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Item Open Access 2-fold structures and homotopy theory(Bilkent University, 2023-01) Haderi, RediShow more It is well-known that correspondences between categories, also known as profunctors, serve in classifying functors. More precisely, every functor F : X → A straightens into a lax mapping χF : A → Catprof from A into a 2-category of categories and profunctors ([45]). We give a conceptual treatment of this fact from the lens of double category theory, contending the latter to be most natural environment to express this result. Then we venture into the world of simplicial sets and prove an analogous theorem. The notion of correspondence is easy to extend to simplicial sets, but a suitable double category may not be formed due to the lack of a natural tensor product. Nonetheless, we show that there is a natural simplicial category structure once we invoke higher correspondences. In proving our result we extend some notions from double category theory into the world of simplicial categories. As an application we obtain a realization of Lurie’s prediction that inner fibrations are classified by mappings into a higher category of correspondences between ∞-categories.Show more Item Open Access Alexander modules of trigonal curves(Bilkent University, 2021-01) Üçer, MelihShow more We classify the monodromy Alexander modules of non-isotrivial trigonal curves.Show more Item Open Access Asymptotics of extremal polynomials for some special cases(Bilkent University, 2017-05) Alpan, GökalpShow more We study the asymptotics of orthogonal and Chebyshev polynomials on fractals. We consider generalized Julia sets in the sense of Br uck-B uger and weakly equilibrium Cantor sets which was introduced in [62]. We give characterizations for Parreau-Widom condition and optimal smoothness of the Green function for the weakly equilibrium Cantor sets. We also show that, for small parameters, the corresponding Hausdor measure and the equilibrium measure of a set from this family are mutually absolutely continuous. We prove that the sequence of Widom-Hilbert factors for the equilibrium measure of a non-polar compact subset of R is bounded below by 1. We give a su cient condition for this sequence to be unbounded above. We suggest de nitions for the Szeg}o class and the isospectral torus for a generic subset of RShow more Item Open Access Bases in banach spaces of smooth functions on cantor-type sets(Bilkent University, 2013) Özfidan, NecipShow more We construct Schauder bases in the spaces of continuous functions C p (K) and in the Whitney spaces E p (K) where K is a Cantor-type set. Here different Cantortype sets are considered. In the construction, local Taylor expansions of functions are used. Also we show that the Schauder basis which we constructed in the space Cp(K), is conditional.Show more Item Open Access Canonical induction, Green functors, lefschetz invariant of monomial G-posets(Bilkent University, 2019-06) Mutlu, HaticeShow more Green functors are a kind of group functor, rather like Mackey functors, but with a further multiplicative structure. They are defined on a category whose objects are finite groups and whose morphisms are generated by maps such as induction, restriction, inflation, deflation. The aim of this thesis is general formulation for canonical induction, suitable for Green functors, optionally equipped with inflations. Let p be a prime number. In Section 3, we apply the Boltje’s theory of canonical induction [1] to p-permutation modules and give a restriction-preserving Z[1/p]- linear canonical induction formula from the inflations of projective modules. In Section 4, we give a general formulation of canonical induction theory for Green biset functors equipped with induction, restriction, inflation maps. Let G be a finite group and C be an abelian group. In Section 5, motivated in part by a search for connection with Peter Symonds’ proof [2] of the integrality of a canonical induction formula, we introduce a Lefschetz invariant for the Cmonomial Burnside ring. These invariants let us to construct generalize tensor induction functors associated to any C-monomial (G, H)-biset from the category of C-monomial G-posets to the category of C-monomial H-posets. We will show that these functors induce well-defined tensor induction maps from BC(G) to BC(H), which in turn gives a group homomorphism BC(G) × → BC(H) × between the unit groups of C-monomial Burnside rings.Show more Item Open Access Character sums, algebraic function fields, curves with many rational points and geometric Goppa codes(Bilkent University, 1997) Özbudak, FerruhShow more In this thesis we have found and studied fibre products of hyperelliptic and superelliptic curves with many rational points over finite fields. We have applied Goppa construction to these curves to get “good” linear codes. We have also found a nontrivial connection between configurations of affine lines in the affine plane over finite fields