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PhD Dissertation Defence - Mgr. Ivan Di Liberti |
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Title: The Scott adjunction
Author: Mgr. Ivan Di Liberti
Defence: Wednesday, September 9, 2020, 13.45, Meeting Room of the department
Supervisor: prof. RNDr. Jiří Rosický, DrSc.
During his doctoral study, the author has mostly dealt with a categorical model theory. The first results were devoted to an understanding of weak Fraiisse limits (see [Kubis]) from the point of view of category theory and were published in \cite{zbMATH07106179}. Later, the author studied codensity monads playing an important role in categorical universal algebra (see, e.g., [Leinster]) and prepared the paper [Lib19]. Both the papers should be considered as a part of this thesis.
Since then, the author has concentrated his efforts to the development of the Scott adjunction, relating accessible categories with directed colimits to topoi. This adjunction has already appeared in the literature in collaboration with Simon Henry [Hen19]. The resulting theory form the most important part of the thesis and will we submitted for a publication in a near future. After establishing basic properties of the Scott adjunction, we study both its applications to model theory and its geometric interpretation. From the geometric point of view, we introduce the categorfied Isbell duality, relating bounded (possibly large) ionads to topoi. The categorified Isbell duality interacts with the Scott adjunction offering a categorfication of the Scott topology over a poset (hence the name). We show that the categorfied Isbell duality is idempotent, similarly to its uncategorified version. From the logical point of view, we use this machinery to provide candidate (geometric) axiomatizations of accessible categories with directed colimits. In particular, we study the $2$-category of accessible categories with directed colimits using the tools and the mind-setting of formal category theory, this formal approach is part of a very general research-motive of the author and is prominently evident in \cite{liberti2019codensity}. We discuss the connection between the above-mentioned adjunctions and the theory of classifying topoi. We relate our framework to the more classical theory of abstract elementary classes. We discuss the relation between atomic topoi and categoricity, providing a more conceptual understanding of our previous contributions to the topic [Di 19], and continuing the research line of [Ros97]. From a more categorical perspective, we show that the $2$-category of topoi is enriched over accessible categories with directed colimits and we relate this result to the Scott adjunction. |
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Last Updated on Monday, 07 September 2020 07:46 |
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GEOMETRY AND APPLICATIONS ONLINE 7.9. - 9.9.2020 |
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CELEBRATING THE 80TH BIRTHDAY OF DMITRI ALEKSEEVSKY
7-9 September 2020
Program of planned online ZOOM talks:
http://prf.uhk.cz/alekseevsky2020/
1st day
13:10 - 14:10 Jose Figueroa-O'Farrill
14:20 - 15:20 Jorge Lauret
15:30 - 16:30 Ivan Izmestiev
16:40 - 17:40 Vicente Cortes
2nd day
13:10 - 14:10 Anna Fino
14:20 - 15:20 Christoph Bohm
15:30 - 16:30 Remco Duits
16:40 - 17:40 Alessandro Sarti
3rd day
13:10 - 14:10 Giovanna Citti
14:20 - 15:20 Simon Salamon
15:30 - 16:30 Jaroslav Hrdina
16:40 - 17:40 Antonio J. Di Scala |
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Last Updated on Monday, 07 September 2020 07:44 |
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Online algebra seminar - July 16, 1pm |
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We will continue online on Thursday, July 16, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:
Charles Walker
Characterization of Lax Orthogonal Factorization Systems
Abstract:
In this talk we will study the lax orthogonal factorization systems (LOFSs) of Clementino and Franco, with a particular focus on finding equivalent definitions of them. In particular, we wish to define them as a pair of classes E and M subject to some conditions. To achieve this, we will reduce the definition of a LOFS in terms of algebraic weak factorization systems (defined as a KZ 2-comonad L and KZ 2-monad R on the 2-category of arrows [2, C] with a 2-distributive law LR ⇒ RL) to a more property-like definition (meaning a definition with less data but more conditions). To do this, we replace strict KZ 2-monads with the property-like definition of KZ pseudomonads in terms of kan-extensions due to Marmolejo and Wood. In addition, pseudo-distributive laws involving KZ pseudomonads have a property- like description which will be used. Thus one can deduce the conditions the classes E and M must satisfy. We will also consider some similarities and differences between LOFSs and (pseudo-)orthogonal factorization systems, and will extend their definitions to include universal fillers for squares which only commute up to a comparison 2-cell. This is joint work with John Bourke, and is currently a work in progress. |
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Last Updated on Wednesday, 15 July 2020 11:34 |
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31th mathematical hike - August 8th 2020 |
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Dear Friends of Hikes and Mathematics,
Extreme hike! The 31st mathematical hike is planned on August 8th. Meet at 5:50 at the Main Train Station of Brno.
We go by train, then 12 hours and 45 km to get to the train back to Brno. Evaluate your strength and join us.
All information and photos can be found at http://conference.math.muni.cz/vylety/. (in CZ)
Have a nice summer, we look forward to you joining us.
Jana Bartoňová and Jonatan Kolegar, organizers
Jan Slovák, Director of the Department of Mathematics and Statistics
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Last Updated on Wednesday, 15 July 2020 11:33 |
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PhD Dissertation Defence |
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Title: Geometric Approach to Segmentation in Diffusion Magnetic Resonance Imaging
Author: Sumit Kaushik
Defence: Monday, 29 June, 2020, 4pm, Meeting Room of the department
Supervisor: Jan Slovák
The anatomy of the biological structures in human brain plays an important role in study and diagnosis of medical conditions. The extraction of these structures in DMR images need appropriate methods and modelling. In this work, two standard segmentation methods are discussed. We have used deformable models and the dimensionality reduction method to project the data from high dimension to very small dimensions. On the projected lower dimensional data, we employed the classical k-means clustering for segmentation. Novelty in this work consists in exploitation of the similarity measures for the voxels based on the properties of Riemannian symmetric spaces, as well as quaternionic representation of transformations and their polar decompositions.
Video , thesis |
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Last Updated on Wednesday, 01 July 2020 16:03 |
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