Відео

We explain how slice quasi-categories are defined, and how they provide a new notion of mapping spaces in a quasi-category. Using this new notion, we give an alternate characterization of equivalences of quasi-categories, and define initial and terminal objects in a quasi- category. Part II of a sequence on Higher Categories (HC).

In a symmetric monoidal category, the set of maps from the monoidal unit to itself inherits additional structure. These scalars in the category are inspired by, for instance, the isomorphism between linear maps from $\mathbb{C}$ to itself and the field $\mathbb{C}$. However, the additive structure and the notion of traces which are key to the computation of quantum probabilities require some de...

The structure involved in a symmetric monoidal category is very broad, which is of benefit to the general theory. However, quantum mechanics, which is typically formulated using Hilbert spaces, requires substantially more structure than is present in an arbitrary SMC. In particular, the notions of conjugation and transpose are of key import in the study of quantum mechanics. In this talk, we in...

Matrix reduction algorithm on a simplicial complex is a fairly new wave in persistent homology due to its implementations on programs like Ripser and many algorithms that have been built upon that. Persistent algorithm dates back to 2002 with a pairing algorithm and its runtime has been shown to be O(N^3). Morozov in his 2005 article gives an explicit example of the existence of this case. In m...

In this talk, we define quasi-categories as simplicial sets satisfying a lifting condition related to both categories and Kan complexes. We describe an adjunction that relates quasi-categories and simplicially enriched categories and explain how it allows us to define some first categorical notions in quasi-categories. Part I of a sequence on Higher Categories.

We will introduce monoidal categories and symmetric monoidal categories (SMCs) as well as the string diagrams which provide graphical calculi for computations in SMCs (Primary reference: [1,Ch. 1], secondary references: [2, §3.1, §3.3], [5, Ch. XI], [3, Ch. 3], [4, Track 1]). We define dual objects and duality data in a SMC (Primary reference: [1, Ch. 3], secondary references: [2,§3.3], [4, Tra...

Partial groups were introduced by Chermak as a tool to approach the issue of the existence and uniqueness of a centric linking system for a saturated fusion system. Roughly speaking, a partial group is a set with a partially defined product (you can still multiply certain strings of elements together, but not always). In this talk, we will discuss the main similarities and differences between t...

In this talk, we will discuss the proof of Thomason's homotopy colimit theorem. The theorem states that given a functor from a small category to the category of small categories, the homotopy colimit construction on the nerves of the categories in the diagram is naturally homotopy equivalent to the nerve space of the Grothendieck Construction.

In this talk, I will revise some aspects and generalizations of the cohomology of small categories introduced by Baues and Wirsching in 1985 developed in more recent work with Neumann and Tonks, such as Thomason cohomology and Gabriel-Zisman cohomology for simplicial sets. Also, I will report about work in progress with Neumann, Paoli and Tonks about the generalization of the above to the frame...

In this talk, we'll introduce fundamental techniques in topological machine learning and showcase their application in two specific contexts. The first application is on computer-aided drug discovery, utilizing Multiparameter Persistence for graph representation learning. Our second application revolves around cancer detection from histopathological images via cubical persistence. We apply our ...

The study of the p-subgroup complexes began motivated by group cohomology and equivariant cohomology of topological spaces "modulo the prime p". For example, Kenneth Brown proved that the reduced Euler characteristic of this complex is divisible by the size of a Sylow p-subgroup, giving rise to a sort of "Homological Sylow theorem". Later, he showed that the mod-p equivariant cohomology of the ...

Topological Data Analysis (TDA) offers a suite of computational tools that provide quantified shape features in high-dimensional data, which can be utilized by modern statistical and predictive machine learning models. In particular, persistent homology (PH) takes in data and derives compact representations of latent topological structures, known as persistence diagrams. PH has been widely adop...

Simplicial quantum contextuality, introduced by Okay, Kharoof and Ipek, is a framework for using topological methods based on simplicial sets to study quantum contextuality. It subsumes earlier approaches - topological (Okay, Roberts, Bartlett, Raussendorf) and sheaf-theoretic (Abramsky, Brandenburger). In this talk we will discuss how group action can be composed into this framework. To this e...

The rational cohomology of the configuration space of commuting elements in a compact Lie group is determined by the action of the Weyl group on the configuration space of its maximal torus. This can be used to determine (co)homological stability phenomena and other unstable computations. In this talk I will begin with some motivation for the study of these spaces and the case of SU(2), where t...

BilTop | Redi Haderi | Colimits of categories, zig-zags and necklaces

BilTop | Walker Stern | A story about spans

BilQC | James Seddon | Stabilizer simulation methods for mixed magic states and noisy channels

BilTop | Victor Antonio Torres Castillo | Stable homotopy type of p-local finite groups

BilTop | Igor Sikora | Segal spaces II

BilTop | Bob Oliver | Relizability of fusion systems

BilQC | Selman Ipek | Trading classical and quantum resources

BilTop | Aziz Kharoof | Simplicial categories II - Dwyer-Kan localizations (part II)

BilTop | Aziz Kharoof | Simplicial categories II - Dwyer-Kan localizations part I

BilTop | Igor Sikora | Simplicial categories

BilQC | Selman Ipek | Quantum universality and magic state distillation

BilQC | Selman Ipek | Universal Quantum Computation with ideal Clifford gates and noisy ancillas

BilTop | Igor Sikora | Model Categories II - Derived functors and Quillen adjunctions

BilQC | Selman Ipek | Improved simulation of stabilizer circuits

BilTop | Igor Sikora | Basic constructions in quasicategories