and fibre products of Rummer extensions giving “good” codes over F,2. Moreover we have calculated an important parameter of a class of towers of algebraic function fields over finite fields, which are studied recently.Show more Item Open Access Characteristic lie algebra and classification of semi-discrete models(Bilkent University, 2009) Pekcan, AslıShow more In this thesis, we studied a differential-difference equation of the following form tx(n + 1, x) = f(t(n, x), t(n + 1, x), tx(n, x)), (1) where the unknown t = t(n, x) is a function of two independent variables: discrete n and continuous x. The equation (1) is called a Darboux integrable equation if it admits nontrivial x- and n-integrals. A function F(x, t, t±1, t±2, ...) is called an x-integral if DxF = 0, where Dx is the operator of total differentiation with respect to x. A function I(x, t, tx, txx, ...) is called an n-integral if DI = I, where D is the shift operator: Dh(n) = h(n + 1). In this work, we introduced the notion of characteristic Lie algebra for semidiscrete hyperbolic type equations. We used characteristic Lie algebra as a tool to classify Darboux integrability chains and finally gave the complete list of Darboux integrable equations in the case when the function f in the equation (1) is of the special form f = tx(n, x) + d(t(n, x), t(n + 1, x)).Show more Item Open Access Code construction on modular curves(Bilkent University, 2003) Kara, OrhunShow more In this thesis, we have introduced two approaches on code construction on modular curves and stated the problems step by step. Moreover, we have given solutions of some problems in road map of code construction. One of the approaches uses mostly geometric and algebraic tools. This approach studies local invariants of the plane model Z0(`) of the modular curve Y0(`) given by the modular equation Φ` in affine coordinates. The approach is based on describing the hyperplane of regular differentials of Z0(`) vanishing at a given Fp 2 rational point. As constructing a basis for the regular differentials of Z0(`), we need to investigate its singularities. We have described the singularities of Z0(`) for prime ` in both characteristic 0 and positive characteristic. We have shown that all singularities of of the affine part, Z0(`), are self intersections. These self intersections are all simple nodes in characteristic 0 whereas the order of contact of any two smooth branches passing though a singular point may be arbitrarily large in characteristic p > 3 where p 6= `. Moreover the self intersections in characteristic zero are double. Indeed, structure of singularities of the affine curve Z0(`) essentially depends on two types of elliptic curves: The singularities corresponding to ordinary elliptic curves and the singularities corresponding to supersingular elliptic curves. The singularities corresponding to ordinary elliptic curves are all double points even though they are not necessarily simple nodes as in the case of characteristic 0. The singularities corresponding to supersingular elliptic curves are the most complicated ones and it may happen that there are more then two smooth branches passing though such kind of a singular point. We have computed the order of contact of any two smooth branches passing though a singular point both for ordinary case and for supersingular case.We have also proved that two points of Z0(`) at ∞ are cusps for odd prime ` which are analytically equivalent to the cusp of 0, given by the equation x ` = y `−1 . These two cusps are permuted by Atkin-Lehner involution. The multiplicity of singularity of each cusp is (`−1)(`−2) 2 . This result is valid in any characteristic p 6= 2, 3. The second approach is based on describing the Goppa codes on modular curve Y (`) as P SL2(F`) module. The main problem in this approach is investigating the structure of a group code as P SL2(F`) module. We propose a way of computing the characters of representations of a group code by using the localization formula. Moreover, we give an example of computing the characters of the code which associated to a canonical divisor on Y (`).Show more Item Open Access Codes on fibre products of Artin-Schreier and Kummer coverings of the projective line(Bilkent University, 2002-08) Shalalfeh, MahmoudShow more In this thesis, we study smooth projective absolutely irreducible curves defined over finite fields by fibre products of Artin-Schreier and Kummer coverings of the projective line. We construct some curves with many rational points defined by the fibre products of Artin-Schreier and Kummer coverings. Then, we apply Goppa construction to the curves that we have found, and obtain long linear codes with good relative parameters.Show more Item Open Access Cohomology of infinite groups realizing fusion systems(Bilkent University, 2019-09) Gündoğan, Muhammed SaidShow more Given a fusion system F defined on a p-group S, there exist infinite group models, constructed by Leary and Stancu, and Robinson, that realize