Topics in Mathematics and Computer Science

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Topics in Mathematics and Computer Science

• CSUCI Masters Seminars
• August 31, 2005
• Dr. AJ Bieszczad
• Dr. Cindy Wyels

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What is Mathematics?

• Mathematics is the study of quantity, structure,
  space, and change. Historically, mathematics
  developed from counting, calculation,
  measurement, and the study of the shapes and
  motions of physical objects, through the use of
  abstraction and deductive reasoning. (Wikipedia)
• Mathematics the abstract science of number,
  quantity, and space studied in its own right
  (pure mathematics) or as applied to other
  disciplines such as physics, engineering, etc.
  (applied mathematics). (Oxford)
• Mathematics the science of numbers and their
  operations, interrelations, combinations,
  generalizations, and abstractions and of space
  configurations and their structure, measurement,
  transformations, and generalizations (Merriam
  Webster)

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What do some experts say?

• Mathematics is the tool specially suited for
  dealing with abstract concepts of any kind and
  there is no limit to its power in this field.
  Paul Dirac (1902 1984)
• Mathematics is the science of patterns. Keith
  Devlin
• Mathematics is a language. Josiah Willard Gibbs
  (1839 1903)
• Mathematics is not a deductive science -- that's
  a cliche. When you try to prove a theorem, you
  don't just list the hypotheses, and then start to
  reason. What you do is trial and error,
  experimentation, guesswork. Paul Halmos

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Some perspectives

• There is no branch of mathematics, however
  abstract, which may not some day be applied to
  phenomena of the real world. Nikolai
  Lobatchevsky (1792 1856)
• Mathematics knows no races or geographic
  boundaries for mathematics, the cultural world
  is one country. David Hilbert (1862 1943)
• Anyone who cannot cope with mathematics is not
  fully human. At best he is a tolerable subhuman
  who has learned to wear shoes, bathe, and not
  make messes in the house. Robert A. Heinlein
• Life is good for only two things, discovering
  mathematics and teaching mathematics. Simeon
  Poisson (1781 1840)

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One scheme for classifying mathematics

• Foundations
• considers questions in logic or set theory --
  the language of mathematics
• Algebra
• symmetry, patterns, discrete sets, and the rules
  for manipulating arithmetic operations
• Geometry
• shapes and sets, properties of shapes and sets
  that are preserved under various kinds of motions
• Analysis
• functions, the real number line, and the ideas
  of continuity and limit
• Probability and Statistics
• can be mathematical and/or experimental
• Computer sciences
• algorithms, information handling, etc.
• Mathematics Education History of Mathematics
• Mathematics used to express ideas in physical
  sciences, engineering, etc.

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The NSF Organizational Scheme

• Algebra and Number Theory
• Topology and Foundations
• Geometric Analysis
• Analysis
• Statistics and Probability
• Computational Mathematics
• Applied Mathematics

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An AMS Scheme for tracking new Ph.D.s

• Logic/Discrete Math/Combinatorics/Computer
  Science (90)
• Algebra and Number Theory (169)
• Geometry and Topology (132)
• Differential, Integral, and Difference Equations
  (98)
• Real, Complex, Functional, and Harmonic Analysis
  (105)
• Numerical Analysis, Approximations (79)
• Probability (51)
• Statistics (269)
• Applied Mathematics (100)
• Linear, Nonlinear Optimization and Control (23)
• Mathematics Education (14)
• Other/Unknown (3)

Numbers from 1999
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Mathematical areas of some local experts

• Logic/Philosophy Dennis Slivinski
• Discrete Math and Combinatorics Cindy Wyels,
  Ron Rieger, Nathaniel Emerson
• Computer Science  Bill Wolfe, AJ Bieszczad, Anna
  Bieszczad
• Algebra Jesse Elliott, Ivona Grzegorczyk,
  Morgan Sherman
• Number Theory  Jesse Elliott
• Geometry Ivona Grzegorczyk, Morgan Sherman
• Topology   Mohamed Ait Nouh, Peter Yi
• Differential Equations and Dynamical Systems
  Nathaniel Emerson
• Real and Complex Analysis  Roger Roybal,
  Nathaniel Reid, Nathaniel Emerson, Jorge Garcia
• Probability Jorge Garcia
• Statistics James Sayre, Matthew Wiers, Jorge
  Garcia
• Applied Mathematics  Aemiro Beyene, Greg Woods
  (Phys), Tabitha Swan-Wood (Phys), Jerry Clifford
  (Phys), Nick Bosco (Phys)
• Imaging/ Pattern Recognition  Geoff Dougherty
  (Phys)
• Linear, Nonlinear Optimization and Control  Ron
  Rieger
• Mathematics Education  Ivona Grzegorczyk,
  Marguerite George, Steven Thomassin, Cindy Wyels,
  Merilyn Buchanan

