$$
% ============================================
% GENERAL / UNIVERSAL
% Used throughout the documentation
% ============================================
% --- Basic Number Systems ---
\newcommand{\IR}{\mathbb{R}} % Real numbers #listed
\newcommand{\IC}{\mathbb{C}} % Complex numbers #listed
\newcommand{\IN}{\mathbb{N}} % Natural numbers #listed
\newcommand{\IZ}{\mathbb{Z}} % Integers #listed
\newcommand{\IQ}{\mathbb{Q}} % Rational numbers #listed
\newcommand{\IA}{\mathbb{A}} % Affine space #listed
\newcommand{\IB}{\mathbb{B}} % Generic field #listed
\newcommand{\ik}{\Bbbk} % Ground field #listed
\newcommand{\ID}{\mathbb{D}} % Generic field #listed
\newcommand{\IF}{\mathbb{F}} % Generic field #listed
\newcommand{\IH}{\mathbb{H}} % Quaternions #listed
\newcommand{\II}{\mathbb{I}} % Generic field #listed
\newcommand{\IL}{\mathbb{L}} % Generic field #listed
\newcommand{\IP}{\mathbb{P}} % Projective space #listed
\newcommand{\IS}{\mathbb{S}} % Sphere #listed
\newcommand{\IT}{\mathbb{T}} % Tensor algebra #listed
\newcommand{\IV}{\mathbb{V}} % Generic vector space #listed
% --- Function Spaces ---
\newcommand{\CINF}{\mathcal{C}^\infty} % Smooth (infinitely differentiable) functions #listed
\newcommand{\CC}{\mathcal{C}} % C^k functions #listed
\newcommand{\Ck}{\mathcal{C}^k} % C^k functions #listed
\newcommand{\CK}{\mathcal{C}^K} % C^k functions #listed
% --- Fundamental Operators ---
\newcommand{\del}{\partial} % Partial derivative #listed
\newcommand{\uDelta}{\underline{\Delta}} % Discrete difference operator #listed
\newcommand{\Shift}{\mathrm{S}_\downarrow} % Shift operator #listed
% --- Logical & Set Operations ---
\newcommand{\IFF}{\Leftrightarrow} % If and only if #listed
\newcommand{\Ind}{\mathbb{1}} % Indicator/characteristic function #listed
\newcommand{\IndA}[1]{\mathbb{1}\lbrace #1 \rbrace} % Indicator with condition #listed
\newcommand{\1}{\mathbb{1}} % Indicator shorthand #listed
\newcommand{\set}[1]{\{#1\}} % Simple set braces #listed
\newcommand{\Set}[2]{\left\{\, #1 \;\vert\; #2 \,\right\}} % Set-builder notation #listed
\newcommand{\CSet}[2]{\#\{\, #1 \;\vert\; #2 \,\right\}} % Cardinality notation #listed
\newcommand{\C}{\,\#} % Cardinality operator #listed
% --- Limits & Categorical ---
\newcommand{\limproj}{\varprojlim} % Inverse limit #listed
\newcommand{\limind}{\varinjlim} % Direct limit #listed
\newcommand{\Hom}{\mathrm{Hom}} % Homomorphism #listed
\newcommand{\End}{\mathrm{End}} % Endomorphism #listed
\newcommand{\Ext}{\mathrm{Ext}} % Ext functor #listed
% --- Arrows & Relations ---
\newcommand{\ra}{\rightarrow} % Right arrow #listed
\newcommand{\lra}{\longrightarrow} % Long right arrow #listed
\newcommand{\xlra}[1]{\overset{#1}{\lra}} % Labeled long arrow #listed
\newcommand{\la}{\leftarrow} % Left arrow #listed
\newcommand{\lla}{\longleftarrow} % Long left arrow #listed
\newcommand{\mono}{\hookrightarrow} % Monomorphism #listed
\newcommand{\epi}{\twoheadrightarrow} % Epimorphism #listed
\newcommand{\isom}{\cong} % Isomorphism #listed
