CIRM - Videos & books Library

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Lattice paths are fundamental combinatorial objects, and their enumeration has strong connections to other fields (physics, computer science). In this talk, we will review enumeration of models of lattice paths with forbidden patterns and with dynamic boundary in both one- and two-dimensional models. We will also examine how automata-based approaches often result in the simplification and classification of enumeration problems.

05A15 ; 05A05 ; 05A19

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Maps decorated by the Ising model are a remarkable instance of a model of non-uniform maps with very nice enumerative properties. In this talk, I will first explain how one can obtain a differential equation for the generating function of Ising-decorated cubic maps in arbitrary genus, related to the Kadomtsev--Petviashvili (KP) hierarchy. In particular, this leads to an efficient algorithm to enumerate Ising cubic maps in high genus. I will also present and compare implementations of this algorithm in Maple and SageMath. This is based on a joint work with Mireille Bousquet-Mélou and Baptiste Louf.[-]

Maps decorated by the Ising model are a remarkable instance of a model of non-uniform maps with very nice enumerative properties. In this talk, I will first explain how one can obtain a differential equation for the generating function of Ising-decorated cubic maps in arbitrary genus, related to the Kadomtsev--Petviashvili (KP) hierarchy. In particular, this leads to an efficient algorithm to enumerate Ising cubic maps in high genus. I will also ...[+]

05A15 ; 82B20 ; 37K10

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Given two algebraic ODEs, is there a differential-algebraic relation between generic tuples of their solutions? In recent work with Freitag and Moosa, we produce a bound on the length of tuples one must look at to f ind a relation. Our proof relies on two ingredients. The first is differential Galois theory, combined with the recent proof by Freitag and Moosa of the Borovik-Cherlin conjecture in algebraically closed fields. The second is some general model theory result which allows us to factor any relation through some minimal ODE. I will give a precise statement of our result and sketch the proof. I will also explain why our bound is tight.[-]

Given two algebraic ODEs, is there a differential-algebraic relation between generic tuples of their solutions? In recent work with Freitag and Moosa, we produce a bound on the length of tuples one must look at to f ind a relation. Our proof relies on two ingredients. The first is differential Galois theory, combined with the recent proof by Freitag and Moosa of the Borovik-Cherlin conjecture in algebraically closed fields. The second is some ...[+]

03C45 ; 14L30 ; 12H05 ; 12L12

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The exponential period conjecture predicts how the Galois group of an exponential motive governs all polynomial relations among its periods. For classical motives (which are special exponential motives) this conjecture specialises to the classical period conjecture. My aim is to present some elementary, yet elucidative examples of exponential motives and periods which illustrate how the exponential period conjecture implies certain popular transcendence conjectures, and how its non classical part is related to the Siegel-Shidlovskii theorem.[-]

The exponential period conjecture predicts how the Galois group of an exponential motive governs all polynomial relations among its periods. For classical motives (which are special exponential motives) this conjecture specialises to the classical period conjecture. My aim is to present some elementary, yet elucidative examples of exponential motives and periods which illustrate how the exponential period conjecture implies certain popular ...[+]

11J91 ; 34M35

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The degree of a dominant rational map $f: \mathbb{P}^n \rightarrow \mathbb{P}^n$ is the common degree of its homogeneous components. By considering iterates of $f$, one can form a sequence $\operatorname{deg}\left(f^n\right)$, which is submultiplicative and hence has the property that there is some $\lambda \geq 1$ such that $\left(\operatorname{deg}\left(f^n\right)\right)^{1 / n} \rightarrow \lambda$. The quantity $\lambda$ is called the first dynamical degree of $f$. We'll give an overview of the significance of the dynamical degree in complex dynamics and describe an example of a birational self-map of $\mathbb{P}^3$ in which this dynamical degree is provably transcendental. This is joint work with Jeffrey Diller, Mattias Jonsson, and Holly Krieger.[-]

The degree of a dominant rational map $f: \mathbb{P}^n \rightarrow \mathbb{P}^n$ is the common degree of its homogeneous components. By considering iterates of $f$, one can form a sequence $\operatorname{deg}\left(f^n\right)$, which is submultiplicative and hence has the property that there is some $\lambda \geq 1$ such that $\left(\operatorname{deg}\left(f^n\right)\right)^{1 / n} \rightarrow \lambda$. The quantity $\lambda$ is called the first ...[+]

