M.S. Theses

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Projective characters of finite groups
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2022., 2022) Yılmaz, Said.; Coşkun, Olcay.
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.
Kadison-Singer problem from a banach algebra perspective
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2022., 2022) Keçkin, Murat.; Tanbay, Betül.
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.
The question of model companionability : positive and negative answers
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2022., 2022) Berksoy, Feyza Nur.; Beyarslan, Özlem.
Model companion ofa universal theory T is the axiomatization of the existentially closed models of T. This thesis studies the concept of model companionability of theories. We present examples of model companions of certain well known theories. We then give examples of theories without model companions. The main focus of this thesis is to elaborate a technique, which we call "the Compactness Argument". Compactness Argument is used to prove that the model companion of a theory does not exist. We apply Compactness Argument to prove that the following theories do not have model companions: the theory of groups, the theory of rings, two examples of the theory of graphs, the theory of fields with two commuting automorphisms, and the theory of dense linear orders with an automorphism. Several proofs are illustrated by original diagrams to provide a better understanding to the reader.
Global well-posedness of NLS equations
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2022., 2022) Yılmaz, Oğuz.; Gürel, Burak.
The thesis is a survey of the I-method. After introducing the method, we discuss the implementation of this method to cubic, defocusing nonlinear Schrödinger equation in the spatial dimension n = 2 and quintic, defocusing nonlinear Schr¨odinger equation in the spatial dimension n = 1 with detailed calculations. We will find out on which type of equations one can use the I-method. We then mention our joint work with Engin Ba¸sako˘glu on the cubic defocusing fourth order nonlinear Schr¨odinger equation in the spatial dimension n = 4. Lastly, we discuss advantage and disadvantage of the method and share our idea of a study plan of the same equation in the spatial dimensions n = 2, 3 as future work.
Monogenic number fields
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2022., 2022) Değirmenci, Pınar.; Beyarslan, Özlem.; Özman, Ekin.
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.
On the hypersurfaces in toric varieties
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2022., 2022) Barış, İlayda.; Coevering, Craig Charles van.; Tanabe, Susumu.
Theory of toric varieties provides fruitful interactions between algebraic geom etry and combinatorics. It is remarkably fertile in terms of connections with many areas of mathematics and has plentiful applications to other disciplines as well. We introduce and study toric varieties and their hypersurfaces in the realm of algebraic geometry with a focus on quasismooth hypersurfaces. This is because quasismooth hy persurfaces are general enough to contain many examples of elements in some special families (e.g. regular hypersurfaces and Calabi-Yau hypersurfaces) that have a fre quent appearance in mirror symmetry, complex and differential geometry, and physics; interesting enough to have special roles in some areas of research such as toric GIT and moduli problems; and easy to characterize using combinatorial tools agreeably to the spirit of toric geometry.
Theory of noncommutative motives
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2022., 2022) Üze, Berkan.; İkeda, Kazım İlhan.
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.
Covariance estimation of spatio-temporal random variables with kronecker product based models
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2022., 2022) Dağıdır, Can Hakan.; Işlak, Ümit.; Baydoğan, Mustafa Gökçe.
Covariance estimation is a widely studied topic. Due to the nature of many prob lems, high-dimensional Spatio-temporal scenarios are considered frequently. In some cases, the true covariance matrices are also expected to have a Kronecker product-based representation. Especially in wind speed analysis, the true covariance matrix can be as sumed to have spatial and temporal Kronecker factors. This thesis studies Kronecker product-based covariance estimation models. Also, a new Kronecker product-based method is proposed with experimentation results showing its performance.
Langlands reciprocity principle for GL(n)
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2021., 2021.) Barbaros, Bahri Fatih.; İkeda, Kazım İlhan.
Langlands program is one of the most significant areas of research in modern mathematics as it asserts a connection between number theory, representation the ory and automorphic forms. Langlands program is basically a study to associate the representations of Galois groups over local and global fields with the automorphic rep resentations of algebraic groups over local fields and adeles, respectively. In this thesis, we particularly focus on the Langlands reciprocity principles and related conjectures for the general linear group GLn which are discussed in two parts: local Langlands correspondence and global Langlands reciprocity conjecture for GLn. We divide the local Langlands correspondence for GLn into two cases whether GLn is over a non archimedean or an archimedean local field. The non-archimedean case is proven thanks to Harris and Taylor [1] for the non-archimedean local fields of zero characteristic in 2001 and thanks to Laumon, Stuhler, Rapoport [2] for those of finite characteristic in 1993. The proof of the archimedean case is given thanks to Langlands [3]. Local Langlands correspondence states a well-defined one to one correspondence of the set of equivalence classes of n-dimensional (Frobenius) semisimple complex Weil-Deligne representations of the Weil group WF of F with the set of equivalence classes of irre ducible admissible representations of GLn(F) where F is a local field. The local L and "-factors attached to these certain representations of WF and of GLn(F) are preserved under this correspondence and the preservations are called the naturality properties. In the local part, our objective is to grasp this correspondence at best. For the global part, we respectively discuss the adele ring AK of a global field K, global class field the ory of K, automorphic and cuspidal representations of GLn(AK), global automorphic L and "-factors and hypothetical Langlands group of K in order to state the global Langlands reciprocity conjecture.
