Trends in Mathematics: How could they Change Education?

1
Trends in Mathematics How could they Change
Education?

• László Lovász
• Eötvös Loránd University
• Budapest

2
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.

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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...)

4
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

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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)

6
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?

7
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, ...
9
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

_at_Stephen Coast
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Applications new areas
Very large graphs
What properties to study?
-Does it have an even number of nodes?

• Internet

-Social networks

• Ecological systems

-How dense is it (average degree)?

• chip design

• Statistical physics

-Is it connected?

• Brain

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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,...

13
New tools computers and IT

• Source of interesting and novel mathematical
  problems gtnew applications
• New tools for research (experimentation,
  collaboration, data bases, word processing, new
  publication tools)

14
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
  dont know?

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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.

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An example diophantine approximation
and continued fractions
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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A mini-history of algorithms 1930s
Mathematical notion of algorithms
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A mini-history of algorithms 1960s
Computers and the significance of running time
simple and complex problems
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A mini-history of algorithms 70-80s
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
PNP?
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A mini-history of algorithms 90s
Increasing sophistication upper and lower
bounds on complexity
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A mini-history of algorithms 90s
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 Erdos 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

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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.