This is the homepage^{1} of Bilkent Math Graduate Seminars.
Location: Zoom
Time: Wednesdays, 4 pm UTC+3
Nov 25: $2$-categories for the working graduate student by Redi Haderi
An ordinary ($1$-)category consists of a collection of objects, morphisms between them and composition of morphisms satisfying some rules. A $2$-category is a structure which has incorporated in it an extra layer of data: morphisms between morphisms. Examples of morphisms between morphisms are natural: indeed we have homotopies between continuous maps, conjugations between group homomorphisms, natural transformations between functors etc. We will briefly illustrate how this extra layer of data allows us to “weaken” usual categorical concepts. In particular we discuss equivalences between objects in a $2$-category.
Dec 2: $2$-categories for the working graduate student by Redi Haderi
Part II
Dec 9: Euler characteristics of Morita equivalent categories by Mustafa Akkaya
In this talk, definition of Euler characteristics of finite categories will be given and some properties will be discussed. We will show that Leinster’s Euler characteristic is invariant under equivalence of categories as we expected, but not under Morita equivalence.
Dec 16: Sheaf-theoretic approach to quantum contextuality by Cihan Okay
I will talk about how sheaf theory can be used to capture the notion of non-locality and contextuality in quantum mechanics following the paper https://arxiv.org/abs/1102.0264. For this I will introduce the distribution monad, the notions of empirical model and hidden-variable model. I will discuss the Bell scenario in this framework.
Dec 23: Sheaves by Pejman Parsizadeh
Part I: We will cover basics of sheaf theory: definition of the category of sheaves, the sheafification functor, and pull-back/push-forward sheaves.
Jan 13: Torsors and first cohomology set of torsors by Pejman Parsizadeh (moved to Feb 17)
Part II
Jan 20: Euler characteristics of Morita equivalent categories by Mustafa Akkaya
Part II: Quillen theorem A
Jan 27: Categorical logic I: Structures in Categories by Kristof Kanalas
The model $M$ of a theory $T$ consists of an underlying set $X$ together with functions $f:X^n \to X$ and subsets $R\subseteq X^m$ which correspond to the function and relation symbols of the language, in such a way that the axioms of $T$ are valid in $M$. If we replace “set” with “object”, “function” with “arrow” and “subset” with “subobject” we get the notion of a structure in the category $\mathcal{C}$, and if this category has enough structure we can also interpret formulas (and decide their validity), hence define the models of $T$ inside $\mathcal{C}$. As a first application we can define the internal language (and theory) of a category: it turns out that “from inside every category looks like $\mathbf{Set}$”, e.g. an arrow $f:A\to B$ of $\mathcal{C}$ is a monomorphism iff the formula $f(a)=f(a') \rightarrow a=a'$ is valid in the internal theory, etc.
Feb 3: Introduction to Double Categories by Redi Haderi
Double categories are a way to do two-dimensional category theory. We will present this structures and some of the main examples, and observe how the theory of double categories is much richer than that of 2-categories. In particular we present the concept of double colimit and (time permitting) equipments.
Feb 10: Defining second cohomology group of a group by using group extensions by Zilan Akbas
In this talk, first we will define what is a group extension, and introduce split and general extensions. By defining factor set (cocycle), we will have several properties which will lead to second cohomology group. Finally, we will show that there exist a bijection between second cohomology group and family of all the equivalence class of extensions of a group.
Feb 17: Torsors and first cohomology set of torsors by Pejman Parsizadeh
Part II
Feb 24: Cech cohomology and its relation with torsors by Pejman Parsizadeh (moved to Apr 7)
Part III
Mar 3: Operads and $E_n$-algebras by Calista Bernard
Suppose we have a space with a multiplication that is not strictly associative, so $(xy)z$ and $x(yz)$ are not equal. If we want to multiply $n$ elements together, we now have many ways to do so depending on where we choose to put our parentheses. Often we have some additional data, such as paths between $(xy)z$ and $x(yz)$ and some coherency between these paths, and we would like to keep track of this data to see how to relate the different ways of multiplying $n$ elements. Operads provide a concise way of encoding this type of data of operations and relations between them. In this talk I will define operads and give examples that determine to what extent a multiplication is associative or commutative up to homotopy. In particular, I will discuss $E_n$-algebras, which are homotopy-commutative objects, and their relationship to $n$-fold loop spaces.
Mar 10: Quantum state certification by Gautam Gopal Krishnan
Quantum state certification deals with the problem of testing whether a quantum mixed state is equal to some unknown mixed state or else is $\epsilon$-far from it. Definitions of quantum mixed states and measurement schemes will be given in this talk and we will talk about some recent developments concerning quantum state certification. We will then illustrate how Schur-Weyl duality and the double commutant theorem in representation theory can be exploited to generalize these results.
Mar 17: Higher order Toda brackets by Azez Kharoof
This will be a preparation for the topology seminar talk on Monday.
Apr 7: An overview of Hochschild cohomology by Pablo Sanchez Ocal (5pm UTC+3)
Taking Hochschild cohomology is a way of algebraically encoding infinitesimal information about an associative algebra. In this talk we will give an unpretentious introduction to this cohomology, we will justify its importance by computing some of the lower degrees, and we will then give explicit applications that advance the understanding of quantum symmetries.
Apr 28: Various Quantum Relative Entropies and some Applications by Sarah Chehade
In Quantum Information Theory, one of the most famous inequalities is called the Data Processing Inequality (DPI). The inequality states that two quantum states become harder to distinguish after they pass through a noisy quantum channel. This inequality holds for a number of distinguishability measures, with the most basic one being the Umegaki relative entropy otherwise known as the quantum relative entropy. We are interested in the case of saturation of DPI for these measures. One of the generalizations of quantum relative entropy is a two-parameter family called the $\alpha-z$ Renyi relative entropy. Recently, the set of parameters has been completely characterized for which the $\alpha-z$ Renyi relative entropy satisfies DPI.
In this talk, I will present a necessary and a sufficient condition for saturation of DPI for $\alpha-z$ Renyi relative entropy. Both conditions are similar to the original condition of recoverability for quantum relative entropy, and coincide when $\alpha=z$, leading to a so-called sandwiched Renyi relative entropy.
Other applications of relative entropy, such as coherence and entanglement measures will also be mentioned and discussed as a future direction of work.
May 3-4-5-10: Mini workshop on Quantum Computing
See the schedule of talks for more information.