\newcommand{\downto}{\searrow} % Diagonal arrow #listed
% --- Tensor & Algebraic Structures ---
\newcommand{\tensor}{\otimes} % Tensor product #listed
\newcommand{\tensors}{\tensor\dots\tensor} % Multiple tensor products #listed
\newcommand{\Tensor}{\bigotimes} % Big tensor product #listed
\newcommand{\stensor}{\odot} % Symmetric tensor product #listed
\newcommand{\vsum}{\oplus} % Direct sum #listed
\newcommand{\Vsum}{\bigoplus} % Big direct sum #listed
% --- Calligraphic Letters (Generic) ---
\newcommand{\KA}{\mathcal{A}}
\newcommand{\KB}{\mathcal{B}}
\newcommand{\KC}{\mathcal{C}}
\newcommand{\KD}{\mathcal{D}}
\newcommand{\KF}{\mathcal{F}}
\newcommand{\KH}{\mathcal{H}}
\newcommand{\KI}{\mathcal{I}}
\newcommand{\KL}{\mathcal{L}}
\newcommand{\KN}{\mathcal{N}}
\newcommand{\KP}{\mathcal{P}}
\newcommand{\KQ}{\mathcal{Q}}
\newcommand{\KR}{\mathcal{R}}
\newcommand{\KS}{\mathcal{S}}
\newcommand{\KV}{\mathcal{V}}
\newcommand{\KZ}{\mathcal{Z}}
% --- Fraktur Letters (Generic) ---
\newcommand{\gc}{\mathfrak{C}}
\newcommand{\gd}{\mathfrak{D}}
\newcommand{\gM}{\mathfrak{M}}
\newcommand{\gm}{\mathfrak{m}}
\newcommand{\gf}{\mathfrak{f}}
\newcommand{\gu}{\mathfrak{U}}
\newcommand{\fa}{\mathfrak{a}}
\newcommand{\fg}{\mathfrak{g}}
\newcommand{\fn}{\mathfrak{n}}
\newcommand{\fk}{\mathfrak{k}}
\newcommand{\fm}{\mathfrak{m}}
\newcommand{\fp}{\mathfrak{p}}
\newcommand{\fP}{\mathfrak{P}}
% \fg already defined above (line 95) as \mathfrak{g}
% --- Text & Formatting ---
\newcommand{\qtext}[1]{\quad\text{#1}\quad} % Quad spaced text #listed
\newcommand{\stext}[1]{\;\text{#1}\;} % Small spaced text #listed
\newcommand{\ssum}[1]{\sum_{\substack{#1}}} % Sum with substacked condition #listed
\newcommand{\half}{\frac{1}{2}} % One-half #listed
\newcommand{\floor}[1]{\lfloor #1 \rfloor} % Floor function #listed
\newcommand{\ceil}[1]{\lceil #1 \rceil} % Ceiling function #listed
\newcommand{\nl}{\\} % Newline #listed
% --- Common Functions ---
\newcommand{\id}{\mathrm{id}} % Identity function #listed
\newcommand{\len}{\mathrm{len}} % Length of a word #listed
\newcommand{\rk}{\mathrm{rk}} % Rank #listed
\newcommand{\diag}{\operatorname{diag}} % Diagonal matrix #listed
\newcommand{\Res}{\operatorname{Res}} % Residue #listed
\newcommand{\Ker}{\mathrm{Ker}} % Kernel #listed
\newcommand{\Diff}{\mathrm{Diff}} % Diffeomorphism group #listed
\newcommand{\Pic}{\mathrm{Pic}} % Picard group #listed
\newcommand{\Spec}{\mathrm{Spec}} % Spectrum #listed
\newcommand{\D}{\mathrm{D}} % Differential operator #listed
\newcommand{\DP}{\mathrm{D_{\!+}}} % Positive differential #listed
\newcommand{\DDP}{\mathrm{D^{\!+}}} % Upper differential #listed
% --- Variants ---
\newcommand{\vphi}{\varphi} % Variant phi #listed
\newcommand{\sphi}{\phi} % Straight phi #listed
\newcommand{\eps}{\varepsilon} % Epsilon variant #listed
\newcommand{\pt}{*} % Point notation #listed
\newcommand{\point}{*} % Point notation alt #listed
% --- Set Operations ---
\newcommand{\union}{\cup} % Union #listed
\newcommand{\Union}{\bigcup} % Big union #listed
