概论
数学
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Andrius Kulikauskas
 ms@ms.lt
 +370 607 27 665
 My work is in the Public Domain for all to share freely.
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Geometry theorems, Geometries, Geometry illustrations, Universal hyperbolic geometry, Triangle geometry, Algebraic geometry, Algebraic topology
几何 _ _ _ _ געאָמעטרי
Overall goals:
 To understand what geometry contributes to the overall map of mathematics.
 To distinguish four geometries: affine, projective, conformal and symplectic.
 To understand the relationship between geometries and logic, the classical Lie groups/algebras, category theory, etc.
 To have a better understanding of mathematical concepts, tools, theorems and examples that would serve me in understanding all branches of mathematics.
I should
 Make a list of geometry theorems and sort them by geometry.
 Make a list of geometries and show how they are related.
 Why are rings important for geometry rather than just groups? Because want to work with ideals and not subrings, because we are dealing with what is not as well as what is, because we are constructing both topdown and bottomup.
 Look at Wildberger's three binormal forms in chromogeometry.
 Think of how transformations act on 0, 1, infinity, for example, translations can take 0 to 1, but infinity to infinity.
 "Grothendieck thought about this very hard and invented his concept of topos, which is roughly a category that serves as a place in which one can do mathematics." A place for figuring things out? What would that mean? Ways of extending the mind by leveraging basic ways of figuring things out and organizing them around a particular observer?
Curvature
 Does the inside of a sphere have negative curvature, and the outside of sphere have positive curvature? And likewise the inside and outside of a torus?
 Why is negative curvature  the curvature inside  more prominent than positive curvature  the outside of a space?
Attach:geometry.png Δ
What is geometry?
 Geometry is the regularity of choice.
 Geometry is about defining equivalence (of shapes), thus the transformations that maintain equivalence, and the symmetries of those transformations.
Geometry is:
 the ways that our expectations can be related, thus how we are related to each other
 the relationship between our old and new search. And search is triggered by constancy, which is the representation of the nullsome which is related to anything and thus to calm and expectations, space and time, etc.
 how to expand our vision (from a smaller space to a larger space) (Tadashi Tokieda)
 how to embed a lower dimensional space into a higher dimensional space
 the ways that a vector space is grounded
 the relationship between two spaces, for example, points, lines, planes
 the construction of sets of roots of polynomials
 a quadratic subject, with quadratic concepts: quadrance and spread. (Norman Wildberger)
 Grothendieck categories
Geometry is the way of fitting a lower dimensional vector space into a higher dimensional vector space.
 Tensors give the embedding of a lower dimension into a higher dimension. Taip pat tensoriai sieja erdvę ir jos papildinį, kaip kad gyvybę ir meilę. Tai vyksta vektoriais (tangent space) ir kovektoriais (normal space?) Tad geometrijos pagrindas būtų Tensors over a ring. Kovektoriai išsako idealią bazę. Tensorius susidaro iš kovektorių ir kokovektorių. Ir tie, ir tie yra tiesiniai funkcionalai. Tikai baigtinių dimensijų vektorių erdvėse kokovektoriai tolygūs vektoriams.
 Dflags explain how to fit a lower dimensional vector space into a higher dimensional vector space.
 A geometric embedding is the right notion of embedding or inclusion of topoi F↪E F \hookrightarrow E, i.e. of subtoposes. Notably the inclusion Sh(S)↪PSh(S) Sh(S) \hookrightarrow P of a category of sheaves into its presheaf topos or more generally the inclusion ShjE↪E Sh_j E \hookrightarrow E of sheaves in a topos E E into E E itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi. Another perspective is that a geometric embedding F↪E F \hookrightarrow E is the localizations of E E at the class W W or morphisms that the left adjoint E→F E \to F sends to isomorphisms in F F.
Definitions of geometry
 Geometry is the study of curvature (Atiyah's video talk on Geometry in 2, 3 and 4 dimensions. Intrinsic and extrinsic curvature. Sphere has constant curvature. Sphere  positive  genus 0. Torus (cylinder)  zero curvature  genus 1. Higher genus  negative curvature.
 2 dimensions  Scalar curvature R
 3 dimensions  Ricci curvature Rij
 4 dimensions  Riemann curvature Rijk
 Geometry in even and odd dimensions is very different (real and complexes). Boundary of n has dimension n1. Icosahedron is the fake sphere in 3dimensions and it is related to nonsolvability of the quintic and to the Poincare conjecture. Icosahedron would be in A5 but reality is given by A4 and so A5 is insolvable!
