Knowledge Map

Created on April 01, 2025

2025   ·   Research



Model Structures


2025 - 2026


Readings


♨️ Zurich



Relevant Coursework

Deep Learning (Python, 25)

3D/4D Computer Vision (26)

3D Shape Modeling and Geometry Processing (C++, 26)

Computational Models of Motion (C++, Robotics, 26)

Computer Vision for Automated Driving (PRS, 26), Christos Sakaridis

(Systems on Chips) (CUDA, HPC, 26)

Visual Computing

(Physically Based Simulation in Computer Graphics)

Doctoral Seminar in Visual Computing

(*Large-Scale AI Engineering, 25)

(Vision Algorithms for Mobile Robotics (L+E))

Mixed Reality (C++, Blender, SUMO, Unreal, 25)

Graph Theory

Information Geometry

(Calculus of Variations)

(Visual Complex Analysis - Tristan Needham)

(Lie Group and Riemannian Geometry)

(Real Analysis)

(Point-Set Topology)

(Discrete Differential Geometry)



A. Fundamentally Speaking, Second-Layer (1950–2000) Problems Still Open in CVPR-Level Vision

Problem Area What the Second-Layer Problem Is Why It Is Not Solved Classical Foundations (1950–2000) Why CVPR Can Still Accept It What Makes It Non-Trivial
Continuous-time vision Formulating vision as a continuous-time inference problem, instead of discretized frames Vision theory is almost entirely discrete-time; continuous formulations are ad-hoc and inconsistent Continuous dynamical systems, variational calculus, continuous-time state-space models Event cameras, IMU fusion, rolling shutter, high-speed robotics Requires correct handling of time, causality, and observability, not just interpolation
Probabilistic vision consistency Clarifying what quantities in CV are true probabilities vs. proxies Most “probabilistic” models violate probability axioms; uncertainty is often meaningless Statistical decision theory, Bayesian inference, consistency theory Uncertainty estimation, safety-critical perception, robustness Forces explicit distinction between likelihood, loss, score, and decision rule
Geometry–learning unification Combining learning with geometry without breaking invariance and identifiability Geometry is treated as a heuristic regularizer, not a first-class structure Differential geometry, Lie groups, invariance theory Pose estimation, SLAM, equivariant networks Requires structure-preserving learning, not post-hoc constraints
Multi-view geometry identifiability Understanding when geometry is identifiable under noise, partial views, or learned components Degenerate and near-degenerate cases lack unified theory Algebraic geometry, identifiability theory Learned SfM, neural multi-view pipelines Leads to impossibility results and failure modes, not just algorithms
Loss function validity Determining when common losses are inconsistent or biased for the intended task Losses are chosen heuristically, not derived from decision theory Statistical risk minimization, decision theory Training stability, benchmark reliability Often produces negative or impossibility results rather than performance gains
Continuous vs. discrete modeling gap Formalizing when discretization changes the problem itself, not just accuracy Discretization is usually assumed harmless without proof Numerical analysis, approximation theory Simulation-to-real, sensor fusion Forces explicit assumptions about time, scale, and sampling
Proxy objective mismatch Showing when benchmark objectives cannot recover the true task Benchmarks optimize surrogates without guarantees Optimization theory, surrogate loss theory Dataset design, evaluation protocols Undermines leaderboard-driven research at a foundational level


B. Key Meta-Observation

Aspect Second-Layer Reality
Nature of contribution Clarification, systematization, or impossibility results
Typical outcome Fewer experiments, stronger definitions
Risk Not aligned with mainstream “model improvement” narratives
Reward Long-term correctness and conceptual cleanliness
Who this suits Researchers already past the “learning new tools” phase


