# Historical Development of Algebraic Geometry

J. DIEUDONNÉ, interpreted by @hannelita

# Who are you?

## Hi! I'm Hanneli (@hannelita)

• Computer Engineer
• Programming
• Electronics
• Math <3 <3
• Physics
• Lego
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## Olá! I'm Hanneli (@hannelita)

• Computer Engineer
• Programming
• Electronics
• Math <3 <3
• Physics
• Lego
• Meetups
• Animals
• Coffee
• Pokémon
• GIFs

# Who is this author?

• Jean Dieudonné - 1906 -1992
• He taught in France, in the USA and also in Brazil!
• Research in abstract algebra, algebraic geometry and functional analysis

# Why this paper?

Algebraic

Geometry

2 + 2 = 4
$2 + 2 = 4$

# What is the relation between curves and programming?

• Computer vision / Computer Graphics
• Linear/non-linear programming (geometric solutions)

Computational

Geometry

GIS, computer vision, computer-aided engineering, Integrated-circuit design

# What is the relation between geometry and programming?

• Computer vision / Computer Graphics
• Linear/non-linear programming (geometric solutions)
• Algorithm complexity
• Elliptic curve cryptography

# I was

According to Wikipedia:

# Historical Development of Algebraic Geometry

## The structure of this paper

and presentation

a. Mention something that happened in history

b. Associate with a field in Mathematics

c. Come up with interesting relations

# Disclaimer

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.

# Goals

Better connection between maths - programming

New fields for research

How are these pieces connected?

Too many fields, too many theorems

## "Algebraic geometry has more open problems than solved ones"

Algebraic Geometry

Classification

Transformation

Infinitely

near points

Extending the Scalars

Extending the space

Analysis and Topology

Commutative Algebra

# 1. The Greeks (The Geometry folks)

Geometry to solve algebraic problems

Apollonius studies conics

Beginning of analytic geometry

~630 BC

# 2. Analytic Geometry

1630 - 1795

Descartes, Fermat, Newton, Leibniz

Parametric representations, surfaces, coordinates

# 2. Analytic Geometry

1630 - 1795

Descartes, Fermat, Newton, Leibniz

Parametric representations, surfaces, coordinates

Classification

Transformation

Infinitely

near points

# 2. Analytic Geometry

1630 - 1795

Descartes, Fermat, Newton, Leibniz

Parametric representations, surfaces, coordinates

Classification

Transformation

Infinitely

near points

Theory of Determinants

## 3. Projective Geometry

1795 - 1850

Development of conics and quadrics (FINALLY!)

## 3. Projective Geometry

1795 - 1850

Development of conics and quadrics (FINALLY!)

Extending the Scalars

Extending the space

## 3. Projective Geometry

1795 - 1850

Development of conics and quadrics (FINALLY!)

Extending the Scalars

Extending the space

Complex points

n-dimensional spaces

## 3. Projective Geometry

1795 - 1850

Klein

Geometry and Group Theory (Galois)

WHAT IS GROUP THEORY?

### 3.1 Group theory made simple

"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

### 3.1 Group theory made simple

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

### 3.1 Group theory made simple

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

### 3.2 Towards projective geometry

Forget the distance. This about the configuration of the points and lines. The metric doesn't matter.

Desargues, Poncelet

## 4. Birrational Geometry

1850 - 1866

Riemann

"... transcendental approach via abelian integrals was important to algebraic geometry"

## 4. Birrational Geometry

1850 - 1866

Riemann

"... transcendental approach via abelian integrals was important to algebraic geometry"

## 4.1 Abel

Me: "I don't understand the meaning of 'abelian integral' "