\newcommand{\dotcup}{\ensuremath{\mathaccent\cdot\cup}} % Disjoint union #listed
\newcommand{\dunion}{\dotcup} % Disjoint union alt #listed
% \< and \> removed - conflict with LaTeX tabbing commands and unused in docs
\newcommand{\inpart}[1]{\in\text{\part}(#1)} % In partition of #listed
\newcommand{\trl}{\triangleleft} % Left triangle #listed
\newcommand{\trr}{\triangleright} % Right triangle #listed
% --- Misc ---
\newcommand{\curly}[1]{\mathcal{#1}}
\newcommand{\op}[1]{\mathrm{#1}}
\newcommand{\Cat}[1]{\mathfrak{#1}}
\newcommand{\cat}[1]{\mathbf{#1}}
\newcommand{\CAT}[1]{\left\{\,\text{#1}\,\right\}}
\newcommand{\CATii}[2]{\left\{\,\begin{array}{c}\text{#1}\\\text{#2}\end{array}\,\right\}}
\newcommand{\Mon}{\mathrm{Mon}}
\newcommand{\Lin}{\mathrm{Lin}}
\newcommand{\ev}{\mathrm{ev}}
\newcommand{\tc}{\prec_{\mathrm{tc}}}
\newcommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\newcommand{\QED}{\square} % End of proof #listed
\renewcommand{\part}{\vdash} % Turnstile #listed (overrides LaTeX \part sectioning)
\newcommand{\opart}{\models} % Semantic entailment #listed
\newcommand{\Def}{\mathrm{Def}} % Bialgebra defect #listed
% ============================================
% CHAPTER 1: DIFFERENTIAL DUALITY
% ============================================
\newcommand{\Jet}{\mathbf{Jet}} % Jet space of C infinity germs #listed
\newcommand{\jet}{\mathrm{jet}} % Jet functor #listed
\newcommand{\E}{\mathbf{E}} % Space of smooth functions #listed
\newcommand{\EE}{\mathbf{E}} % Space of smooth functions alt #listed
% ============================================
% CHAPTER 2: DISCRETE DUALITY
% ============================================
% --- Multisets and Iterated Structures ---
\newcommand{\KM}{\mathcal{M}} % Multiset / multi-index set #listed
\newcommand{\ualpha}{\underline{\alpha}} % Underline alpha (iterated multi-index) #listed
\newcommand{\ubeta}{\underline{\beta}} % Underline beta (iterated multi-index) #listed
\newcommand{\ugamma}{\underline{\gamma}} % Underline gamma (iterated multi-index) #listed
\newcommand{\uS}{\underline{S}} % Underline S (iterated subset) #listed
\newcommand{\uT}{\underline{T}} % Underline T (iterated subset) #listed
% --- From Affine Cubes ---
\newcommand{\BC}{\mathbf{B}} % Affine Cube space #listed
% --- From Cubes.md and affine cube space ---
\newcommand{\BB}{\mathbb{B}} % Boolean lattice #listed
\newcommand{\minelt}{\hat{0}} % Minimal element #listed
\newcommand{\maxelt}{\hat{1}} % Maximal element #listed
\newcommand{\sleq}{\subseteq} % Subset relation #listed
\newcommand{\drk}{\mathrm{rk_{\Delta}}} % Discrete rank #listed
\newcommand{\Fun}{\operatorname{F}} % Function space #listed
\newcommand{\Cube}{\operatorname{Cube}} % Geometric cubes #listed
\newcommand{\Cu}{\mathrm{Cube}} % Cube spaces (NC calculus); symbol choice revisable #listed
\newcommand{\Fl}{\operatorname{Fl}} % Flow (NC calculus) #listed
\newcommand{\Aff}{\operatorname{Aff}} % Affine cubes #listed
\newcommand{\Exp}{\operatorname{exp}} % Geometric realization (zeta transform) #listed
