the associated Borel $\sigma$--algebras are obvious candidates for our measure-theoretic examination. A central focus will be on identifying sufficient conditions under which definable sets and relevant functions are Borel measurable. These considerations yield a learnability result for o-minimal expansions of the reals. Time permitting, we will further discuss measurability subtleties that emerge beyond the tame setting.
This is based on [2] and [3], which are submitted for publication and are part of my doctoral research project supervised by Professor Salma Kuhlmann and Dr. Lothar Sebastian Krapp at Universität Konstanz.
A list of registered participants will be available closer to the conference dates.
Full schedule will be available closer to the conference dates.
The conference will be held in room B3.03 of the Zeeman building (Coventry CV4 7EZ) of the University of Warwick, UK
The nearest airport to the university is Birmingham International (BHX). To get to the University of Warwick from BHX, taxis are available, as is Uber; should take mildly upwards of 20 minutes to get to campus. Also, trains go from BHX to Coventry station. You can also fly into one of the London airports. If you fly into Heathrow, you can take a train to Watford Junction (Railair), and you can take another train from Watford Junction to Coventry station. From Coventry, several bus lines go to campus regularly, for example 11 or 12X, see the map and the timetables.
You can take long distance buses instead of trains; there is a direct one from Heathrow to Coventry.
It is a 10 minute walk from the University Interchange bus station to the Zeeman Building, which is not necessarily the closest bus station, but the easiest.
For any inquiries, please contact us at omigawd2025[hopefully at]gmail[on the dot]com.
The conference is organized by Abhiram Natarajan and the administrative staff of the Warwick Mathematics Institute, and Martin Lotz, Harry Schmidt.
Abhiram Natarajan gratefully acknowledges financial support for this event from his EPSRC Grant EP/V003542/1, and also from University of Warwick's Mathematics Research Centre (MRC). Banner image made using InsMind with image taken from here.