32H50

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The density theorem of Schlesinger ensures that the monodromy group of a differential system with regular singular points is Zariski-dense in its differential Galois group. We have analogs of this result for difference systems such as q-difference and Mahler systems, whose only assumption is the regular singular condition. Moreover, solutions of difference or differential systems with regular singularities have good analytical properties. For example, the solutions of differential systems which are regular singular at 0 have moderate growth at 0. We have general algorithms for recognizing regular singularities and they apply to many systems such as differential and q-difference systems. However, they do not apply to the Mahler case, systems that appear in many areas like automata theory. We will explain how to recognize regular singularities of Mahler systems. It is a joint work with Colin Faverjon.[-]

The density theorem of Schlesinger ensures that the monodromy group of a differential system with regular singular points is Zariski-dense in its differential Galois group. We have analogs of this result for difference systems such as q-difference and Mahler systems, whose only assumption is the regular singular condition. Moreover, solutions of difference or differential systems with regular singularities have good analytical properties. For ...[+]

39A06 ; 68W30 ; 11B85

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In the first part of the talk I will sketch how tame geometry results can be of relevance in quantum systems that are described by quantum field theory. I will highlight some mathematical questions that arise in these applications. I will then turn to using tame geometry in quantum gravity, and specifically in string theory, and stress that it is a powerful framework that allows one to address finiteness questions that were posed in these fields.

81-XX ; 83-XX

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We explain recent developments in sharply o-minimal structures. Specifically, we explain how to obtain a "sharp" cellular decompostion in a general o-minimal structures.

03C64 ; 11Gxx ; 30Dxx

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This talk outlines the major experimental challenges when conducting a 3D Texture tomography. It aims a providing an outline on how to design an experiment, prepare samples and carry out data acquisition and data inversion.

74E15

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In the o-minimal setting, parametrizations of definable sets form a key component of the Pila-Wilkie counting theorem. A similar strategy based on parametrizations was developed by Cluckers-Comte-Loeser and Cluckers-Forey-Loeser to prove an analogue of the Pila-Wilkie theorem for subanalytic sets in p-adic fields. In joint work with R. Cluckers and I. Halupczok, we prove the existence of parametriza- tions for arbitrary definable sets in Hensel minimal fields, leading to a counting theorem in this general context. [-]

In the o-minimal setting, parametrizations of definable sets form a key component of the Pila-Wilkie counting theorem. A similar strategy based on parametrizations was developed by Cluckers-Comte-Loeser and Cluckers-Forey-Loeser to prove an analogue of the Pila-Wilkie theorem for subanalytic sets in p-adic fields. In joint work with R. Cluckers and I. Halupczok, we prove the existence of parametriza- tions for arbitrary definable sets in Hensel ...[+]

14G05 ; 03C98 ; 11D88

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Research talks 1 815 résultats

Lattice paths with flexible boundaries: patterns, automata, and counting - Selkirk, Sarah (Auteur de la Conférence) | CIRM H Nouveau

A powerful differential equation for Ising-decorated maps in arbitrary genus - Carrance, Ariane (Auteur de la Conférence) | CIRM H Nouveau

Relations between solutions of ODEs and model theory - Jimenez, Léo (Auteur de la Conférence) | CIRM H Nouveau

Examples around the exponential period conjecture - Jossen, Peter (Auteur de la Conférence) | CIRM H Nouveau

Transcendental dynamical degrees of birational maps - Bell, Jason (Auteur de la Conférence) | CIRM H Nouveau

Regular singularities of Mahler systems - Poulet, Marina (Auteur de la Conférence) | CIRM H Nouveau

Tame geometry in quantum field theory and gravity - Grimm, Thomas (Auteur de la Conférence) | CIRM H Nouveau

Sharply o-minimal structures and sharp cell decomposition - Zak, Benny (Auteur de la Conférence) | CIRM H Nouveau

Experimental challenges in texture tomography - Grünewald, Tilman (Auteur de la Conférence) | CIRM H Nouveau

Parametrizations in valued fields - Vermeulen, Floris (Auteur de la Conférence) | CIRM H Nouveau