\newcommand{\Log}{\operatorname{log}} % Tangent logarithm (Möbius transform) #listed
\newcommand{\aprod}{\star} % Anchored product of cubes #listed
\newcommand{\acoprod}{\Delta} % Anchored coproduct of cubes #listed
\newcommand{\tcoprod}{\Delta^\tau} % Transport coproduct of cubes #listed
\newcommand{\tprod}{\star^\tau} % Transport product of cubes #listed
\newcommand{\Part}{\mathrm{Part}} % Partitions of a set #listed
\newcommand{\Cov}{\mathrm{Cov}} % Coverings #listed
\DeclareMathOperator{\lf}{\mathrm{leaf}} % Leaf support #listed
\newcommand{\CT}{\mathbf{CT}} % Covering trees #listed
\newcommand{\BH}{\mathbf{IH}} % Iterated hypergraphs #listed
\newcommand{\IHCov}{\mathbf{IHCov}} % Iterated hypergraph covers #listed
\newcommand{\vac}{{|0\rangle}} % Empty partition #listed
% --- From Filtered Vector Spaces.md ---
\newcommand{\gr}{\mathrm{gr}} % Graded/associated graded object #listed
% ============================================
% CHAPTER 3: ULTRA CALCULUS
% ============================================
\newcommand{\POS}{\mathbf{Pos}} % Growth domain spaces #listed
\newcommand{\U}{\mathbf{U}} % Ultra regulator quotient #listed
\newcommand{\IU}{\mathbb{U}} % Ultra regulator #listed
\newcommand{\IG}{\mathbb{G}} % Growth profile #listed
\newcommand{\Tame}{\mathbf{Tame}} % Tame growth functions #listed
\newcommand{\Scale}{\mathrm{Scale}} % Scaling operator #listed
\newcommand{\Bell}{\mathcal{B}} % Bell polynomials #listed
\newcommand{\uexp}{\exp_+} % Exponential generating series #listed
% --- Ultra Structures ---
\newcommand{\Sym}{\mathbf{S}} % Symmetric functions/algebra #listed
\newcommand{\SYM}{\mathrm{Sym}} % Symmetric algebra, roman #listed
\renewcommand{\SS}{\Sym} % Symmetric functions alt #listed (overrides LaTeX \SS)
\newcommand{\SP}{\Sym^+} % Positive symmetric functions #listed
\newcommand{\SH}{\hat{\Sym}} % Completed symmetric functions #listed
\newcommand{\SPH}{\hat{\Sym}^+} % Completed positive symmetric #listed
% --- Completed tangent/cotangent spaces (DDC paper) ---
\newcommand{\STH}{\hat{S}T} % Completed symmetric cotangent space #listed
\newcommand{\CTH}{\hat{C}T} % Completed co-cube space #listed
% --- Ultra Operators ---
\newcommand{\MIX}{\mathrm{Mix}} % Mixing operator #listed
\newcommand{\BMIX}{\mathrm{BMix}} % Boolean mixing #listed
\newcommand{\TMIX}{\mathrm{TMix}} % Transport mixing #listed
\newcommand{\TBMIX}{\mathrm{TMix}} % Transport boolean mixing #listed
\newcommand{\TRANS}{\mathrm{Trans}} % Transport operator #listed
% --- Gauge & Equivalence ---
\newcommand{\gaugeleq}{\preccurlyeq} % Gauge less-or-equal #listed
\newcommand{\gaugeeq}{\asymp} % Gauge equivalence #listed
\newcommand{\gaugegeq}{\preccurlygeq} % Gauge greater-or-equal #listed
\newcommand{\tstar}{\circledast} % Tight star product #listed
% --- Forward Differences ---
\newcommand{\FD}{\blacktriangle} % Forward difference #listed
\newcommand{\fd}{\FD} % Forward difference alt #listed
\newcommand{\AC}{\square} % Associated character #listed
% ============================================