Negative dependence in discrete probability :|notions, models and consequences
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2021., 2021.) Çiçeksiz, Recep Altar.; Ravichandran, Mohan.
In this thesis we study the negative dependence properties of random cluster measure. First, we introduce the notions of negative correlation and how they relate to each other. We observe that this important notion is difficult to confirm. The uniform spanning tree measure and the uniform spanning forest measure is the measures that we mainly focus as the crucial examples that shows negative dependence properties. Later, we focus on the properties of the random cluster measure which generalizes the uniform spanning tree and uniform spanning forest measures. We prove negative edge dependence of the random cluster measure on the complete graph, giving a partial solution to a well known conjecture of Grimmett, Winkler and Wagner. We than consider a natural problem concerning correlations between collection of connectivity events in graphs with respect to the random cluster measure and relate this natural problem to the Grimmett, Winkler and Wagner conjecture.
Stationary distributions and convergence rates for the edge flipping process
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2021., 2021.) Demirci, Yunus Emre.; Işlak, Ümit.
The edge flipping process is a random walk over the set of all possible color patterns of a graph. Each time the two endpoints of the selected edge are colored the same color. This color is blue with probability p and red with probability 1 − p. In the vertex flipping process, we choose vertices instead of edges. All the neighbors of the selected vertex and itself are colored in the same color. The eigenvalues of this random walk are indexed by all subsets of the vertices of the graph. Thanks to this indexing, we have obtained information about eigenvalues and, as a result, converge rates in the graph classes we are working on. In some simple graph classes such as complete bipartite graph and caterpillar tree, we have obtained results related to where this random walk converges after a while. In general, we are looking for answers to the two questions: where we converge and how fast it occurs.
Asymptotically anti-de Sitter spacetimes in three dimensions
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2021., 2021.) Deral, Ceren Ayşe.; Değer, Nihat Sadık.
In this thesis, we reviewed several aspects of asymptotically anti-de Sitter (AAdS) spacetimes in three dimensional Einstein gravity by following some important historical work. Starting with a brief introduction to anti-de Sitter (AdS) spacetimes where also the BTZ black hole solution is given we de ned Noether-Wald charges using Noether theorems. Next, we compared di erent de nitions of AAdS spacetimes. Here, we adopted the Fefferman-Graham coordinates and solved Einstein equations order by order to prove that the Fefferman-Graham expansion of AAdS spacetimes terminates at second order in three dimensions, as first shown by Skenderis and Solodukhin. Lastly, we considered two sets of boundary conditions and presented their asymptotic symmetry algebras and charge algebras. Imposing Brown-Henneaux boundary conditions we arrived at Banados metric, which is the most general metric for AAdS spacetimes under these conditions. Then we showed that the asymptotic symmetry algebra is two copies of the Virasoro algebra. Under the Compere-Song-Strominger boundary conditions, we calculated the most general metric and showed the charge algebra is a semidirect sum of Virasoro and Kac-Moody algebras. We concluded with some comments and future research directions.
Relative endo-trivial modules
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2021., 2021.) Budak, Doğa Ulaş.; Coşkun, Olcay.
We first develop the necessary tools from the theory of G-algebras and endomorphism rings. Then the basic results of Green's theory of vertices and sources for kG-modules is given. Building on these, we retrace the steps of Lassueur and introduce the group of relative endo-trivial modules, which gives a generalization of the Dade group. Lastly, we introduce a related group in terms of a specific type of algebras. We describe the Jacobson radical of such algebras.
Supersymmetric non-linear sigma models in D=2+1 dimensional spacetime
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2021., 2021.) Bardavid, Metin.; Değer, Nihat Sadık.
In this thesis we study the theory of Supersymmetric Non-Linear a-Models. We first review the free scalar field theory which can be considered as a Linear a-model. We then discuss the Non-Linear a-model as well as the symmetries of the model with an emphasis on the role of Killing vector fields. After that we turn our attention to study the possible target spaces of these models; these being Kahler and homogeneous manifolds. Finally we introduce the 3-dimensional Non-Linear a-Model with N = 1, 2 and 4 supersymmetries. We show that the D = 3 and N = 2 NLaM has to admit a Kahler target manifold while D = 3 and N = 4 model has to admit a HyperKahler target manifold.
