thesis.tex

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%\usepackage{fixltx2e}   % fixes various  bugs in pre 2015 LaTeX 
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\usepackage{microtype}  % microtypographical enhancements
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\usepackage{amsmath, amssymb, amsthm, amsfonts}
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\usepackage{pdfpages} % for inserting pdfs of published articles
\usepackage{twoopt} % for convenient commands
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\usepackage{minted} % for pretty printing of source code

\usepackage[nottoc]{tocbibind} % makes sure that bibliography and the lists
			    % of figures/tables are included in the table
			    % of contents
\usepackage{dcolumn}        % improved alignment of table columns
\usepackage{booktabs}       % improved horizontal lines in tables
\usepackage{paralist}       % improved enumerate and itemize
\usepackage[justification=centering]{caption} % center multiline captions or e.g. [format=hang] 

%\usepackage{natbib}         % citation style AUTHOR (YEAR), or AUTHOR [NUMBER]
%\usepackage[style=alphabetic, backend=biber]{biblatex}
\usepackage[style=iso-alphabetic, backend=biber]{biblatex} 
\addbibresource{thesis.bib}

\usepackage{float}
\usepackage[toc]{appendix}
\usepackage[matrix,arrow]{xy}    % for diagrams
\usepackage{euscript}

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\input{diagrams/tikz_settings.tex}

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\usepackage{csquotes}

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\makeatletter
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% This macro defines a chapter, which is not numbered, but is included
% in the table of contents.
\def\chapwithtoc#1{
\chapter*{#1}
\addcontentsline{toc}{chapter}{#1}
}

\input{definitions}

\begin{document}

% Trochu volnější nastavení dělení slov, než je default.
\lefthyphenmin=2
\righthyphenmin=2

%%% Basic information on the thesis

% Thesis title in English (exactly as in the formal assignment)
\def\ThesisTitle{Invariant differential operators for $1$-graded geometries}

% Author of the thesis
\def\ThesisAuthor{Vít Tuček}

% Year when the thesis is submitted
\def\YearSubmitted{2017}

% Name of the department or institute, where the work was officially assigned
% (according to the Organizational Structure of MFF UK in English,
% or a full name of a department outside MFF)
\def\Department{Mathematical Institute of Charles University}

% Is it a department (katedra), or an institute (ústav)?
\def\DeptType{Institute}

% Thesis supervisor: name, surname and titles
\def\Supervisor{prof. RNDr. Vladimír Souček, DrSc.}

% Supervisor's department (again according to Organizational structure of MFF)
\def\SupervisorsDepartment{Mathematical Institute of Charles University}

% Study programme and specialization
\def\StudyProgramme{Mathematics}
\def\StudyBranch{4M2 \begin{small}Geometry and topology, global analysis\end{small}}

% An optional dedication: you can thank whomever you wish (your supervisor,
% consultant, a person who lent the software, etc.)
\def\Dedication{%

I wholeheartedly thank my supervisor, prof. Vladimír Souček, for his infinite patience. I am grateful to Jirka and Petra who stood beside me all these years and to all my other friends who provided moral support. I thank Sváťa and Libor for mathematical discussions. Finally, I'm indebted to all who contributed to Sage.

\vspace{5cm}

\begin{center}
I dedicate this work to my parents.
\end{center}
}

% Abstract (recommended length around 80-200 words; this is not a copy of your thesis assignment!)
\def\Abstract{%
In this thesis we classify singular vectors in scalar parabolic Verma modules for those pairs $(\lie{sl}(n, \mathbb{C}), \lie{p})$  of complex Lie algebras where the homogeneous space $\mathrm{SL}(n, \mathbb{C}) / P$ is the Grassmannian of $k$-planes in $\mathbb{C}^n.$ We calculate cohomology of nilpotent radicals with values in certain unitarizable highest weight modules. According to \cite{boe_kostant_2009} these modules have  BGG resolutions with weights determined by this cohomology. Such resolutions induce complexes of invariant differential operators on sections of associated bundles over Hermitian symmetric spaces. We describe formal completions of unitarizable highest weight modules that one can use to modify method from \cite{calderbank_differential_2001} that constructs sequences of differential operators over any $1$-graded (aka almost Hermitian) geometry. We suggest uniform description of octonionic planes that could serve as a basis for better understanding of the exceptional Hermitian symmetric space for group $\mathrm{E}_6.$
}

% 3 to 5 keywords (recommended), each enclosed in curly braces
\def\Keywords{%
{Hermitian symmetric space}, {unitarizable highest weight module}, {nilpotent Lie algebra cohomology}, {octonionic plane}
}

%% The hyperref package for clickable links in PDF and also for storing
%% metadata to PDF (including the table of contents).
%% Most settings are pre-set by the pdfx package.
\hypersetup{unicode}
\hypersetup{breaklinks=true}
%\hypersetup{pdftitle=Invariant differential operators}
%\hypersetup{pdfauthor=Vít Tuček}

\input{titlepage}

%%% A page with automatically generated table of contents of the doctoral thesis

\tableofcontents

\hbadness=10000 % silence underfull warnings

%%% Jednotlivé kapitoly práce jsou pro přehlednost uloženy v samostatných souborech
\include{introduction}
\include{algebras}
\include{spaces}
\include{unitarizable_modules}
%\include{invariant}
%\include{parabolic_geometries}
\include{conclusion}

%%% Seznam použité literatury
\addcontentsline{toc}{chapter}{Bibliography}
\printbibliography
%\bibliographystyle{plain}
%\bibliography{thesis}

%%% Figures used in the thesis (consider if this is needed)
\listoffigures

%%% Tables used in the thesis (consider if this is needed)
%%% In mathematical theses, it could be better to move the list of tables to the beginning of the thesis.
\listoftables

%%% Do not include figures and tables from appendices in list of figures and list of tables
%%% from https://tex.stackexchange.com/questions/213523/remove-appendix-tables-and-figures-from-list-of-figure-tables
\let\svaddcontentsline\addcontentsline
\renewcommand\addcontentsline[3]{%
  \ifthenelse{\equal{#1}{lof}}{}%
  {\ifthenelse{\equal{#1}{lot}}{}{\svaddcontentsline{#1}{#2}{#3}}}}

\include{appendix}

%%% Tabulky v disertační práci, existují-li.
% \chapwithtoc{List of Tables}

%%% Použité zkratky v disertační práci, existují-li, včetně jejich vysvětlení.
% \chapwithtoc{List of Abbreviations}

%%% Přílohy k disertační práci, existují-li (různé dodatky jako výpisy programů,
%%% diagramy apod.). Každá příloha musí být alespoň jednou odkazována z vlastního
%%% textu práce. Přílohy se číslují.
% \chapwithtoc{Attachments}

\openright
\end{document}