F. We study these models when F is a fusion system of a finite group G. If the fusion system is given by a finite group, then it is known that the cohomology of the fusion system and the Fp-cohomology of the group are the same. However, this is not true in general when the group is infinite. For the fusion system F given by finite group G, the first main result gives a formula for the difference between the cohomology of an infinite group model realizing the fusion F and the cohomology of the fusion system. The second main result gives an infinite family of examples for which the cohomology of the infinite group obtained by using the Robinson model is different from the cohomology of the fusion system. The third main result gives a new method for the realizing fusion system of a finite group acting on a graph. We apply this method to the case where the group has p-rank 2, in which case the cohomology ring of the fusion system is isomorphic to the cohomology of the group.Show more Item Open Access Complete intersection monomial curves and non-decreasing Hilbert functions(Bilkent University, 2008) Şahin, MesutShow more In this thesis, we first study the problem of determining set theoretic complete intersection (s.t.c.i.) projective monomial curves. We are also interested in finding the equations of the hypersurfaces on which the monomial curve lie as set theoretic complete intersection. We find these equations for symmetric Arithmetically Cohen-Macaulay monomial curves. We describe a method to produce infinitely many s.t.c.i. monomial curves in P n+1 starting from one single s.t.c.i. monomial curve in P n . Our approach has the side novelty of describing explicitly the equations of hypersurfaces on which these new monomial curves lie as s.t.c.i.. On the other hand, semigroup gluing being one of the most popular techniques of recent research, we develop numerical criteria to determine when these new curves can or cannot be obtained via gluing. Finally, by using the technique of gluing semigroups, we give infinitely many new families of affine monomial curves in arbitrary dimensions with CohenMacaulay tangent cones. This gives rise to large families of 1-dimensional local rings with arbitrary embedding dimensions and having non-decreasing Hilbert functions. We also construct infinitely many affine monomial curves in A n+1 whose tangent cone is not Cohen Macaulay and whose Hilbert function is nondecreasing from a single monomial curve in A n with the same property.Show more Item Open Access Concrete sheaves and continuous spaces(Bilkent University, 2015) Özkan RecepShow more In algebraic topology and differential geometry, most categories lack some good ”convenient” properties like being cartesian closed, having pullbacks, pushouts, limits, colimits... We will introduce the notion of continuous spaces which is more general than the concept of topological manifolds but more specific when compared to topological spaces. After that, it will be shown that the category of continuous spaces have ”convenient” properties we seek. For this, we first define concrete sites, concrete sheaves and say that a generalized space is a concrete sheaf over a given concrete site. Then it will be proved that a category of generalized spaces (for a given concrete site) has all limits and colimits. At the end, it will be proved that the category of continuous spaces is actually equivalent to the category of generalized spaces for a specific concrete site.Show more Item Open Access A conjecture on square-zero upper triangular matrices and Carlsson's rank conjecture(Bilkent University, 2018-09) Şentürk, BerrinShow more A well-known conjecture states that if an elementary abelian p-group acts freely on a product of spheres, then the rank of the group is at most the number of spheres in the product. Carlsson gives an algebraic version of this conjecture by considering a di erential graded module M over the polynomial ring A in r variables: If the homology of M is nontrivial and nite dimensional over the ground eld, then N := dimAM is at least 2r. In this thesis, we state a stronger conjecture concerning varieties of square-zero upper triangular N N matrices with entries in A. By stratifying these varieties via Borel orbits, we show that the stronger conjecture holds when N < 8 or r < 3. As a consequence, we obtain a new proof for many of the known cases of Carlsson's conjecture as well as novel results for N > 4 and r = 2.Show more Item Open Access A correspondence of simple alcahestic group functors(Bilkent University, 2008) Coşkun, OlcayShow more Representation theory of finite groups associates two classical constructions to a group G, namely the representation ring of G and the Burnside ring of G. These rings share a special structure that comes from three classical maps, namely restriction, conjugation, and transfer maps. These are not the only objects having this structure and the theory of Mackey functors, introduced by Green, unifies the treatment