% OPERATORS (DeclareMathOperator)
% ============================================
\DeclareMathOperator{\Supp}{\mathrm{Supp}} % Support #listed
\DeclareMathOperator{\supp}{\mathrm{Supp}} % Support #listed
\newcommand{\esssupp}{\operatorname*{ess-supp}} % Support #listed
\DeclareMathOperator{\Alt}{\Lambda} % Alternating/exterior #listed
\DeclareMathOperator{\ad}{ad} % Adjoint representation #listed
\DeclareMathOperator{\ch}{ch} % Chern character #listed
\DeclareMathOperator{\td}{td} % Todd class #listed
\DeclareMathOperator{\TD}{TD} % Todd operator #listed
\DeclareMathOperator{\pr}{pr} % Projection #listed
\DeclareMathOperator{\Map}{Map} % Mapping space #listed
\DeclareMathOperator{\Pol}{Pol} % Polarization
% ============================================
% ENVIRONMENT SIDEBARS
% Plain neutral-grey left rule on theorem-like environments,
% marking structure without web-style color coding.
% ============================================
\usepackage{xcolor}
\usepackage[framemethod=TikZ]{mdframed}
\usepackage{booktabs} % pandoc emits \toprule/\midrule/\bottomrule for pipe tables
\definecolor{envRule}{HTML}{666666}
\mdfdefinestyle{envstyle}{%
linewidth=0.8pt,
topline=false,
bottomline=false,
rightline=false,
leftline=true,
leftmargin=-8pt,
rightmargin=0pt,
innerleftmargin=7.2pt,
innerrightmargin=0pt,
innertopmargin=0pt,
innerbottommargin=0pt,
skipabove=1em,
skipbelow=1em,
linecolor=envRule,
}
\surroundwithmdframed[style=envstyle]{theorem}
\surroundwithmdframed[style=envstyle]{proposition}
\surroundwithmdframed[style=envstyle]{lemma}
\surroundwithmdframed[style=envstyle]{corollary}
\surroundwithmdframed[style=envstyle]{definition}
\surroundwithmdframed[style=envstyle]{example}
\surroundwithmdframed[style=envstyle]{remark}
\surroundwithmdframed[style=envstyle]{proof}
$$
Dr. Heinrich Hartmann
Heinrich Hartmann is an independent self-funded mathematician whose work spans
algebraic geometry, applied mathematics, and large-scale observability systems.
Follow my research on Mastodon: @hhartmann
Positions
Publications
Algebraic Geometry & Mirror Symmetry
-
Cusps of the Kähler Moduli Space and Stability Conditions on K3 Surfaces 36 citations
Mathematische Annalen 354(1), 2012.
Relates boundary points (“cusps”) in the K3 moduli space to Bridgeland stability conditions on derived categories, giving a precise picture of how stability behaves near the boundary. The appendix has become a standard reference for perfect complexes and complex base-change; Proposition 6.4 is frequently cited as the canonical base-change result.
-
Period- and Mirror-Maps for the Quartic K3 13 citations
manuscripta mathematica 141(3), 2013.
Gives a complete, explicit treatment of mirror symmetry for the quartic K3, computing period maps and Picard–Fuchs equations and matching complex and Kähler moduli. It is widely used as the standard reference for the quartic K3 mirror example in later work on K3 surfaces.
Applied Mathematics & Engineering
Digital Democracy & Computational Social Science
Fellowships
