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This thesis is concerned with three distinct, but closely related, research topics focusing on the unipotent elements of a connected reductive algebraic group G, over an algebraically closed field k, and nilpotent elements in the Lie algebra g = LieG. The first topic is a determination of canonical forms for unipotent classes and nilpotent orbits of G. Using an original approach, we begin by obtaining a new canonical form for nilpotent matrices, up to similarity, which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi;j) -> (xn+1-j;n+1-i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group, we thus obtain a unified approach to computing representatives for nilpotent orbits for all classical groups G. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary, we obtain a complete set of generic canonical representatives for the unipotent classes of the finite general unitary groups GUn(Fq) for all prime powers q. Our second topic is concerned with unipotent pieces, defined by G. Lusztig in [Unipotent elements in small characteristic, Transform. Groups 10 (2005), 449-487]. We give a case-free proof of the conjectures of Lusztig from that paper. This presents a uniform picture of the unipotent elements of G, which can be viewed as an extension of the Dynkin-Kostant theory, but is valid without restriction on p. We also obtain analogous results for the adjoint action of G on its Lie algebra g and the coadjoint action of G on g*. We also obtain several general results about the Hesselink stratification and Fq-rational structures on G-modules. Our third topic is concerned with generalised Gelfand-Graev representations of finite groups of Lie type. Let u be a unipotent element in such a group GF and let Γu be the associated generalised Gelfand-Graev representation of GF . Under the assumption that G has a connected centre, we show that the dimension of the endomorphism algebra of Γu is a polynomial in q (the order of the associated finite field), with degree given by dimCG(u). When the centre of G is disconnected, it is impossible, in general, to parametrise the (isomorphism classes of) generalised Gelfand-Graev representations independently of q, unless one adopts a convention of considering separately various congruence classes of q. Subject to such a convention, we extend our result. We also present computational data related to the main theoretical results. In particular, tables of our canonical forms are given in the appendices, as well as tables of dimension polynomials for endomorphism algebras of generalised Gelfand-Graev representations, together with the relevant GAP source code.

This work constructs and compares different kinds of categorified cohomology of a locally contractible topological space X. Fix a commutative ring k of characteristic 0 and also denote by k the differential graded category with a single object and endomorphisms k. In the Morita model structure k is weakly equivalent to the category of perfect chain complexes over k. We define and compute derived global sections of the constant presheaf k considered as a presheaf of dg-categories with the Morita model structure. If k is a field this is done by showing there exists a suitable local model structure on presheaves of dg-categories and explicitly sheafifying constant presheaves. We call this categorified Cech cohomology Morita cohomology and show that it can be computed as a homotopy limit over a good (hyper)cover of the space X. We then prove a strictification result for dg-categories and deduce that under mild assumptions on X Morita cohomology is equivalent to the category of homotopy locally constant sheaves of k-complexes on X. We also show categorified Cech cohomology is equivalent to a category of ∞-local systems, which can be interpreted as categorified singular cohomology. We define this category in terms of the cotensor action of simplicial sets on the category of dg-categories. We then show ∞-local systems are equivalent to the category of dg-representations of chains on the loop space of X and find an explicit method of computation if X is a CW complex. We conclude with a number of examples.

A generalization of a theorem of N.R. 0'Brian, zeroes of holomorphic vector fields and the Grothendieck residue, Bull. London Math. Soc, 7 (1975) is given. The theorem of Riemann-Roch and Hirzebruch for V-equivariant holomorphic vector bundles is obtained, via holomorphic vector fields, in the case all zeroes of the holomorphic vector field V are isolated. The Bruhat decomposition of G/B is obtained from the G -action on G/B . It is shown that a theorem of A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. 98, 480-497 (1973) is the generalization of the Bruhat decomposition on G/B , which was a conjecture of B. Iversen. The existence of a G -action on G/P with only one fixed a point is proved, where G is a connected linear algebraic group defined over an algebraically closed field k of characteristic zero and P is a parabolic subgroup of G . The following is obtained P = N[sub G](Pu) = {geG: Adg(Pu) = Pu} where G is a connected linear algebraic group, P is a parabolic subgroup of G and P^ is the tangent space of the set of unipotent elements of P at the identity. An elementary proof of P = N[sub G](P) = {geG: gPg ⁻¹=P} is given, where G is a connected linear algebraic group and P is a parabolic subgroup of G . / Science, Faculty of / Mathematics, Department of / Graduate

