Lecture

Universal Coating: Understanding and Proof

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In course

MATH-225: Topology II - fundamental groups

On étudie des notions de topologie générale: unions et quotients d'espaces topologiques; on approfondit les notions de revêtements et de groupe fondamental,et d'attachements de cellules et on démontre le Théorème de Seifert-van Kampen. Des exemples de surf ...

Description

This lecture covers the concept of universal coating, focusing on the proof that a given function is universal. Starting with the definition of a lace and its relation to x, the instructor explains the process step by step, emphasizing the importance of showing specific equations. Through examples and detailed explanations, the lecture demonstrates the uniqueness and significance of universal coating in various scenarios, highlighting its role in mathematical proofs and applications.

Instructors (2)

Jérôme Scherer

Teaching & PhD PhD Students Virgile Simon Constantin Past EPFL PhD Students Samuel Lavenir Past EPFL PhD Students as codirector Kay Remo Werndli, Lyne Moser Courses Topology IV.a - Cohomology rings MATH-506 Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative algebra. We study an algebraic version, namely group cohomology, and compare both approa

Dimitri Stelio Wyss

I did my PhD under the supervision of Tamas Hausel at EPFL and the Institue of Science and Technology Austria . From 2017-2019 I was a postdoctoral researcher at the Sorbonne University Paris 6 with François Loeser. Since September 2019 I hold the chair of arithmetic geometry at EPFL as a tenure track assistant professor. For more information see my CV.I work in algebraic and arithmetic geometry. My research uses ideas from p-adic and motivic integration to study the interplay between geometric and arithmet

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Related concepts (32)

Proof theory

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Error function

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: Some authors define without the factor of . This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real.

Proof (truth)

A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition.

Proof by contradiction

In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.

Gamma function

In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.

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