Dept. of Mathematics - Ph.D. / Sc.D.

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  • ItemOpen Access
    2-fold structures and homotopy theory
    (Bilkent University, 2023-01) Haderi, Redi
    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.
  • ItemOpen Access
    Alexander modules of trigonal curves
    (Bilkent University, 2021-01) Üçer, Melih
    We classify the monodromy Alexander modules of non-isotrivial trigonal curves.
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    Deformations of some biset-theoretic categories
    (Bilkent University, 2020-09) Öğüt, İsmail Alperen
    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.
  • ItemOpen Access
    Generic initial ideals of modular polynomial invariants
    (Bilkent University, 2020-07) Danış, Bekir
    We study the generic initial ideals (gin) of certain ideals that arise in modular invariant theory. For all the cases where an explicit generating set is known, we calculate the generic initial ideal of the Hilbert ideal of a cyclic group of prime order for all monomial orders. We also clarify gin for the Klein four group and note that its Hilbert ideals are Borel fixed with certain orderings of the variables. In all the situations we consider, there is a monomial order such that the gin of the Hilbert ideal is equal to its initial ideal. Along the way we show that gin respects a permutation of the variables in the monomial order.
  • ItemOpen Access
    Cohomology of infinite groups realizing fusion systems
    (Bilkent University, 2019-09) Gündoğan, Muhammed Said
    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.
  • ItemOpen Access
    Canonical induction, Green functors, lefschetz invariant of monomial G-posets
    (Bilkent University, 2019-06) Mutlu, Hatice
    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.
  • ItemOpen Access
    A conjecture on square-zero upper triangular matrices and Carlsson's rank conjecture
    (Bilkent University, 2018-09) Şentürk, Berrin
    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.
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    Dilations of doubly invariant kernels valued in topologically ordered *- spaces
    (Bilkent University, 2018-06) Ay, Serdar
    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.
  • ItemOpen Access
    Codes on fibre products of Artin-Schreier and Kummer coverings of the projective line
    (Bilkent University, 2002-08) Shalalfeh, Mahmoud
    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.
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    Asymptotics of extremal polynomials for some special cases
    (Bilkent University, 2017-05) Alpan, Gökalp
    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 R
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    Extension problem and bases for spaces of infinitely differentiable functions
    (Bilkent University, 2017-04) Merpez, Zeliha Ural
    We examine the Mityagin problem: how to characterize the extension property in geometric terms. We start with three methods of extension for the spaces of Whitney functions. One of the methods was suggested by B. S. Mityagin: to extend individually the elements of a topological basis. For the spaces of Whitney functions on Cantor sets K( ), which were introduced by A. Goncharov, we construct topological bases. When the set K( ) has the extension property, we construct a linear continuous extension operator by means of suitable individual extensions of basis elements. Moreover, we use local Newton interpolations to contruct an extension operator. In the end, we show that for the spaces of Whitney functions, there is no complete characterization of the extension property in terms of Hausdorff measures or growth of Markov's factors.
  • ItemOpen Access
    Monoid actions, their categorification and applications
    (Bilkent University, 2016-12) Erdal, Mehmet Akif
    We study actions of monoids and monoidal categories, and their relations with (co)homology theories. We start by discussing actions of monoids via bi-actions. We show that there is a well-defined functorial reverse action when a monoid action is given, which corresponds to acting by the inverses for group actions. We use this reverse actions to construct a homotopical structure on the category of monoid actions, which allow us to build the Burnside ring of a monoid. Then, we study categorifications of the previously introduced notions. In particular, we study actions of monoidal categories on categories and show that the ideas of action reversing of monoid actions extends to actions of monoidal categories. We use the reverse action for actions of monoidal categories, along with homotopy theory, to define homology, cohomology, homotopy and cohomotopy theories graded over monoidal categories. We show that most of the existing theories fits into our setting; and thus, we unify the existing definitions of these theories. Finally, we construct the spectral sequences for the theories graded over monoidal categories, which are the strongest tools for computation of cohomology and homotopy theories in existence.
  • ItemOpen Access
    Deformation classes of singular quartic surfaces
    (Bilkent University, 2016-12) Aktaş, Çisem Güneş
    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.
  • ItemOpen Access
    Concrete sheaves and continuous spaces
    (Bilkent University, 2015) Özkan Recep
    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.
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    Distance between a maximum point and the zero set of an entire function
    (Bilkent University, 2006) Üreyen, Adem Ersin
    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.
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    Modular vector invariants
    (Bilkent University, 2006) Madran, Uğur
    Vector invariants of finite groups (see the introduction for definitions) provides, in general, counterexamples for many properties of the invariant theory when the characteristic of the ground field divides the group order. Noether number is such property. In this thesis, we improve a lower bound for Noether number given by Richman in 1996: namely, we give a lower bound depending on the Jordan canonical form of an element of order equal to characteristic of the field. This method yields an effective bound by means of simple arithmetic arguments. The results are valid for any faithful representation of the group, including reducible and irreducible ones. Also they are extended to any algebraic field extensions provided the characteristic of the field divides the group order.
  • ItemOpen Access
    Extension operators for spaces of infinitely differentiable functions
    (Bilkent University, 2005) Altun, Muhammed
    We start with a review of known linear continuous extension operators for the spaces of Whitney functions. The most general approach belongs to PawÃlucki and Ple´sniak. Their operator is continuous provided that the compact set, where the functions are defined, has Markov property. In this work, we examine some model compact sets having no Markov property, but where a linear continuous extension operator exists for the space of Whitney functions given on these sets. Using local interpolation of Whitney functions we can generalize the PawÃlucki-Ple´sniak extension operator. We also give an upper bound for the Green function of domains complementary to generalized Cantor-type sets, where the Green function does not have the H¨older continuity property. And, for spaces of Whitney functions given on multidimensional Cantor-type sets, we give the conditions for the existence and non-existence of a linear continuous extension operator.
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    Representations of functions harmonic in the upper half-plane and their applications
    (Bilkent University, 2003) Gergün, Seçil
    In this thesis, new conditions for the validity of a generalized Poisson representation for a function harmonic in the upper half-plane have been found. These conditions differ from known ones by weaker growth restrictions inside the halfplane and stronger restrictions on the behavior on the real axis. We applied our results in order to obtain some new factorization theorems in Hardy and Nevanlinna classes. As another application we obtained a criterion of belonging to the Hardy class up to an exponential factor. Finally, our results allowed us to extend the Titchmarsh convolution theorem to linearly independent measures with unbounded support.
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    Code construction on modular curves
    (Bilkent University, 2003) Kara, Orhun
    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 (`).
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    Recursion operator and dispersionless Lax representation
    (Bilkent University, 2002) Zheltukhin, Kostyantyn
    We give a general method for constructing recursion operators for some equations of hydrodynamic type, admitting a dispersionless Lax representation. We consider a polynomial and rational Lax function. We give several examples containing the equations of shallow water waves, polytropic gas dynamics and a degenerate bi-Hamiltonian system with a recursion operator. We also discuss a reduction of N + 1 systems to N systems of some new integrable equations of hydrodynamic type.