M.S. Theses
http://digitalarchive.boun.edu.tr/handle/123456789/11662
2024-05-30T19:15:50ZProjective characters of finite groups
http://digitalarchive.boun.edu.tr/handle/123456789/21475
Projective characters of finite groups
Yılmaz, Said.
The purpose of this thesis to analyze the basic facts about projective characters of finite groups and compare them to the facts about ordinary characters of finite groups. We start with review of basic facts about the twisted group algebras and projective representations of finite groups over a field. Then we study the properties of the projective characters. Finally, we will study these properties on the complex projective characters.
2022-01-01T00:00:00ZKadison-Singer problem from a banach algebra perspective
http://digitalarchive.boun.edu.tr/handle/123456789/19902
Kadison-Singer problem from a banach algebra perspective
Keçkin, Murat.
In 1959, Kadison and Singer asked whether every pure state of the diagonal subspace D(ℓ2) of B(ℓ2) has a unique pure state extension to B(ℓ2). This problem has remained open until 2013; in 2013 it has been solved by a team of computer scientists. In my Master thesis, which is largely based on a paper by Akemann, Tanbay and Ulger, ¨ I have tried to learn this problem and the approach considered in this paper. We identify D(ℓ2) with C(βN). For t in βN, δt is the Dirac measure at t considered as a functional on C(βN). We denote by [δt ] the set of the states of B(ℓ2) that extend δt . Our main aim is to understand how large the set [δt ] is. Using the fact that the von Neumann algebra B(ℓ2) has the Pelczýnski’s property (V ), it is proven that either the set [δt ] lies in a finite dimension subspace of B(ℓ2) ∗ or, in its weak-star topology, it contains a homeomorphic copy of βN. We study this result under the so far directly unproven knowledge that [δt ] is a singleton.
2022-01-01T00:00:00ZTheory of noncommutative motives
http://digitalarchive.boun.edu.tr/handle/123456789/19898
Theory of noncommutative motives
Üze, Berkan.
The theory of motives was originally conceived by Alexander Grothendieck as a universal cohomology theory for algebraic varieties. In the decades since it was first introduced, it has become a vast and profoundly sophisticated subject systematically developed in many directions spanning algebraic and arithmetic geometry, homotopy theory and higher category theory. The quest for a fully developed theory of motives as envisioned by Grothendieck drove a great deal of fundamental research in the aforementioned disciplines, while delivering fantastic and long-promised results and settling classical questions as it reached maturity in the past decades. This quest is arguably not complete, since the abelian category of mixed motives, originally established by Grothendieck himself as the ultimate desideratum of a satisfactory theory of motives, has proven elusive. However, ideas of motivic nature as a programmatic approach to cohomology theories and invariants have proven extremely useful in a variety of other contexts. Noncommutative algebraic geometry is precisely one of these contexts. Following ideas of Maxim Kontsevich, Goncalo Tabuada and Marco Robalo independently developed theories of "noncommutative" motives which fully encompasses the classical theory of motives and helps assemble so-called additive invariants such as Algebraic K-Theory, Hochschild Homology and Topological Cyclic Homology into a motivic formalism in the very precise sense of the word. In this expository work, we will review the fundamental concepts at work, which will inevitably involve a foray into the formalism of enhanced and higher categories. We will then discuss Kontsevich's notion of a noncommutative space, sharpened and made precise over the years by Toen, Tabuada, Robalo and others and introduce noncommutative motives as "universal additive invariants" of noncommutative spaces. We will conclude by offering a brief sketch of Robalo's construction of the noncommutative stable homotopy category.
2022-01-01T00:00:00ZMonogenic number fields
http://digitalarchive.boun.edu.tr/handle/123456789/19900
Monogenic number fields
Değirmenci, Pınar.
Determining whether the ring of integers OK of an algebraic number field K of degree n admits a power integral basis is one of the classic problems in algebraic number theory. In other words, we want to determine whether there exists α ∈ OK such that {1, α, . . . , αn−1} is a Q-basis for K. This question dates back to the 1960s and was introduced by a German mathematician, Helmut Hasse. In this thesis, we will study the monogenicity of cubic number fields and their lift to monogenic sextic number fields. After recalling some background material on algebraic number theory and related topics, we will focus on specific cubic fields such as pure cubic fields and cyclic cubic fields. Next, we will study the lifting of all monogenic cyclic cubic fields to monogenic sextic fields. This thesis was supported by Bo˘gazi¸ci University Research Fund Grant Number 19082.
2022-01-01T00:00:00Z