Variational methods for nonlinear elliptic partial differential equations with nonlocal terms
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2007., 2007.) Topaloğlu, İhsan Ata.; Eden, Alp,
In this thesis, existence of standing waves for the DaveyStewartson (DS) and generalized DaveyStewartson (GDS) systems are established using variational methods. Since both the DS system and the GDS system reduce to a non-linear Schr¨odinger (NLS) equation with the only difference in their non-local term, arguments used in this thesis apply to a larger class of equations which include the DS and GDS systems as special cases. Existence of standing waves for an NLS equation is investigated in two ways: by considering an unconstrained minimization problem and a constrained minimization problem. These two variational methods apply to the GDS system as well and here the sufficient conditions on the existence of standing wave solutions for the GDS system which are imposed by these methods and the minimizers obtained are investigated in comparison.
Central-firing of type A2n with initial weight 0
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2020., 2020.) Cin, Uğur.; Boysal, Arzu.
In this thesis a variant of the chip- ring game introduced by Hopkins, McConville and Propp in [1], called the labeled chip- ring on Z, is studied. We will rst explore the basic properties and examples of this game. We will then show, how one can see this game as a binary relation on the weight lattice of Type A root system. It is then a natural step to generalize it to other root systems, which was done by Galashin, Hopkins, McConville and Postnikov in [2] and [3]. After reviewing the basics of central- ring introduced in these papers, we examine the central- ring of type A2n with initial weight 0 in Chapter 5. Finally, we study the restrictions in Lemma 12 of [1] in more detail, and conjecture that the number of permutations with maximum number of inversions allowed by this lemma is given by the Catalan numbers.
On online and approximate cover time problems
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2020., 2020.) Yıldız, Mehmet Akif.; Işlak, Ümit.
As a generalization of the classical coupon collector problem in the probability theory, the cover time in random walks on Markov chains has been investigated in numerous studies in the literature. Especially, there are several results for the cover time of a simple random walk on connected and undirected graphs. In this thesis, we study two new problems about the cover time of graphs. Firstly, we build an on-line model where there is a walker moving with random time intervals on a graph growing in time. We initiate this study by examining the number of vertices covered up to a fixed time for a simple model, and we discuss further research directions. Secondly, we generalize the classical cover time definition in order to understand the differences between the partial covering and the full covering. We initiate this study with the investigation of the approximate covering time on specific graph families such as path graphs and complete graphs, and our main motivation is to explore the structure of the graphs allowing easy partial covering in terms of the order of magnitude. For the sake of completeness, we also give some preliminary results about the classical cover time problem and several variations of the problem from the literature such as edge covering and dynamic versions in the thesis.
Algebraic overtwised contact structures on 3-sphere
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2020., 2020.) Karadereli, Şeyma.; Öztürk, Ferit.
It is known that all of the complex analytic singularity links and the associated Milnor open books on the 3-sphere correspond to a single contact structure, which is the unique tight structure of the 3-sphere. The main question of this thesis is whether the overtwisted contact structures on the 3-sphere are real algebraic. We will de ne the notion of real algebraicity in the introduction of the thesis. We explicitly construct a family of real algebraic multilinks in the 3-sphere which are the bindings of planar Milnor open book decompositions supporting overtwisted contact structures. Furthermore, we prove that all the overtwisted contact structures with non-negative 3-dimensional invariants are obtained in this family.
Complex representations of finite general linear groups
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2019., 2019.) Küçük, Ebru Beyza.; Coşkun, Olcay.
In this thesis, we determine complex irreducible representations of GL(2,K), the group of 2 by 2 invertible matrices over a ﬁnite ﬁeld K. Actually, this is done by Ilya Piatetski-Shapiro in 1983. In his article [1], Shapiro classiﬁes the irreducible representations of the group GL(2,K) by using the deﬁnition of induced module de pends as a space of functions. The aim of this thesis is to rewrite the article using the induction module deﬁnition constructed by a tensor product. We start the thesis by reminding some basic deﬁnitions and theorems related to our topic. Then we de termine the commutator subgroup of GL(2,K) and introduce some special subgroups of GL(2,K). The number of irreducible representations of a ﬁnite group is equal to the number of conjugacy classes of that group. Hence we calculate the conjugacy classes of GL(2,K). We determine irreducible representations of GL(2,K) through irreducible representations of the subgroups of it and quotient groups.
ABC conjecture and its implications
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2019., 2019.) Koç, Tuğçe.; Özman, Ekin.
In this thesis, our aim is to state the importance of abc conjecture and prove the strong results we obtain with the help of abc conjecture. First, we give necessary notions and tools which are used throughout the thesis. Then we introduce Hall conjecture, Fermat’s last theorem and Mordell conjecture, and their relations with abc conjecture. In particular, we give the eﬀective proof of Mordell conjecture using abc conjecture, given in the article of Noam Elkies, [1], and also get another height bound by combining with a diﬀerent theorem. Finally, we give three examples where we use both of the height bounds. This thesis was supported by T¨UB˙ITAK Project 117F274.

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