of such objects. The above constructions share a further structure that comes from two other maps, the inflation map and the deflation map. Unified treatment of the objects having this further structure was introduced by Bouc [4]. These objects are called biset functors. Between Mackey functors and biset functors there lies more natural constructions, for example the functor of group (co)homology. In order to handle these intermediate structures, Bouc introduced another concept, now known as globallydefined Mackey functors, a name given by Webb. In this thesis, we unify the above theories by considering the algebra whose module category is equivalent to the category of biset functors and by introducing alcahestic group functors. Our main results classify and describe simple alcahestic group functors and give a criterion of semisimplicity for the categories of these functors.Show more Item Open Access Deformation classes of singular quartic surfaces(Bilkent University, 2016-12) Aktaş, Çisem GüneşShow more We study complex spatial quartic surfaces with simple singularities and give their classication up to equisingular deformation. Simple quartics are K3-surfaces and as such they can be studied by means of the global Torelli theorem and the surjectivity of the period map combined with Nikulin's theory of discriminant forms. We reduce the classification problem to a certain arithmetical problem concerning lattice extensions. Then, based on Nikulin's existence criterion, we list all sets of simple singularities realized by non-special quartics; the result is stated in terms of perturbations of a few extremal sets. For each realizable set of singularities, we use Miranda{Morrison's theory to give a complete description of the connected components of the corresponding equisingular stratum.Show more Item Open Access Deformations of some biset-theoretic categories(Bilkent University, 2020-09) Öğüt, İsmail AlperenShow more We define the subgroup category, a category on the class of finite groups where the morphisms are given by the subgroups of the direct products and the composition is the star product. We also introduce some of its deformations and provide a criteria for their semisimplicity. We show that biset category can be realized as an invariant subcategory of the subgroup category, where the composition is much simpler. With this correspondence, we obtain some of the deformations of the biset category. We further our methods to the fibred biset category by introducing the subcharacter partial category. Similarly, we also realize the fibred biset category and some of its deformations in a category where the composition is more easily described.Show more Item Open Access Dilations of doubly invariant kernels valued in topologically ordered *- spaces(Bilkent University, 2018-06) Ay, SerdarShow more An ordered *-space Z is a complex vector space with a conjugate linear involution *, and a strict cone Z+ consisting of self adjoint elements. A topologically ordered *-space is an ordered *-space with a locally convex topology compatible with its natural ordering. A VE (Vector Euclidean) space, in the sense of Loynes, is a complex vector space equipped with an inner product taking values in an ordered *-space, and a VH (Vector Hilbert) space, in the sense of Loynes, is a VE-space with its inner product valued in a complete topologically ordered *-space and such that its induced locally convex topology is complete. On the other hand, dilation type theorems are important results that often realize a map valued in a certain space as a part of some simpler elements on a bigger space. Dilation results today are of an extraordinary large diversity and it is a natural question whether most of them can be uni*ed under general theorems. We study dilations of weakly positive semide*nite kernels valued in (topologically) ordered *-spaces, which are invariant under left actions of *-semigroups and right actions of semigroups, called doubly invariant. We obtain VE and VHspaces linearisations of such kernels, and on equal foot, their reproducing kernel spaces, and operator representations of the acting semigroups. The main results are used to unify many of the known dilation theorems for invariant positive semide*nite kernels with operator values, also for kernels valued in certain algebras, as well as to obtain some new dilation type results, in the context of Hilbert C*-modules, locally Hilbert C*-modules and VH-spaces.Show more Item Open Access Distance between a maximum point and the zero set of an entire function(Bilkent University, 2006) Üreyen, Adem ErsinShow more We obtain asymptotical bounds from below for the distance between a maximum modulus point and the zero set of an entire function. Known bounds (Macintyre, 1938) are more precise, but they are valid only for some maximum modulus points. Our bounds are valid for all maximum modulus points and moreover, up to a constant factor, they are unimprovable. We consider entire functions of