 Wikipedia defines geometry as "concerned with questions of shape, size, relative position of figures, and the properties of space".
 MathWorld defines geometry as "the study of figures in a space of a given number of dimensions and of a given type", and formally, as "a complete locally homogeneous Riemannian manifold".
 nLab seems to define it as part of an Isbell duality between geometry (presheaves) and algebra (copresheaves) where presheaves (contravariant functors C>Set) and copresheaves (functors on C) are identified with each other and thus glued together (for some category C).
 At its roots, geometry consists of a notion of congruence between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a Cartan geometry is a generalization of this notion of congruence to allow for curvature to be present. The flat Cartan geometries — those with zero curvature — are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.
 Algebraic geometry is the study of spaces of solutions to algebraic equations.
Dimension
 Notions of dimension d (Mathematical Companion):
 locally looks like ddimensional space
 the barrier between any two points is never more than d1 dimensional
 can be covered with sets such that no more than d+1 of them ever overlap
 the largest d such that there is a nontrivial map from a ddimensional manifold to a substructure of the space
 the sum of dth powers of the diameter of squares that cover the object, with d such that the sum is between zero and infinity
Plane Geometry
Videos
Affine and Projective Geometry
 Introduction to Geometry by Coxeter.
 Norman Wildberger
 Geometry at CuttheKnot
 Eccentricity defines a conic as the points such that a fixed multiple (the eccentricity) times the distance to a line (directrix) is equal to the distance to a point (focus). The conic is thus a lens (God the Spirit) that relates the line (God the Father) and the point (God the Son). The conic thus relates a higher dimension (the line) with a lower dimension (the point), thus expanding one's perspective. Also, the directrix and focus bring to mind Appolonian polarity.
 Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.
 Sylvain Poirier: We can understand the stereographic projection as the effect of the
projective transformation of the space, which changes the sphere into
a paraboloid, itself projected into an affine space.
 (1 + ti)(1 + ti) = (1  t2) + (2t) i is the rational parametrization of the circle.
 What about the sphere? The stereographic projection of the circle onto the plane in Cartesian coordinates is given by (1 + xi + yj)(1 + xi + yj) where ij + ji = 1, that is, i and j anticommute.
 Note also that infinity is the flip side of zero  they make a pair. They are alternate ways of linking together the positive and negative values.
 squarerootofpi is gammaofnegativeonehalf (relate this to the volume of an odddimensional ball: pitothen/2 over (n/2)!
Modern Algebraic Geometry
Intuition
Ravi Vakil: The intuition for schemes can be built on the intuition for affine complex varieties. Allen Knutson and Terry Tao have pointed out that this involves three different simultaneous generalizations, which can be interpreted as three large themes in mathematics.
 (i) We allow nilpotents in the ring of functions, which is basically analysis (looking at nearsolutions of equations instead of exact solutions).
 (ii) We glue these affine schemes together, which is what we do in differential geometry (looking at manifolds instead of coordinate patches).
 (iii) Instead of working over C (or another algebraically closed field), we work more generally over a ring that isn’t an algebraically closed field, or even a field at all, which is basically number theory (solving equations over number fields, rings of integers, etc.).
http://drmichaelrobinson.net/sheaftutorial/index.html
Ideas
Videos
Books
Sheaves
Schemes
Homology and Cohomology
 Weibel, Homological Algebra
 Coherent sheaf cohomology
 Motives and Universal cohomology. Weil cohomology theory and the four classical Weil cohomology theories (singular/Betti, de Rham, ladic, crystalline)
 spectrum  topology, cohomology
 Our Father relates a left exact sequence and a right exact sequence.
 Divisions of everything are given by finite exact sequences which start from a State of Contradiction and end with that State.
 Short exact sequence: kernel yra tuo pačiu image. Tai, matyt, yra pagrindas trejybės poslinkio, išėjimo už savęs.
 Long exact sequence from short exact sequence: derived functors.
 Dievas žmogui yra skylė gyvenime, prasmė  neaprėpiamumo, kurios ieško pasaulyje, panašiai, kaip savyje jaučia laisvės tėkmę. Atitinkamai dieviška yra skylė matematikoje  homologijoje.
 Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes?
Other Geometry
影片
Noncommutative geometry
Books
Relating Geometries
History of Geometry
Books
Organizing Geometry
Intuition
Symmetry
Different geometries
Geometry challenges
 Dimension 3: relate Jones quantum invariants (knots, any manifold) with PerlmanThurston.
 Dimension 4: understand the structure of simplyconnected 4manifolds and the relation to physics.
