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University of Oxford

1. Klein, Florian. Additive higher representation theory.

Degree: PhD, 2014, University of Oxford

This thesis is devoted to the study of higher representation theory as introduced in [Rou4]. As this theory is in its early days, it is essential to seek out modules that can rightfully be named building blocks and allow one to express as much of the structure of arbitrary modules as possible in their terms. We contribute towards this undertaking in the case of additive higher representation theory. Inspiration is drawn from Soergel bimodules which categorify the Hecke algebra. We introduce functorially cyclic modules as well as (strongly) universal cell modules. Examples include the minimal categorifications of [Rou4]. Properties of such modules are discussed and universal properties in terms of representable 2-functors are established. This leads to constructions and classifications in terms of split Frobenius objects, using a new variant of the Barr-Beck theorem for additive categories. Furthermore, we encounter a new class of modules so called coinvariant modules which arise from automorphism group actions. We also construct canonical cofiltrations and demonstrate why the Jordan-Hölder theory of [Rou4] does not readily generalise. Throughout, we comment on the succession [MaMi1]-[MaMi5] that tackles the same questions, however arrives at different conclusions. As applications, we first show that the 2-category of singular Soergel bimodules of [Wi2] arises naturally within the additive higher representation theory of Soergel bimodules. Second, we establish (weak) equivalences between certain associated universal cell modules together with a categorification of cell module homomorphisms of the Hecke algebra. Third, we show that singular Soergel bimodules constructed with a faithful representation categorify the Schur algebroid, generalising the main result of [Li]. Fourth given a group and a subgroup, we recover the additive monoidal category of representations of the subgroup from the corresponding category for the group without invoking Tannakian formalism.

Subjects/Keywords: 510; Representations of groups; Jordan-Ho¨lder Theorem; Higher Representation Theory; Barr-Beck Theorem; Soergel Bimodule; Category Theory

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APA (6th Edition):

Klein, F. (2014). Additive higher representation theory. (Doctoral Dissertation). University of Oxford. Retrieved from https://ora.ox.ac.uk/objects/uuid:b6af63fb-09f6-458b-ac48-01ede93e462f ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.730610

Chicago Manual of Style (16th Edition):

Klein, Florian. “Additive higher representation theory.” 2014. Doctoral Dissertation, University of Oxford. Accessed August 23, 2019. https://ora.ox.ac.uk/objects/uuid:b6af63fb-09f6-458b-ac48-01ede93e462f ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.730610.

MLA Handbook (7th Edition):

Klein, Florian. “Additive higher representation theory.” 2014. Web. 23 Aug 2019.

Vancouver:

Klein F. Additive higher representation theory. [Internet] [Doctoral dissertation]. University of Oxford; 2014. [cited 2019 Aug 23]. Available from: https://ora.ox.ac.uk/objects/uuid:b6af63fb-09f6-458b-ac48-01ede93e462f ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.730610.

Council of Science Editors:

Klein F. Additive higher representation theory. [Doctoral Dissertation]. University of Oxford; 2014. Available from: https://ora.ox.ac.uk/objects/uuid:b6af63fb-09f6-458b-ac48-01ede93e462f ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.730610


Universidade Estadual de Campinas

2. Rampasso, Giane Casari, 1991-. On the regularity theory for a class of state-dependent degenerate diffusions : Teoria de regularidade para difus?es degeneradas do tipo estado-dependentes .

Degree: 2019, Universidade Estadual de Campinas

Resumo: Esta tese contempla dois trabalhos originais que analisam a teoria da regularidade para equa??es el?pticas degeneradas/singulares com expoente vari?vel. No primeiro trabalho, analisamos uma equa??o el?ptica degenerada/singular com expoente vari?vel na forma n?o-divergente. Atrav?s de t?cnicas de aproxima??o e ferramentas de an?lise geom?trica, provamos estimativas 𝓒1,α para solu??es de viscosidade para equa??es el?pticas totalmente n?o lineares com expoente vari?vel. No segundo trabalho, estudamos equa??es el?pticas degeneradas com expoente vari?vel na forma divergente. Considerando o expoente vari?vel, dado por uma fun??o θ:B1 → ℝ apenas limitada e mensur?vel, provamos exist?ncia e regularidade Hölder para solu??es fracas. Tamb?m, se consideramos o expoente vari?vel θ de classe 𝓒1, obtemos ganhos de regularidade para as solu??es fracas. De fato, neste caso mostramos que solu??es fracas s?o assintoticamente de classe 𝓒1,1loc; Abstract: This thesis contemplates two original works which analyze the regularity theory for variable-exponent degenerate/singular elliptic equations. In the first one, we investigate a degenerate/singular elliptic equation with variable exponent in the non-divergence form. Arguing through approximation techniques and geometrical analysis tools we prove 𝓒1,α-estimates for viscosity solutions to variable exponent fully nonlinear elliptic equations. In the second work, we consider degenerate elliptic equations with variable exponent in the divergence form. In this setting, we prove existence and Hölder regularity of weak solutions to roughly degenerate diffusions. Also, if we consider the variable exponent θ of class 𝓒1, we obtain gains of regularity for the solutions. Namely, we show that weak solutions are asymptotically of class 𝓒1,1loc Advisors/Committee Members: Bronzi, Anne Caroline, 1984- (advisor), Pimentel, Edgard Almeida (advisor), Montenegro, Marcelo da Silva (committee member), Sirakov, Boyan Slavchev (committee member), Imbert, Cyril Claude (committee member), Teixeira, Eduardo Vasconcelos Oliveira (committee member).