## 4.1 Abel

Abelian integral

f(z,w) = z + (3w^2 - w + 2) = 0
$f(z,w) = z + (3w^2 - w + 2) = 0$
\int_{0}^{1} \frac{z}{w} dz
$\int_{0}^{1} \frac{z}{w} dz$
w(z) = \frac{1}{6(1 + \sqrt{23 - 12z})}
$w(z) = \frac{1}{6(1 + \sqrt{23 - 12z})}$
\int_{0}^{1}\frac{z}{\frac{1}{6(1 + \sqrt{23 - 12z})}} dz
$\int_{0}^{1}\frac{z}{\frac{1}{6(1 + \sqrt{23 - 12z})}} dz$

## 4.1 Abel

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

## 4.2 Back to group theory

Function with this property

Holomorphic

(Complex function differentiable in every point of its domain)

Except on single "peaks"(poles)

Meromorphic

Function

## 4.2 Riemann's idea

Function with this property

Holomorphic

(Complex function differentiable in every point of its domain)

Except on single "peaks"(poles)

Meromorphic

Function

(Ex: Gamma function ^)

### Random question: Given two functions, are they isomorphic (map one to another)?

How can I compare them in different spaces?

Take Abel's rational functions idea - Birrational geometry

The object is always a rational

## 4.2 Riemann's idea

Riemann notices some invariants on these functions (genus)

Classification

# Break - Breathe!

## 5. Development and Chaos

1866 - 1920

Riemann inserted several ideas for algebraic geometry. Different schools tried to extend them with different approaches

4 different attempts:

## 5. Development and Chaos

1866 - 1920

a) The algebraic approach - Kronecher, Dedeking and Weber

Algebraic geometry and Number theory

## 5. Development and Chaos

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

## 5. Development and Chaos

1866 - 1920

a) The algebraic approach - Kronecher, Dedeking and Weber

Algebraic geometry and Number theory

Commutative Algebra

## 5. Development and Chaos

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)

## 5. Development and Chaos

1866 - 1920

a) The algebraic approach - Kronecher, Dedeking and Weber

Algebraic geometry and Number theory

Commutative Algebra

(call these objects 'rings')

## 5. Development and Chaos

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?*

## 5. Development and Chaos

Why is commutative algebra important?

Statistics and combinatory - think about ML and AI

a) Algebraic geometry and Number theory

## 5. Development and Chaos

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.

## 5. Development and Chaos

Cayley, Clebsch and Noether

c) Integrals of differential forms on higher dimensional varieties

Back to rational functions (ROLLBACK!)

## 5. Development and Chaos

Cayley, Clebsch and Noether

c) Integrals of differential forms on higher dimensional varieties

Back to rational functions (ROLLBACK!)

Analysis and Topology

Infinitely

near points

## 5. Development and Chaos

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?

## 5. Development and Chaos

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

## 5. Development and Chaos

d) Linear systems and the Italian School

Castelnuovo, Enriques and Severi

Claim: "Purely geometric", no "abstract"

Limitations to some transformations

## 5. Development and Chaos

d) Linear systems and the Italian School

Castelnuovo, Enriques and Severi

Claim: "Purely geometric", no "abstract"

Limitations to some transformations

Classification

## 6. New Structures

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)

## 6. New Structures

a) Kahlerian varieties

Differential

Geometry

Riemann geometry (curved surfaces) + Calculus

## 6. New Structures

b) Abstract Algebraic Geometries

Commutative Algebra

Notion of abstract struct (ring, group, field) is solid

Noether, Krull, van de Waerden and F.K. Schmidt

## 6. New Structures

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

## BREAK!

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"

## BREAK!

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

## 6. New Structures

c) Bonus: Zeta Functions and correspondences

A. Weil

## 6. New Structures

c) Zeta Functions and correspondences

A. Weil

Infinitely

near points

Extending the Scalars

\zeta(s) = \sum_{1}^{\infty}\frac{1}{n^s}
$\zeta(s) = \sum_{1}^{\infty}\frac{1}{n^s}$

## 7. Sheaves and Schemes

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)

## 7. Sheaves

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

## 8. PHEW!

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

## 9. What I learned

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.

## 9. What I learned

Keep it simple! :)

## Thank you :)

Questions?

hannelita@gmail.com

@hannelita