A. Fundamental Open Problems in Pure Mathematics

Problem Core Question (Pure Math Level) Why It Is Still Unsolved Who Formulated / Popularized It Why It Matters
Riemann Hypothesis Are all non-trivial zeros of the Riemann zeta function on the critical line $\Re(s)=1/2$? Deep connection between primes and complex analysis; no method controls global zero distribution Bernhard Riemann (1859) Governs prime number distribution; foundational to analytic number theory
P vs NP Is every problem whose solution can be verified in polynomial time also solvable in polynomial time? No known techniques separate complexity classes; diagonalization and relativization fail Stephen Cook, Leonid Levin (1971) Defines the limits of computation and algorithmic feasibility
Navier–Stokes Existence and Smoothness Do smooth initial conditions always yield smooth solutions for all time? Nonlinear PDE with energy cascade and potential singularities Claude-Louis Navier, George Stokes (19th c.), modern formulation by Clay Institute Fundamental to fluid dynamics and PDE theory
Hodge Conjecture Are certain cohomology classes always algebraic? Requires bridging topology, geometry, and algebra in high dimensions W. V. D. Hodge (1941) Central to algebraic geometry and geometry–topology relations
Birch and Swinnerton-Dyer Conjecture Does the rank of an elliptic curve equal the order of vanishing of its L-function at $s=1$? Deep arithmetic–analytic correspondence not understood Bryan Birch, Peter Swinnerton-Dyer (1960s) Key to Diophantine equations and arithmetic geometry
Yang–Mills Existence and Mass Gap Does quantum Yang–Mills theory exist and have a mass gap? Nonlinear gauge theory lacks rigorous constructive definition C. N. Yang, Robert Mills (1954) Foundation of quantum field theory
Smooth 4D Poincaré Conjecture Is every smooth 4D manifold homotopy-equivalent to $S^4$ diffeomorphic to it? Dimension 4 is uniquely pathological; smooth and topological categories diverge Henri Poincaré; modern work by Freedman, Donaldson Fundamental to 4-manifold topology
ABC Conjecture How tightly are addition and prime factorization related? Involves deep properties of arithmetic height; proofs remain controversial Joseph Oesterlé, David Masser (1985) Explains why many Diophantine equations have few solutions
Langlands Program (Global Correspondence) Can automorphic forms and Galois representations be fully unified? Requires unifying analysis, geometry, and arithmetic across dimensions Robert Langlands (1967) A “grand unification” of number theory
Collatz Conjecture Does every positive integer reach 1 under the $3n+1$ iteration? Simple definition but chaotic behavior; no invariant or monotone structure Lothar Collatz (1937) Illustrates limits of elementary reasoning in arithmetic dynamics
Goldbach Conjecture Is every even integer greater than 2 the sum of two primes? Requires fine control of prime correlations Christian Goldbach (1742) Basic additive structure of primes
Twin Prime Conjecture Are there infinitely many primes $p$ such that $p+2$ is also prime? Distribution of prime gaps not fully understood Euclid (implicit), modern form by Hardy–Littlewood Prime structure and randomness
Unique Games Conjecture How hard are certain constraint satisfaction problems? Complexity–geometry interplay unresolved Subhash Khot (2002) Central to hardness of approximation
Measure Rigidity Conjectures When are invariant measures uniquely determined by symmetry? Requires controlling entropy and ergodicity in high dimensions Marina Ratner, later generalizations Dynamical systems and number theory
Grothendieck’s Standard Conjectures Can positivity properties in algebraic geometry be proven? Requires tools beyond current cohomology theories Alexander Grothendieck (1960s) Foundations of modern algebraic geometry


B. Meta-Level Classification

Aspect Explanation
What these problems share They involve global structure, not local computation
Why they resist solution Existing tools fail to bridge key domains (analysis–algebra–geometry–logic)
What will not solve them More computation, more data, more engineering
What might solve them New mathematical frameworks, not refinements
Typical timeline Decades to centuries
Relation to applied fields Mostly indirect; influence appears only after resolution



Appendix - Unified Problem Hierarchy Across Pure Math, Theoretical Physics, and CVPR

Level 0 — Foundational Laws (Discovery of Principles)

Time Period Field Problem Type Core Question Who / Where Why It Emerged
1600–1900 Pure Mathematics Foundational structures What are numbers, space, continuity, infinity? Newton, Euler, Gauss, Riemann Need to formalize calculus, geometry, and analysis
1900–1950 Theoretical Physics Fundamental laws What are the laws governing matter, energy, spacetime? Einstein, Dirac, Schrödinger, Heisenberg Empirical anomalies in classical physics
1900–1950 Mathematics & Physics Interface Structural unification Can physical laws be written as invariant mathematical objects? Hilbert, Noether Symmetry and conservation laws demanded formalization


Level 1 — Formalization and Systematization

Time Period Field Problem Type Core Question Who / Where Why It Emerged
1930–1970 Pure Mathematics Axiomatization Can mathematics be made logically complete and rigorous? Gödel, Bourbaki Need for consistency and abstraction
1950–1980 Theoretical Physics Field theories Can physical laws be written as consistent mathematical theories? Yang, Mills, Feynman Quantum field inconsistencies
1950–2000 Mathematics Global structures How do local rules imply global behavior? Grothendieck, Atiyah Geometry, topology, and algebra unification


Level 2 — Identifiability, Limits, and Impossibility

Time Period Field Problem Type Core Question Who / Where Why It Emerged
1960–1990 Pure Mathematics Limits of reasoning What cannot be proven, computed, or classified? Gödel, Turing, Church Logical self-reference
1970–2000 Theoretical Physics Non-perturbative limits When do theories fail to define reality? Wilson, ’t Hooft Divergences, renormalization
1970–2000 Computer Science Complexity limits Which problems are fundamentally hard? Cook, Levin Algorithmic explosion


