Dept. of Mathematics - Ph.D. / Sc.D.No Descriptionhttps://hdl.handle.net/11693/138982024-05-28T19:45:58Z2024-05-28T19:45:58Z4912-fold structures and homotopy theoryHaderi, Redihttps://hdl.handle.net/11693/1120132024-01-24T08:21:01Z2023-01-01T00:00:00Zdc.title: 2-fold structures and homotopy theory
dc.contributor.author: Haderi, Redi
dc.description.abstract: 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.
dc.description: Cataloged from PDF version of article.; Thesis (Ph.D.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2023.; Includes bibliographical references (leaves 128-131).
2023-01-01T00:00:00ZAlexander modules of trigonal curvesÜçer, Melihhttps://hdl.handle.net/11693/550422024-01-24T07:31:22Z2021-01-01T00:00:00Zdc.title: Alexander modules of trigonal curves
dc.contributor.author: Üçer, Melih
dc.description.abstract: We classify the monodromy Alexander modules of non-isotrivial trigonal curves.
dc.description: Cataloged from PDF version of article.; Thesis (Ph.D.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2021.; Includes bibliographical references (leaves 139-140) and index.
2021-01-01T00:00:00ZDeformations of some biset-theoretic categoriesÖğüt, İsmail Alperenhttps://hdl.handle.net/11693/540442024-01-24T07:23:17Z2020-09-01T00:00:00Zdc.title: Deformations of some biset-theoretic categories
dc.contributor.author: Öğüt, İsmail Alperen
dc.description.abstract: 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.
dc.description: Cataloged from PDF version of article.; Thesis (Ph.D.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2020.; Includes bibliographical references (leaves 48-49).
2020-09-01T00:00:00ZGeneric initial ideals of modular polynomial invariantsDanış, Bekirhttps://hdl.handle.net/11693/539272024-01-24T08:13:14Z2020-07-01T00:00:00Zdc.title: Generic initial ideals of modular polynomial invariants
dc.contributor.author: Danış, Bekir
dc.description.abstract: 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.
dc.description: Cataloged from PDF version of article.; Thesis (Ph.D.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2020.; Includes bibliographical references (leaves 40-41).
2020-07-01T00:00:00ZCohomology of infinite groups realizing fusion systemsGündoğan, Muhammed Saidhttps://hdl.handle.net/11693/524442024-01-24T08:22:54Z2019-09-01T00:00:00Zdc.title: Cohomology of infinite groups realizing fusion systems
dc.contributor.author: Gündoğan, Muhammed Said
dc.description.abstract: 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.
dc.description: Cataloged from PDF version of article.; Thesis (Ph.D.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2019.; Includes bibliographical references (leaves 66-68).
2019-09-01T00:00:00ZCanonical induction, Green functors, lefschetz invariant of monomial G-posetsMutlu, Haticehttps://hdl.handle.net/11693/521472024-01-24T07:23:16Z2019-06-01T00:00:00Zdc.title: Canonical induction, Green functors, lefschetz invariant of monomial G-posets
dc.contributor.author: Mutlu, Hatice
dc.description.abstract: 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.
dc.description: Cataloged from PDF version of article.; Thesis (Ph.D.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2019.; Includes bibliographical references (leaves (101-102).
2019-06-01T00:00:00ZA conjecture on square-zero upper triangular matrices and Carlsson's rank conjectureŞentürk, Berrinhttps://hdl.handle.net/11693/478862024-01-24T08:21:02Z2018-09-01T00:00:00Zdc.title: A conjecture on square-zero upper triangular matrices and Carlsson's rank conjecture
dc.contributor.author: Şentürk, Berrin
dc.description.abstract: 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.
dc.description: Cataloged from PDF version of article.; Thesis (Ph.D.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2018.; Includes bibliographical references (leaves 66-68).
2018-09-01T00:00:00ZDilations of doubly invariant kernels valued in topologically ordered *- spacesAy, Serdarhttps://hdl.handle.net/11693/477092024-01-24T07:38:03Z2018-06-01T00:00:00Zdc.title: Dilations of doubly invariant kernels valued in topologically ordered *- spaces
dc.contributor.author: Ay, Serdar
dc.description.abstract: 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.
dc.description: Cataloged from PDF version of article.; Thesis (Ph.D.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2018.; Includes bibliographical references (leaves 102-107).
2018-06-01T00:00:00ZCodes on fibre products of Artin-Schreier and Kummer coverings of the projective lineShalalfeh, Mahmoudhttps://hdl.handle.net/11693/356672024-01-24T08:14:29Z2002-08-01T00:00:00Zdc.title: Codes on fibre products of Artin-Schreier and Kummer coverings of the projective line
dc.contributor.author: Shalalfeh, Mahmoud
dc.description.abstract: 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.
dc.description: Cataloged from PDF version of article.; Thesis (Ph.D.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2002.; Includes bibliographical references (leaves 68-72).
2002-08-01T00:00:00ZAsymptotics of extremal polynomials for some special casesAlpan, Gökalphttps://hdl.handle.net/11693/331962024-01-24T07:38:32Z2017-05-01T00:00:00Zdc.title: Asymptotics of extremal polynomials for some special cases
dc.contributor.author: Alpan, Gökalp
dc.description.abstract: 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
dc.description: Cataloged from PDF version of article.; Thesis (Ph.D.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2017.; Includes bibliographical references (leaves 129-141).
2017-05-01T00:00:00Z