Category theory | Algebraic topology
In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane. The reverse of categorification is the process of decategorification. Decategorification is a systematic process by which isomorphic objects in a category are identified as equal. Whereas decategorification is a straightforward process, categorification is usually much less straightforward. In the representation theory of Lie algebras, modules over specific algebras are the principal objects of study, and there are several frameworks for what a categorification of such a module should be, e.g., so called (weak) abelian categorifications. Categorification and decategorification are not precise mathematical procedures, but rather a class of possible analogues. They are used in a similar way to the words like 'generalization', and not like 'sheafification'. (Wikipedia).
Representation Theory & Categorification II - Catharina Stroppel
2021 Women and Mathematics - Uhlenbeck Course Lecture Topic: Representation Theory & Categorification II Speaker: Catharina Stroppel Affiliation: University of Bonn Date: May 25, 2021 In modern representation theory we often study the category of modules over an algebra, in particular i
From playlist Mathematics
Representation Theory & Categorification III - Catharina Stroppel
2021 Women and Mathematics - Uhlenbeck Course Lecture Topic: Representation Theory & Categorification III Speaker: Catharina Stroppel Affiliation: University of Bonn Date: May 27, 2021 In modern representation theory we often study the category of modules over an algebra, in particular
From playlist Mathematics
Metacognition and speaking | Introduction | Part 1
In this video, I provide an overview of metacognition and discuss its role in speaking.
From playlist Metacognition
1.9 factors and categorical variables in R | statistical analysis and data science course in Rstudio
In this chapter of the video series in the crash course in statistics and data science with R / Rstudio we will see the definition, creation, and importance of using factors in R. We discuss among others: - Why use factors in our analysis and models ? - nominals and categorical variabl
From playlist R Tutorial | Rstudio
IAML2.7: Categorical (nominal) attributes
From playlist Thinking about Data
Representation Theory & Categorification IV - Catharina Stroppel
2021 Women and Mathematics - Uhlenbeck Course Lecture Topic: Representation Theory & Categorification IV Speaker: Catharina Stroppel Affiliation: University of Bonn Date: May 28, 2021 In modern representation theory we often study the category of modules over an algebra, in particular i
From playlist Mathematics
In this video, I define a cool operation called the symmetrization, which turns any matrix into a symmetric matrix. Along the way, I also explain how to show that an (abstract) linear transformation is one-to-one and onto. Finally, I show how to decompose and matrix in a nice way, sort of
From playlist Linear Transformations
Coherent categorification of quantum loop sl(2) - Peng Shan
Virtual Workshop on Recent Developments in Geometric Representation Theory Topic: Coherent categorification of quantum loop sl(2) Speaker: Peng Shan Tsinghua University; Member, School of Mathematics Date: November 17, 2020 For more video please visit http://video.ias.edu
From playlist Virtual Workshop on Recent Developments in Geometric Representation Theory
Peng Shan: Coherent categorification of quantum loop sl(2)
'Coherent categorification of quantum loop sl(2)' Peng Shan (Tsinghua University) Abstract: We explain an equivalence of categories between a module category of quiver Hecke algebras associated with the Kronecker quiver and a category of equivariant perverse coherent sheaves on the nilpot
From playlist SMRI Algebra and Geometry Online
Christian Voigt: Clifford algebras, Fermions and categorification
Given the fundamental role of Dirac operators in K-homology and K-theory, it can be argued that "categorified Dirac operators" should be crucial for the understanding of elliptic cohomology. However, the actual construction of such operators seems a challenging open problem. In this talk
From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"
Ed Witten -- From Gauge Theory to Khovanov Homology Via Floer Theory
Edward Witten lecture entitled "From Gauge Theory to Khovanov Homology Via Floer Theory" as part of the Banff International Research Station conference "Perspectives on Knot Homology". The Banff International Research Station will host the "Perspectives on Knot Homology" workshop in Banf
From playlist Research Lectures
2-Verma modules - Gregoire Naisse; Pedro Vaz
Virtual Workshop on Recent Developments in Geometric Representation Theory Topic: 2-Verma modules Speakers: Gregoire Naisse; Pedro Vaz Affiliation: University College London; University College London Date: November 18, 2020 For more video please visit http://video.ias.edu
From playlist Virtual Workshop on Recent Developments in Geometric Representation Theory
Lars Thorge Jensen: Cellularity of the p-Kazhdan-Lusztig Basis for Symmetric Groups
After recalling the most important results about Kazhdan-Lusztig cells for symmetric groups, I will introduce the p-Kazhdan-Lusztig basis and give a complete description of p-cells for symmetric groups. After that I will mention important consequences of the Perron-Frobenius theorem for p-
From playlist Workshop: Monoidal and 2-categories in representation theory and categorification
R Programming: Introduction: Factors (R Intro-04)
[my R script is here https://github.com/bionicturtle/youtube/tree/master/r-intro] Factors are categorical vectors. Specifically, they are (integer) vectors that store categorical values, or ordinal values. Ordinal values are *ranked* categories (but they are not intervals).Factors can only
From playlist R Programming: Intro
Representation Theory & Categorification - Catharina Stroppel
2021 Women and Mathematics - Uhlenbeck Course Lecture Topic: Representation Theory & Categorification Speaker: Catharina Stroppel Affiliation: University of Bonn Date: May 24, 2021 For more video please visit https://www.ias.edu/video
From playlist Mathematics
Ivan Di Liberti - Towards higher topology
Talk at the school and conference “Toposes online” (24-30 June 2021): https://aroundtoposes.com/toposesonline/ Slides: https://aroundtoposes.com/wp-content/uploads/2021/07/DiLibertiSlidesToposesOnline.pdf We categorify the adjunction between locales and topological spaces, this amounts t
From playlist Toposes online
Ben Elias: Categorifying Hecke algebras at prime roots of unity
Thirty years ago, Soergel changed the paradigm with his algebraic construction of the Hecke category. This is a categorification of the Hecke algebra at a generic parameter, where the parameter is categorified by a grading shift. One key open problem in categorification is to categorify He
From playlist Workshop: Monoidal and 2-categories in representation theory and categorification