$$
% ============================================
% GENERAL / UNIVERSAL
% Used throughout the documentation
% ============================================
% --- Basic Number Systems ---
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\newcommand{\IF}{\mathbb{F}} % Generic field #listed
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\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 ---
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\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 ---
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% --- Logical & Set Operations ---
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% --- Limits & Categorical ---
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% --- Arrows & Relations ---
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% --- Tensor & Algebraic Structures ---
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\newcommand{\Vsum}{\bigoplus} % Big direct sum #listed
% --- Calligraphic Letters (Generic) ---
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\newcommand{\KR}{\mathcal{R}}
\newcommand{\KS}{\mathcal{S}}
\newcommand{\KV}{\mathcal{V}}
\newcommand{\KZ}{\mathcal{Z}}
% --- Fraktur Letters (Generic) ---
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% \fg already defined above (line 95) as \mathfrak{g}
% --- Text & Formatting ---
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% --- Common Functions ---
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\newcommand{\Pic}{\mathrm{Pic}} % Picard group #listed
\newcommand{\Spec}{\mathrm{Spec}} % Spectrum #listed
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\newcommand{\DDP}{\mathrm{D^{\!+}}} % Upper differential #listed
% --- Variants ---
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\newcommand{\sphi}{\phi} % Straight phi #listed
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% --- Set Operations ---
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% \< and \> removed - conflict with LaTeX tabbing commands and unused in docs
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% --- Misc ---
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% ============================================
% CHAPTER 1: DIFFERENTIAL DUALITY
% ============================================
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\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 ---
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\newcommand{\ualpha}{\underline{\alpha}} % Underline alpha (iterated multi-index) #listed
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% --- From Affine Cubes ---
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% --- From Cubes.md and affine cube space ---
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% --- 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 ---
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% --- 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 ---
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\newcommand{\gaugeeq}{\asymp} % Gauge equivalence #listed
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% --- Forward Differences ---
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\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
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\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
$$
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
Recent Articles
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
Selected Blog Posts
-
The Calculus of Local Smooth Functions (2023)
Develops differential calculus on germs of smooth functions, emphasizing local operators, composition rules, and a jet-style view of Taylor series.
-
Effective Rank Decomposition of Linear Maps (2021)
Revisits the rank decomposition theorem with an explicit constructive proof, algorithms, and NumPy implementations, highlighting practical aspects often glossed over in textbooks.
-
Quantile Mathematics (2019)
Explains quantiles and quantile estimation from a mathematical perspective and links them to real-world latency and SLO analysis in observability systems.
-
Natural Operators in Linear Algebra (2021, PDF)
A structuralist treatment of linear maps organized by their naturality under basis changes, written in the style of advanced lecture notes.
Fellowships