Level 3 — Engineering Reality on Top of Theory

Time Period Field Problem Type Core Question Who / Where Why It Emerged
1980–2000 Physics & Engineering Approximation regimes How do ideal laws survive noise and scale? Applied physics community Real systems are messy
1990–2005 Computer Vision (pre-deep) Geometric modeling How does vision arise from projection and motion? Hartley, Faugeras, Kanade Robotics and perception needs
2000–2010 Statistics / ML Learning theory When does data approximate truth? Vapnik Data replaces models


Level 4 — Data-Driven Construction (Modern CVPR Core)

Time Period Field Problem Type Core Question Who / Where Why It Emerged
2010–present CVPR / ML Empirical performance Can we build systems that work at scale? Large labs, industry Compute + data abundance
2015–present CVPR Representation learning Which architectures work best empirically? AI labs Engineering pressure
2020–present CVPR System integration How do models survive deployment? Industry-academia Real-world constraints



Core Branches of Pure Mathematics - Timeline, Proposers, and Motivation

Era (Approx.) Branch Key Figures Why It Was Proposed (Core Motivation)
~300 BCE Euclidean Geometry 📐 Euclid To formalize spatial reasoning and deduction from axioms; birth of rigorous proof
1600s Calculus Newton, Leibniz To describe motion, change, and physical laws using infinitesimal reasoning
1700s Classical Analysis Euler, Lagrange To systematize calculus and infinite processes
1800–1850 Linear Algebra Gauss, Hamilton To solve systems of equations and understand linear structure
1800–1900 Abstract Algebra Galois, Cayley To understand symmetry and solvability of equations
1850–1900 Real Analysis Weierstrass, Dedekind To remove ambiguity from calculus via rigorous limits
1850–1900 Complex Analysis Cauchy, Riemann To study functions with extraordinary rigidity and structure
1850–1900 Differential Geometry 🌐 Gauss, Riemann To formalize curvature and smooth spaces beyond Euclid
1880–1930 Set Theory Cantor To define infinity and provide foundations for all mathematics
1900–1930 Mathematical Logic 🧠 Hilbert, Gödel To formalize proof, consistency, and limits of reasoning
1900–1950 Topology Poincaré To study properties invariant under continuous deformation
1900–1950 Functional Analysis Banach, Hilbert To understand infinite-dimensional spaces and operators
1910–1950 Measure Theory Lebesgue To rigorously define integration and size of sets
1920–1960 Algebraic Geometry Zariski, Weil To unify polynomial equations and geometry
1930–1970 Homological Algebra Cartan, Eilenberg To systematically measure failure of exactness
1940–1970 Probability Theory 🎲 Kolmogorov To axiomatize randomness and stochastic processes
1950–1980 Partial Differential Equations Hörmander To rigorously analyze equations governing physics
1950–1980 Representation Theory Harish-Chandra To understand how abstract symmetries act concretely
1950–1980 Category Theory 🧩 Eilenberg, Mac Lane To unify mathematics via compositional structure
1950–1980 Calculus of Variations Tonelli, Morrey To study minimization principles underlying physics
1960–1990 Algebraic Topology Serre, Bott To connect topology with algebraic invariants
1960–1990 Dynamical Systems Smale To study long-term behavior of evolving systems
1960–1990 Ergodic Theory Kolmogorov, Sinai To connect dynamics with statistical behavior
1960–2000 Number Theory (Modern) 🔢 Weil, Langlands To unify arithmetic via geometry and analysis
1960–present Arithmetic Geometry Grothendieck To merge number theory and geometry at a structural level
1960–present Noncommutative Geometry Connes To generalize geometry beyond classical spaces
1970–present Combinatorics Erdős To study discrete structure and extremal phenomena
1970–present Graph Theory Tutte To understand networks and connectivity
1970–present Operator Algebras von Neumann To formalize quantum observables
1980–present Stochastic Analysis Itô To analyze continuous-time randomness
1990–present Higher Category Theory Grothendieck (ideas) To reason about structures between structures
2000–present Homotopy Type Theory Voevodsky To unify logic, topology, and computation
2000–present Derived Geometry Lurie To repair limitations of classical geometry
2000–present Arithmetic Langlands Program Langlands To unify symmetry, geometry, and arithmetic
2000–present Motivic Theory Grothendieck (vision) To explain why cohomology theories agree




Post-Training Techniques

❄️ Efficient Adaptation

  • Adapter (reduce training parameters)
  • Distillation (transfer knowledge)
  • Pruning (delete redundant structure)

❄️ Representation Learning

  • Self-Supervised Learning - SSL

❄️ Model Manipulation

  • Model Editing
  • Model Merging

❄️ Generalization

  • Few-shot Learning
  • Zero-shot Learning




Check List

  • Reading, Oxygen, Fruits No sugar, Protein
    • Gym, Jogging, Tennis
    • Chatting
    • 🌊 Water
  • Take a Walk 🗣️




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