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Online algebra seminar - May 14, 1pm |
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We will continue online on Thursday, May 14, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:
Soichiro Fujii
A unified framework for notions of algebraic theory
Abstract:
Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads. These variants of universal algebra are called notions of algebraic theory. In this talk, we present a unified framework for them. The key observation is that each notion of algebraic theory can be identified with a monoidal category, in such a way that algebraic theories correspond to monoid objects therein. To incorporate semantics, we introduce a categorical structure called metamodel, which formalises a definition of models of algebraic theories. We also define morphisms between notions of algebraic theory, which are a monoidal version of profunctors. Every strong monoidal functor gives rise to an adjoint pair of such morphisms, and provides a uniform method to establish isomorphisms between categories of models in different notions of algebraic theory. A general structuresemantics adjointness result and a double categorical universal property of categories of models are also shown. |
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Last Updated on Tuesday, 12 May 2020 08:59 |
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Online algebra seminar - May 7, 1pm |
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We will continue online on Thursday, May 7, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:
Christian Espindola
Topos-theoretic completeness theorems
Abstract:
In this talk we will delve into the background details of the previous talk by introducing syntactic proof systems and their categorical semantics, including the construction of syntactic categories and $\kappa$-classifying toposes, as well as the role of certain properties of Grothendieck topologies and Kripke-Joyal semantics. We will then study some topos-theoretic completeness theorems for certain infinitary logics that generalize results of Deligne and Joyal.
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Last Updated on Monday, 04 May 2020 10:39 |
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Online algebra seminar - April 30, 1pm |
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We will continue online on Thursday, April 30, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:
Christian Espindola
A topos theoretic proof of Shelah’s eventual categoricity conjecture
Abstract:
Assuming the Generalized Continuum Hypothesis (GCH), we give a topos-theoretic proof of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC) with amalgamation. The proof is based on infinitary generalizations of Deligne and Joyal's completeness theorems for certain infinitary intuitionistic logics, by means of which it is possible to attack the question of categoricity in a topos-theoretic way. Using recent results on the Scott adjunction developed by Henry and Di Liberti, we prove under GCH that an AEC with amalgamation which is categorical in a pair of cardinals is also categorical in all cardinals in between. Under some extra natural assumptions on the AEC, we also deduce a new downward categoricity transfer. We also explain how these methods can be adapted to shed some light on the categoricity spectrum of more general accessible categories. |
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Last Updated on Friday, 24 April 2020 10:04 |
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Record from public habilitation talk: Mgr. Lenka Zalabová, Ph.D. |
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Wednesday, March 11, 2020, at 16.00 in M1
Recordings: slides and camera
Symmetric spaces and their filtered generalizations
Abstract: We focus on the role of symmetries in geometry and geometric control theory. We introduce symmetric spaces as important examples of geometries with many symmetries and we study consequences of existence of special symmetries. Finally we study various generalizations of symmetric spaces. |
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Last Updated on Friday, 13 March 2020 13:44 |
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Public habilitation talk: RNDr. Dana Černá, Ph.D. |
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Tuesday, March 10, 2020, at 14:00 in M5
Wavelet methods for operator equations
Wavelet methods are a useful tool for the numerical solution of various types of operator equations. We introduce the concept of a wavelet basis and two examples of spline wavelet bases.Then we study the wavelet-Galerkin method, namely the existence and uniqueness of the numerical solution and the convergence rate of the method. Furthermore, we also focus on the properties of the resulting systems of linear algebraic equations, especially on their structures and the condition numbers of the discretization matrices. Numerical experiments are provided for integral and integro-differential equations, and as practical application, the jump-diffusion option pricing model is solved. |
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Last Updated on Wednesday, 04 March 2020 11:31 |
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