Juan Carlos Ponce Campuzano
Independent Mathematics Educator
Juan Carlos Ponce Campuzano
I acknowledge the Traditional Owners and their custodianship of the lands on which we meet today and pay my respect to their Ancestors and their descendants.
Image: Digital reproduction of A guidance through timeΒ by Casey Coolwell and Kyra Mancktelow
What is Mathematical Analysis?
What is Mathematical Analysis?
Essentially, it is the study of limts (and related theories) using the axiomatic properties of real numbers.
This course is proof oriented instead of an application oriented course
What are Axioms?
Axioms
Definitions
All Mathematical Theories
Non-Euclidian Geometry
Group Theory
Algebra
Linear Algebra
Game Theory
Topology
Category Theory
Euclidian Geometry
Calculus
Number Theory
What are Axioms?
Euclidian Geometry
Euclid's Elements (~2300 years old)
Definitions
Point
Straight line
Circle
Right angle (\(90^\circ\))
What are Axioms?
Euclidian Geometry
Euclid's Elements (~2300 years old)
Axioms
What are Axioms?
Euclidian Geometry
Euclid's Elements (~2300 years old)
Axioms
Proposition 47, Book I
What are Axioms?
Euclidian Geometry
Euclid's Elements (~2300 years old)
Axioms
Elisha S. Loomis (1940)
Mathematical Analysis
We are going to prove statements like:
Mathematical Analysis
Traditionall, a rigorous first corse in Analysis progresses (more or less) in the follwoing order:
In the other hand, the historical development of these subjects ocurred in reverse order:
sets,
real numbers
limits,
continuous functionns
derivatives
integration
Cantor 1874,
Dedekin
Cauchy 1823,
Riemann,
Weiestrass
Newton 1669
Leibniz 1684
Archimedes
3rd B.C.
Archimedes
c. 287 BC - c. 212 BC
0
1600
Liu Hui
c. 220 - c. 280
200
Hindu-Arabic numerical system
Algebra: Muhammad ibn Musa al-Khwarizmi
Analytic Geometry:
Descartes & Fermat
-200
Time line
Archimedes
c. 287 BC - c. 212 BC
1669
Newton
1684
Leibniz
1872
Dedekin
1874
Cantor
1854
Riemann
1755
Euler
...
0
1600
1900
1700
1800
2000
Cauchy
1823
Liu Hui
c. 220 - c. 280
1748
Agnesi
1861
Weiestrass
Time line
(roughly)
13 Weeks
Good news!
You already know most of the content.
Lectures
Tutorials
- Every week you will practice with a set of problems.
It covers core material not covered in lectures!
Lectures
- We will cover the theory with examples.
Most weeks there will be at most
one complementary reading π π
Important:
Available in Blackboard under Learning Resources
Most weeks there will be at most
one complementary reading π π
Important:
via weekly modules:
π
Why should I read the complementary material? π€
Most weeks there will be at most
one complementary reading π π
Important:
Ingrid Deubechies
Maryam Mirzakhaniβ
You will gain some experience to read maths papers:
Tutorials
Consult your timetable.
π Tutorials begin in week 2.
Ask questions about assignments, content and
problem sheets (available in Blackboard).
π You should attempt problems beforehand.
Link in Blackboard
MATH2400 & MATH7400
Sumbission will be online through Blackboard
Under Assessment look for the folder corresponding
to your course code MATH2400 or MATH7400
Check the Submission Instructions βΌοΈ
Assignments
Final Exam
Invigilated on-campus & over Zoom
Β for internal and external students, respectively
You will have two hours to
solve problems and formulate proofs
Final Exam
In which you may encounter some beautiful expressions such as:
Final Exam
But before, we need to practice a lot π βοΈ
π
By Juan Carlos Ponce Campuzano