The affine Grassmannian associated to a reductive group G is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. In this work, we study their quantum connection. We use the stable envelopes of D. Maulik and A. Okounkov[MO2] to write an explicit formula for this connection. In order to do this, we construct a recursive relation for the stable envelopes in the G = PSL_2 case and compute the first-order correction in the general case. The computation of the purely quantum part of the multiplication is done based on the deformation approach of A. Braverman, D. Maulik and A. Okounkov[BMO]. For the case of simply-laced G, we identify the quantum connection with the trigonometric Knizhnik-Zamolodchikov equation for the Langlands dual group G^\vee.

"A topological ring can be defined as a Hausdorff space which is a ring whose operations are continuous in both variables simultaneously. The purpose of this paper is to present a development of a branch of topological rings, normed rings. Chapter I presents the algebraic concepts and Chapter II, the topological concepts which are basic to the understanding of the remaining chapters. Chapter III is devoted to theory of Banach spaces and linear transformations. Chapter IV gives a treatment of normed rings, particularly the rings of continuous functions of a compact topological space. The latter part of this chapter presents a characterization of these rings"--Introduction. / Typescript. / "August, 1955." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Advisor: M. J. Walsh, Professor Directing Paper. / Includes bibliographical references (leaf 51).

In this thesis we solve three problems about derived categories of algebraic varieties: We prove the conjecture [EL21, Conjecture 4.13] of Elagin and Lunts; we positively answer a question raised by the conjecture [Orl09, Conjecture 10] of Orlov, proving new cases of that conjecture in the process; and we extend Orlov's theorem [Orl97, Theorem 2.2] from smooth projective varieties to smooth proper algebraic spaces. These results go toward answering the questions: How rigid is the (triangulated) derived category of coherent sheaves on an algebraic variety, and how much information does it possess about the variety? Our techniques are general and work for algebraic spaces just as well as they do for projective varieties.

<p>Unlike matrix rank, hypermatrix rank is not lower semi-continuous. As a result, optimal low rank approximations of hypermatrices may not exist. Characterizing hypermatrices without optimal low rank approximations is an important step in implementing algorithms with hypermatrices. The main result of this thesis is an original coordinate-free proof that real 2 by 2 by 2 tensors that are rank three do not have optimal rank two approximations with respect to the Frobenius norm. This result was previously only proved in coordinates. Our coordinate-free proof expands on prior results by developing a proof method that can be generalized more readily to higher dimensional tensor spaces. Our proof has the corollary that the nearest point of a rank three tensor to the second secant set of the Segre variety is a rank three tensor in the tangent space of the Segre variety. The relationship between the contraction maps of a tensor generalizes, in a coordinate-free way, the fundamental relationship between the rows and columns of a matrix to hypermatrices. Our proof method demonstrates geometrically the fundamental relationship between the contraction maps of a tensor. For example, we show that a regular real or complex tensor is tangent to the 2 by 2 by 2 Segre variety if and only if the image of any of its contraction maps is tangent to the 2 by 2 Segre variety. </p>

A dissertation submitted to the Department of Mathematics, Science, and Technology in fulfilment of the requirements for the degree of Master of Education (Mathematics Education) in the Faculty of Education at the University of Zululand, 2019. / Algebra is a mathematical concept that explains the rules of symbol operations, equations, and inequality. Algebra is a combination of logic and language; hence common mistakes and conceptions are either attributed to logic or language problems, or both. There is also ongoing debate about the fact that learners come to class with different ideas that result in errors and misconceptions when they solve algebraic equations and expressions. Based on this debate concerning both errors and misconceptions in solving algebraic equations and expressions, the purpose of this study was to investigate the errors and misconceptions committed by learners when learning Algebra. The study answered the following research questions: What are the types and the sources of errors and misconceptions committed by Grade 9 learners in Algebra learning? How do the types and the sources of errors and misconceptions influence errors in Grade 9 learners’ cognition when learning Algebra? Which strategies work to avoid errors? What are the sources of the errors and misconceptions in Algebra? Unlike the predominant existing studies, which are urban-based, this study was based in rural schools in the King Cetshwayo District of UMlalazi and Mtunzini Municipality. The structure of the observed learning outcome (SOLO) theory was adopted to observe, examine and analyse learners’ misconceptions in rural-based secondary schools.

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