Subjects/Keywords: Equa??es diferenciais elipticas; Equa??es diferenciais n?o-lineares; Regularidade ?tima; H?lder, Espa?os de

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Rampasso, Giane Casari, 1. (2019). On the regularity theory for a class of state-dependent degenerate diffusions : Teoria de regularidade para difus?es degeneradas do tipo estado-dependentes . (Thesis). Universidade Estadual de Campinas. Retrieved from http://repositorio.unicamp.br/jspui/handle/REPOSIP/334015

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Rampasso, Giane Casari, 1991-. “On the regularity theory for a class of state-dependent degenerate diffusions : Teoria de regularidade para difus?es degeneradas do tipo estado-dependentes .” 2019. Thesis, Universidade Estadual de Campinas. Accessed August 23, 2019. http://repositorio.unicamp.br/jspui/handle/REPOSIP/334015.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Rampasso, Giane Casari, 1991-. “On the regularity theory for a class of state-dependent degenerate diffusions : Teoria de regularidade para difus?es degeneradas do tipo estado-dependentes .” 2019. Web. 23 Aug 2019.

Vancouver:

Rampasso, Giane Casari 1. On the regularity theory for a class of state-dependent degenerate diffusions : Teoria de regularidade para difus?es degeneradas do tipo estado-dependentes . [Internet] [Thesis]. Universidade Estadual de Campinas; 2019. [cited 2019 Aug 23]. Available from: http://repositorio.unicamp.br/jspui/handle/REPOSIP/334015.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Rampasso, Giane Casari 1. On the regularity theory for a class of state-dependent degenerate diffusions : Teoria de regularidade para difus?es degeneradas do tipo estado-dependentes . [Thesis]. Universidade Estadual de Campinas; 2019. Available from: http://repositorio.unicamp.br/jspui/handle/REPOSIP/334015

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation


University of Southern California

3. Dreyer, Guillaume. Geometric properties of Anosov representations.

Degree: PhD, Mathematics, 2012, University of Southern California

Let S be a connected, closed, oriented surface of negative Euler characteristic, we consider the PSLn(R)-character variety RepPSLn(R)(S). An interesting connected component of the latter space is the Hitchin space HPSLn(R)(S): it contains a copy of the Teichm uller space T(S), and hence is regarded as the higher rank Teichm uller space in the case of PSLn(R). In order to study the elements in the Hitchin space HPSLn(R)(S), F.Labourie introduced the notion of Anosov representation. In particular, he proved that every Hitchin representation is discrete and injective, some properties already shared by Teichm uller representations. In this dissertation, we extend to Anosov representations several classic tools from hyperbolic geometry designed to study Teichm uller representations: we generalize Thurston's length function and cataclysm deformation, and analyze how these two notions relate to each other. We then explain how these techniques illuminate the geometry of Anosov representations, and provide crucial information about a new system of coordinates on the Hitchin space HPSLn(R)(S). Advisors/Committee Members: Bonahon, Francis (Committee Chair), Honda, Ko (Committee Member), Asok, Aravind (Committee Member), Haydn, Nicolai T. A. (Committee Member), Däppen, Werner (Committee Member), Dappen, Werner (Committee Member), Daeppen, Werner (Committee Member).

Subjects/Keywords: surface group representation; Anosov representation; Hitchin representation; Hitchin component; Anosov flow; ; lder geodesic current; length function; cataclysm deformation; geodesic lamination; Thurston'; s intersection number

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Dreyer, G. (2012). Geometric properties of Anosov representations. (Doctoral Dissertation). University of Southern California. Retrieved from http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/74253/rec/3030

Chicago Manual of Style (16th Edition):

Dreyer, Guillaume. “Geometric properties of Anosov representations.” 2012. Doctoral Dissertation, University of Southern California. Accessed August 23, 2019. http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/74253/rec/3030.

MLA Handbook (7th Edition):

Dreyer, Guillaume. “Geometric properties of Anosov representations.” 2012. Web. 23 Aug 2019.

Vancouver:

Dreyer G. Geometric properties of Anosov representations. [Internet] [Doctoral dissertation]. University of Southern California; 2012. [cited 2019 Aug 23]. Available from: http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/74253/rec/3030.

Council of Science Editors:

Dreyer G. Geometric properties of Anosov representations. [Doctoral Dissertation]. University of Southern California; 2012. Available from: http://digitallibrary.usc.edu/cdm/compoundobject/collection/p15799coll3/id/74253/rec/3030

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