 Atiyah speculation: Space + Circle = 4 dimensions (Riemannian). Donaldson theory > geometric models of matter? Signature of 4manifold = electric charge. Second Betti number = number of protons + neutrons.
Construction of the continuum
 Start with 0 dimension: a point. Every point is the same point. Then consider embedding a point in 1 dimension. The point does not yet distinguish between the two sides because there is no orientation. A distinction comes with the arisal of a second point. But whether the second point distinguishes the two sides depends on global knowledge. So there must be a third point. This is the relationship between "persons": I, You, Other. Either the dimension is a closed curve or it is an open line. This is "global knowledge". So there is the distinction between local knowledge and global knowledge. But basically geometry is a construction of the continuum, either locally or globally. The construction takes place through infinite sequences, through completion. This completion is not relevant for all constructions.
A System of Geometries
Ideas
 Each kind of geometry is based on a different tool set for constructions, on different symmetries, and on a different relationship between zero and infinity. And a different way of relating two dimensions.
 Each geometry is the action of a monoid, thus a language. But that monoid may contain an inverse, which distinguishes the projective geometry from the affine geometry.
 In a free monoid the theorems are equations and they are determined by what can be done with associativity. This is first order logic. A second order logic or higher order logic would be given by what can be expressed, for example, by counting various possibilities.
 Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distancepreserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a Gstructure, where G is a Lie group of "local" symmetries of a manifold.
Distinct Geometries
Special geometries
 Euclidean geometry: empty space + tools: straightedge, compass, area measurer
 most important theorem: Pythagoras q=q1+q2
 (q1+q2+q3)2 = 2(q1^2 + q2^2 + q2^3)
 Ordered geometry features the concept of intermediacy. It is a common foundation for affine, Euclidean, absolute geometry and hyperbolic geometry, but not projective geometry. Like projective geometry, it omits the notion of measurement.
 Absolute geometry, also known as neutral geometry, is based on the axioms of Euclidean geometry (including the first four of Euclid's axioms) but with the parallel postulate removed.
These geometries show how to relate (ever more tightly) two distinct dimensions.
How do we assign this needed structure? Such a local structure could provide a measure of ‘distance’ between points (in the case of a metric structure), or ‘area’ of a surface (as is speciWed in the case of a symplectic structure, cf. §13.10), or of ‘angle’ between curves (as with the conformal structure of a Riemann surface; see §8.2), etc. In all the examples just referred to, vectorspace notions are what are needed to tell us what this local geometry is, the vector space in question being the ndimensional tangent space Tp of a typical point p of the manifold M (where we may think of Tp as the immediate vicinity of p in M ‘infinitely stretched out’; see Fig. 12.6). Penrose, Road to Reality, page 293, §14.1.
Notes
 Try to understand asymmetric functions, for example, by setting {$q_3^2=0$}.
 Space arises with bundles, which separate the homogeneous choice, relevant locally, from the position, given globally. And then what does this say about time?
 A geometry (like hyperbolic geometry) allows for a presentation of a bundle, thus a perspective on a perspective (atsitokėjimas  atvaizdas). Compare with: įsijautimasaplinkybė.
 Geometry is concrete, it is a manifestation. Thus group actions can represent an abstract structure (of actions) in terms of a geometrical space (set, vector space, topological space). And this makes for understanding of the abstract in terms of the concrete.
 Euclidean space  (algebraic) coordinate systems  define left, right, front, backwards  and this often makes sense locally  but this does not make sense globally on a sphere, for example
 Quadratic forms are key for geometry. Moebius transformation involves complex numbers and 2 x 2 matrix. Whereas Clifford algebra involves quadratic form  as a lens for perspectives? Think of a linear form (proportion) as a plain sheet of glass, and a quadratic form as a convex and/or concave lens.
Notes
 David Corfield: Spatial notions of cohesion as the basis for geometry. A fourfold adjunction: components {$\dashv$} discrete {$\dashv$} points {$\dashv$} codiscrete, and a threefold adjunction of modalities based on that, originally due to Lawvere.
John Baez about toposes and Lawvere:
1999. I asked God which questions I should think over so as to understand why good will makes way for good heart. He responded:
 What captures attention and guides it? mažėjantis laisvumas
 What drops down upon reality and bounces away in random paths? didėjantis laisvumas
 What is wound in one direction, and lives through spinning in the opposite direction? prasmingas  kodėl
 What falls as rain day and night until there sprout and grow plants that will bear fruit? pastovus  kaip
 What like a ray reflects off of society and does not return? betarpiškas  koks
 What by its turning (in the direction of winding) commands our attention and then slips away to the side? tiesus  ar