regular growth and obtain better bounds for these functions. We separately study the functions which have very slow growth. We show that the growth of these functions can not be very regular and obtain precise bounds for their growth irregularity. Our bounds are expressed in terms of some smooth majorants of the growth function. These majorants are defined by using orders, types, (strong) proximate orders of entire functions.Show more Item Open Access Essays in collective decision making(Bilkent University, 2014-10) Derya, Ayşe MutluShow more Four different problems in collective decision making are studied, all of which are either formulated directly in a game-theoretical context or are concerned with neighboring research areas. The rst two problems fall into the realm of cooperative game theory. In the first one, a decomposition of transferable utility games is introduced. Based on that decomposition, the structure of the set of all transferable utility games is analyzed. Using the decomposition and the notion of minimal balanced collections, a set of necessary and sufficient conditions for a transferable utility game to have a singleton core is given. Then, core selective allocation rules that, when confronted with a change in total cost, not only distribute the initial cost in the same manner as before, but also treat the remainder in a consistent way are studied. Core selective rules which own a particular kind of additivity that turns out to be relevant in this context are also characterized. In the second problem, different notions of merge proofness for allocation rules pertaining to transferable utility games are introduced. Relations between these merge proofness notions are studied, and some impossibility as well as possibility results for allocation rules are established, which are also extended to allocation correspondences. The third problem deals with networks. A characterization of the Myerson value with two axioms is provided. The first axiom considers a situation where there is a change in the value function at a network g along with all networks containing g. At such a situation, the axiom requires that this change is to be divided equally between all the players in g who are not isolated. The second axiom requires that if the value function assigns zero to each network, then each player gets zero payo at each network. Modifying the rst axiom, along a characterization of the Myerson value, a characterization of the position value is also provided. Finally, the fourth problem is concerned with social choice theory which deals with collective decision making in a society. A characterization of the Borda rule for a given set of alternatives with a variable number of voters is studied on the domain of weak preferences, where indi erences between alternatives are allowed at agents' preferences. A new property, which we refer to as degree equality, is introduced. A social choice rule satis es degree equality if and only if, for any two pro les of two nite sets of voters, equality between the sums of the degrees of every alternative under these two pro les implies that the same alternatives get chosen at both of them. The Borda rule is characterized by the conjunction of faithfulness, reinforcement, and degree equality on the domain of weak preferences.Show more Item Open Access Essential cohomology and relative cohomology of finite groups(Bilkent University, 2009) Aksu, Fatma AltunbulakShow more In this thesis, we study mod-p essential cohomology of finite p-groups. One of the most important problems on essential cohomology of finite p-groups is finding a group theoretic characterization of p-groups whose essential cohomology is non-zero. This is an open problem introduced in [22]. We relate this problem to relative cohomology. Using relative cohomology with respect to the collection of maximal subgroups of the group, we define relative essential cohomology. We prove that the relative essential cohomology lies in the ideal generated by the essential classes which are the inflations of the essential classes of an elementary abelian p-group. To determine the relative essential cohomology, we calculate the essential cohomology of an elementary abelian p-group. We give a complete treatment of the module structure of it over a certain polynomial subalgebra. Moreover we determine the ideal structure completely. In [17], Carlson conjectures that the essential cohomology of a finite group is finitely generated and is free over a certain polynomial subalgebra. We also prove that Carlson’s conjecture is true for elementary abelian p-groups. Finally, we define inflated essential cohomology and in the case p > 2, we prove that for non-abelian p-groups of exponent p, inflated essential cohomology is zero. This also shows that for those groups, relative essential cohomology is zero. This result gives a partial answer to a particular case of the open problem in [22].Show more

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