Madrid - April, 2017
J. DIEUDONNÉ, interpreted by @hannelita
Algebraic
Geometry
Computational
Geometry
GIS, computer vision, computer-aided engineering, Integrated-circuit design
According to Wikipedia:
and presentation
a. Mention something that happened in history
b. Associate with a field in Mathematics
c. Come up with interesting relations
I don't understand 100% of the definitions on this paper :')
We won't cover every detail on this extensive study
No formalism - sorry! Simplicity.
Better connection between maths - programming
New fields for research
How are these pieces connected?
Too many fields, too many theorems
Algebraic Geometry
Classification
Transformation
Infinitely
near points
Extending the Scalars
Extending the space
Analysis and Topology
Commutative Algebra
Geometry to solve algebraic problems
Apollonius studies conics
Beginning of analytic geometry
~630 BC
1630 - 1795
Descartes, Fermat, Newton, Leibniz
Parametric representations, surfaces, coordinates
1630 - 1795
Descartes, Fermat, Newton, Leibniz
Parametric representations, surfaces, coordinates
Classification
Transformation
Infinitely
near points
1630 - 1795
Descartes, Fermat, Newton, Leibniz
Parametric representations, surfaces, coordinates
Classification
Transformation
Infinitely
near points
Theory of Determinants
1795 - 1850
Development of conics and quadrics (FINALLY!)
1795 - 1850
Development of conics and quadrics (FINALLY!)
Extending the Scalars
Extending the space
1795 - 1850
Development of conics and quadrics (FINALLY!)
Extending the Scalars
Extending the space
Complex points
n-dimensional spaces
1795 - 1850
Klein
Geometry and Group Theory (Galois)
WHAT IS GROUP THEORY?
"Group Theory tries to collect patterns in mathematical objects and put them in such way that we can analyse symmetries"
TAVANTE, H. 2016 (me)
https://medium.com/@hannelita/a-summary-of-topics-in-mathematics-ff573e520986
If I take this object, what happens if I rotate it clockwise? Is there any other element with the same behaviour?
Group theory helps you to detect patterns.
If I make some transformations on this object, it will be identical to another structure
If I take this object, what happens if I rotate it clockwise? Is there any other element with the same behaviour?
Group theory helps you to detect patterns.
If I make some transformations on this object, it will be identical to another structure
Transformation
Forget the distance. This about the configuration of the points and lines. The metric doesn't matter.
Desargues, Poncelet
1850 - 1866
Riemann
"... transcendental approach via abelian integrals was important to algebraic geometry"
1850 - 1866
Riemann
"... transcendental approach via abelian integrals was important to algebraic geometry"
Me: "I don't understand the meaning of 'abelian integral' "
Abelian integral
Why is it important?
Multi-valued expression that is easy to manipulate!
Multi-valued expressions can denote surfaces
Riemann comes in! Using Abel ideas, he changes the initial object
Function with this property
Holomorphic
(Complex function differentiable in every point of its domain)
Except on single "peaks"(poles)
Meromorphic
Function
Function with this property
Holomorphic
(Complex function differentiable in every point of its domain)
Except on single "peaks"(poles)
Meromorphic
Function
(Ex: Gamma function ^)
How can I compare them in different spaces?
Take Abel's rational functions idea - Birrational geometry
The object is always a rational
Riemann notices some invariants on these functions (genus)
Classification
1866 - 1920
Riemann inserted several ideas for algebraic geometry. Different schools tried to extend them with different approaches
4 different attempts:
1866 - 1920
a) The algebraic approach - Kronecher, Dedeking and Weber
Algebraic geometry and Number theory
1866 - 1920
a) The algebraic approach - Kronecher, Dedeking and Weber
Algebraic geometry and Number theory
"adding and product of integers (module a prime number) has
the algebraic structure of a finite field" - credits by Luis L., attendee
1866 - 1920
a) The algebraic approach - Kronecher, Dedeking and Weber
Algebraic geometry and Number theory
Commutative Algebra
1866 - 1920
a) The algebraic approach - Kronecher, Dedeking and Weber
Algebraic geometry and Number theory
Commutative Algebra
(Study of objects where the multiplication is commutative)
1866 - 1920
a) The algebraic approach - Kronecher, Dedeking and Weber
Algebraic geometry and Number theory
Commutative Algebra
(call these objects 'rings')
1866 - 1920
a) The algebraic approach - Kronecher, Dedeking and Weber
Algebraic geometry and Number theory
Commutative Algebra
*What is a non-example of commutative ring?*
Why is commutative algebra important?
Statistics and combinatory - think about ML and AI
a) Algebraic geometry and Number theory
M. Noether and Brill
b) Theory of linear systems of points of a curve
Infinitely
near points
Extending the space
No rational functions - any polynomials.
Quadratic transformations
Cayley, Clebsch and Noether
c) Integrals of differential forms on higher dimensional varieties
Back to rational functions (ROLLBACK!)
Cayley, Clebsch and Noether
c) Integrals of differential forms on higher dimensional varieties
Back to rational functions (ROLLBACK!)
Analysis and Topology
Infinitely
near points
c) Integrals of differential forms on higher dimensional varieties
What would it be a good invariant to observe in a surface?
"Number of holes"
How do you measure that in an n-dimensional scenario?
c) Integrals of differential forms on higher dimensional varieties
Betti number: "number of k-dimensional holes on a topological surface"
By H. Poincaré
Consider the Betti number on further analysis
d) Linear systems and the Italian School
Castelnuovo, Enriques and Severi
Claim: "Purely geometric", no "abstract"
Limitations to some transformations
d) Linear systems and the Italian School
Castelnuovo, Enriques and Severi
Claim: "Purely geometric", no "abstract"
Limitations to some transformations
Classification
1920 - 1950
Unification of mathematics through structures
Manifold
"some space such that if you zoom in, it looks like flat euclidean space."
4 New approaches for algebraic geometry (we will see 2 cases)
a) Kahlerian varieties
Differential
Geometry
Riemann geometry (curved surfaces) + Calculus
b) Abstract Algebraic Geometries
Commutative Algebra
Notion of abstract struct (ring, group, field) is solid
Noether, Krull, van de Waerden and F.K. Schmidt
b) Abstract Algebraic Geometries
Commutative Algebra
Ring on an Ascending Chain Condition (ex: the field of Rational numbers is a Noetherian ring)
Noether, Krull, van de Waerden and F.K. Schmidt
It reveals surprising connections
Why are we studying these topics?
It is fun (???)
History
"Unentangle" - "Unfortunately, the complexity of the Italian definitions was such that it was often impossible to be sure that the same words meant the same thing in two different papers"
We don't know!
Where are we going?
But look! Now geometry is "abstract", "complex", it has algebra, invariants, transformations and several other components far from those we learn in High School
c) Bonus: Zeta Functions and correspondences
A. Weil
c) Zeta Functions and correspondences
A. Weil
Infinitely
near points
Extending the Scalars
1950 - now
Sheaf
"A tool which provides a unified approach
to establishing connections between local
and global properties of topological
spaces"
(topological space: space of points and their neighbourhoods satisfying properties)
Associate algebraic objects with other distinct mathematical objects is the study of homology
You can map different types of objects.
commutative group
(abelian group)
Topological space
Cohomology
Why is it important?
Functions on topological space - homotopy
We have functions in programming (see my session about Type Theory :D )
And we can represent them under a topological perspective
But we need the tools from Algebraic Geometry
Maths - even if it sounds useless, it might be useful at some point
We need more content connecting the pieces
Analogy: math fields are like bad-planned micro services.
Keep it simple! :)
Questions?
hannelita@gmail.com
@hannelita