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Open Problem Geometry and Logic

•  
• Give criteria that may be used to determine
  whether the plane may be tiled with a random
  tile.
• (Suggested here by Ivona Grzegorczyk)

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Open Problem Operator theory

•   Consider polynomials of one variable that take
  on only non-negative values. It has been shown
  that such polynomials are each a sum of the
  squares of at most two polynomials.
• But, if we consider polynomials in two or more
  variables, there exist positive polynomials that
  are not the sum of any number of squares.
• An important problem is to find a
  complete algebraic characterization of all such
  polynomials.
• A solution would have important ramifications in
  real algebraic geometry, operator theory, medical
  imaging, and general mathematical awesomeness.
• (Suggested here by Roger Roybal)

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Open Problem Complex Dynamics

•  
• Is there a Julia set of a polynomial with
  positive area (Lebesgue measure)?  For all
  polynomials where the area is known, it is 0.
• (Suggested here by Nathaniel Emerson)

pictures borrowed from Wikipedia
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Open Problem Geometry and Number Theory

•  
• How many points can you find on the (half)
  parabola y x2, x gt 0, so that the distance
  between any pair of these points is rational?
• This sounds like geometry, but it is likely to
  require techniques in number theory we dont
  really know!
• Source Nate Dean at DIMACS http//dimacs.rutger
  s.edu/hochberg/undopen/

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Open Problem Number Theory

•  
• The Twin Prime Conjecture
• There are infinitely many twin primes.
• Or you might prefer
• Are there infinitely many primes of the form
  n21?

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Open Problem Combinatorial Game Theory

•   Gales Vingt-et-un game
• Cards numbered 1 through 10 are laid on the
  table. L chooses a card. Then R chooses cards
  until his total of chosen cards exceeds the card
  chosen by L. Then L chooses until her cumulative
  total exceeds that of R, etc. The first player
  to get 21 wins. Who is it?

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Open Problem Graph Theory

•  
• What is the crossing number of the complete
  bipartite graph K(9, 9)?
• Equivalently Place 9 red points in the plane
  and 9 blue points in the plane, and then connect
  each red point to each blue point with curves (81
  curves in all). What is the minimum number of
  crossing points that must appear in your drawing?
• It is conjectured to be 256, but nobody knows.
• An example for K(4,4) is shown to the right,
  with 8 crossings. Actually, this graph can be
  drawn with just 4 crossings...can you find it?
• Source Robert Hochberg at DIMACS
  http//dimacs.rutgers.edu/hochberg/undopen/

K4,4
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Is any of this stuff practical?
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References for further investigation

• The Mathematical Atlas a fantastic source for
  learning more about various subfields of
  mathematics. Very comprehensive.
  http//www.math.niu.edu/rusin/known-math/index/to
  ur.html
• Math on the Web (by the AMS) web material
  organized by mathematical topics. Extensive.
  http//www.ams.org/mathweb/mi-mathbytopic.html
• The Math Archives web resources grouped by
  mathematical topic. http//archives.math.utk.edu/
  topics/
• MathPages a series of well-written articles
  outlining interesting subtopics/ problems,
  organized by mathematical topic. Very extensive.
  http//www.mathpages.com/home/index.htm
• SIAM, Mathematics in Industry report examines
  applications of mathematics to problems arising
  in industry, government, and business.
  Thought-provoking. http//www.siam.org/mii/
• Unsolved problems this site gives links to many
  pages giving unsolved problems.
  http//www.mathsoft.com/mathsoft_resources/unsolve
  d_problems/1999.asp