László Lovász
Eötvös Loránd University
Budapest
1

General trends in mathematical activity
• The size of the community and of mathematical research activity increases exponentially.
• New areas of application, and their increasing significance.
• New tools: computers and information technology.
• New forms of mathematical activity.
2

Size of the community and of research
Mathematical literature doubles in every 25 years
Impossible to keep up with new results: need of more efficient cooperation and better dissemination of new ideas.
Larger and larger part of mathematical activity must be devoted to communication (conferences with expository talks only, survey volumes, internet encyclopedias, multiple authors of research papers...)
3

Size of the community and of research
Challenges in education:
Difficult to identify ``core'' mathematics
Two extreme solutions:
New results, theories, methods belong to Masters/PhD programs
- Leave out those areas that are not in the center of math research today
4

Size of the community and of research
Challenges in education:
Difficult to identify ``core'' mathematics
Focus on mathematical competencies
(problem solving, abstraction, generalization and specialization, logical reasoning, mathematical formalism)
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Size of the community and of research
Challenges in education:
Exposition style mathematics in education: teach students to explain mathematics to
“outsiders” and to each other, to summarize results and methods,...
teach some mathematical material
“exposition style”?
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Applications: new areas
Traditional areas of application: physics, astronomy and engineering.
Use: analysis, differential equations.
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Applications: new areas
Biology: genetic code population dynamics protein folding
Physics: elementary particles, quarks, etc.
(Feynman graphs) statistical mechanics
(graph theory, discrete probability )
Economics: indivisibilities
(integer programming, game theory)
Computing: algorithms, complexity, databases, networks, VLSI, ...
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Applications: new areas
Traditional areas of application: physics, astronomy and engineering.
Use: analysis, differential equations.
New areas: computer science, economics, biology, chemistry, ...
Use: most areas (discrete mathematics, number theory, probability, algebra,...)
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Applications: new areas
Very large graphs:
-Internet
@Stephen Coast
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Applications: new areas
Very large graphs:
-Internet
Social networks
-Ecological systems
-chip design
-Statistical physics
-Brain
What properties to study?
-Does it have an even number of nodes?
-How dense is it
(average degree)
-Is it connected?
?
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Applications: new areas and significance
Challenges in education:
- Explain new applications programming, modeling,...
- Train for working with non-mathematicians interdisciplinary projects, modeling,...
12

New tools: computers and IT
Source of interesting and novel mathematical problems >new applications
New tools for research (experimentation, collaboration, data bases, word processing, new publication tools)
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New tools: computers and IT
Challenges in education:
- Students are very good in using some of these tools. How to utilize this? nonstandard mathematical activities
- How to make them learn those tools that they don’t know?
14

New forms of mathematical activity
Algorithms and programming
Algorithm design is classical activity
(Euclidean Alg, Newton's Method,...) but computers increased visibility and respectability.
15

An example: diophantine approximation and continued fractions
Given
  such that |
,
/ find rational approximation
 
/ q and q

  m
 n


| m
  n
 p | | q
 p | q

1/


  a
0
   a
0
?

1
 a
0
 a
1



1
 a
0


  a
0

1 a
1 continued fraction expansion
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New forms of mathematical activity
Algorithms and programming
Algorithm design is classical activity
(Euclidean Alg, Newton's Method,...) but computers increased visibility and respectability.
Algorithms are penetrating math and creating new paradigms.
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1930’s
Mathematical notion of algorithms recursive functions, Λ-calculus, Turing-machines
Church, Turing, Post algorithmic and logical undecidability
Church, Gödel
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1960’s
Computers and the significance of running time simple and complex problems sorting searching arithmetic
…
Travelling Salesman matching network flows factoring
…
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7080’s
Complexity theory
Time, space, information complexity
Nondeterminism, good characteriztion, completeness
Polynomial hierarchy
Classification of many real-life problems into P vs. NP-complete
Randomization, parallelism
P=NP?
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90’s
Increasing sophistication: upper and lower bounds on complexity algorithms negative results factoring volume computation semidefinite optimization topology algebraic geometry coding theory
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90’s
Approximation algorithms positive and negative results
Probabilistic algorithms
Markov chains, high concentration, phase transitions
Pseudorandom number generators from art to science: theory and constructions
Cryptography state of the art number theory
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New forms of mathematical activity
Challenges in education:
Balance of algorithms and theorems
Algorithms and their implementation develop collections of examples, problems...
No standard way to describe algorithms: informal? pseudocode? program?
develop a smooth and unified style for describing and analyzing algorithms
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New forms of mathematical activity
Problems and conjectures
Paul Erd ős: the art of raising conjectures
Best teaching style of mathematics emphasizes discovery, good teachers challenge students to formulate conjectures.
Challenges in education:
Preserve this!!
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New forms of mathematical activity
Mathematical experiments
Computers turn mathematics into an experimental subject.
Can be used in the teaching of analysis, number theory, optimization, ...
Challenges in education:
Lot of room for good collection of problems and demo programs
25

New forms of mathematical activity
Modeling
First step in successful application of mathematics.
Challenges in education:
Combine teaching of mathematical modeling with training in team work and professional interaction.
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New forms of mathematical activity
Exposition and popularization
Growing very fast in the research community.
Notoriously difficult to talk about math to non-mathematicians.
Challenges in education:
Teach students at all levels to give presentations, to write about mathematics.
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