$$
% ============================================
% 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
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\newcommand{\IB}{\mathbb{B}} % Generic 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{\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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\newcommand{\uDelta}{\underline{\Delta}} % Discrete difference operator #listed
\newcommand{\Shift}{\mathrm{S}_\downarrow} % Shift operator #listed
% --- Logical & Set Operations ---
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\newcommand{\CSet}[2]{\#\{\, #1 \;\vert\; #2 \,\right\}} % Cardinality notation #listed
\newcommand{\C}{\,\#} % Cardinality operator #listed
% --- Limits & Categorical ---
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\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
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% --- Tensor & Algebraic Structures ---
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\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) ---
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\newcommand{\KB}{\mathcal{B}}
\newcommand{\KC}{\mathcal{C}}
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\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}}
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\newcommand{\fn}{\mathfrak{n}}
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\newcommand{\fm}{\mathfrak{m}}
\newcommand{\fp}{\mathfrak{p}}
% --- Text & Formatting ---
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\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{\rk}{\mathrm{rk}} % Rank #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
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\newcommand{\<}{\langle} % Left angle bracket #listed
\newcommand{\>}{\rangle} % Right angle bracket #listed
\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}}
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\newcommand{\cat}[1]{\mathbf{#1}}
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\newcommand{\QED}{\square} % End of proof #listed
\newcommand{\part}{\vdash} % Turnstile #listed
\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
% ============================================
% --- From Affine Cubes ---
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% --- From Cubes.md and affine cube space ---
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% --- From Filtered Vector Spaces.md ---
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% ============================================
% 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{\SS}{\Sym} % Symmetric functions alt #listed
\newcommand{\SP}{\Sym^+} % Positive symmetric functions #listed
\newcommand{\SH}{\hat{\Sym}} % Completed symmetric functions #listed
\newcommand{\SPH}{\hat{\Sym}^+} % Completed positive symmetric #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{\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
$$
Dr. Heinrich Hartmann
Dr. Heinrich Hartmann
Mathematics & Engineering
Heinrich Hartmann is an independent self-funded mathematician whose work spans
algebraic geometry, applied mathematics, and large-scale observability systems.
Positions
Selected Publications
Algebraic Geometry & Mirror Symmetry
-
Cusps of the Kähler Moduli Space and Stability Conditions on K3 Surfaces 32 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
Research Career
Heinrich Hartmann is an independent self-funded mathematician whose work spans
algebraic geometry, applied mathematics, and large-scale observability systems.
In pure mathematics, Heinrich worked at the intersection of algebra, geometry,
and theoretical physics, specifically on derived categories, stability
conditions, K3 surfaces, and mirror symmetry. He published two influential
papers in high-ranking journals (Mathematische Annalen and manuscripta
mathematica). Both have become well-cited references in their respective areas,
combining conceptual insight with technical foundations that are now used in a
variety of subsequent works.
Working with Theo Schlossnagle at Circonus, he became one of the early pioneers
of histogram-based data structures for observability and telemetry. His work on
log-linear histograms and percentile estimation helped shape emerging industry
standards and directly influenced the histogram implementations of Prometheus and OpenTelemetry.
In computational social science, Hartmann collaborated with Christian Kling and
others on empirical studies of liquid democracy, focusing on the Pirate Party’s
LiquidFeedback platform. Their ICWSM 2015 paper on voting behaviour and power
concentration has accumulated close to 100 citations and is a standard
empirical reference on delegation dynamics and power concentration in online
democracies.
In parallel with his industry work, Hartmann has maintained a steady stream of
mathematical writing on his personal blog, focusing on the foundations of
analysis, statistics, and the structures underpinning numerical methods and
machine learning. These notes and essays extend his research profile beyond
formal publications and document his ongoing independent work on the interface
between pure and applied mathematics.