Revelations in Mathematics

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Revelations in Mathematics

• Presented by
• Catherine Sedalnick

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The many uses of Mathematics

• Within the last few decades, there have been many
  discoveries in mathematics.
• Some have had practical uses
• Some are better left as interesting theories

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The Four Color Theorem

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Four Muah ha ha!!!

• AKA The Four Color Map Theorem
• A map that is separated into regions can only use
  four different colors
• Adjacent regions cannot be the same color
• Adjacent means that a border is shared
• First Theorem to be proven by a computer

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Things that make you go hmmm

• In 1852, Francis Guthrie came up with the 4 color
  conjecture
• Several early failed attempts at proving the
  theorem
• Attempts were made by
• Alfred Kempe in 1879 (proven to be wrong in
  1890)
• Peter Guthrie Tait in 1880 (proven to be wrong in
  1891)
• Heinrich Heesch attempted to use a computer
  (1960s and 1970s)
• Kenneth Appel and Wolfgang Haken finally proved
  the four-color conjecture in 1976

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What does this mean?

• Efficient algorithms have been made
• This is not really used for cartographers
• It is not practical because many things such as
  lakes have to be the same color

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To Fermat or not to Fermat

• 350 Years Later, Fermat's Last Theorem Finally
  Proven

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Lots of Information

• Pierre de Fermat came up with a theorem in the
  1630s that, until recently, had not been solved.
• Fermat claimed that there are no positive
  integers for which xn yn zn when n is greater
  than 2.
• Andrew Wiles, from Princeton University, finally
  did it!!! Yay!!! ?
• He was intrigued from the time he was a teenager
  and decided to pursue a career in mathematics.

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Second times a charm

• In 1986, mathematician Ken Ribet and the
  Taniyama-Shimura conjecture
• June 23, 1993, Wiles announced his solution
• In 1994, the problem was presented again with the
  help of a colleague, Richard Taylor.

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• The new manuscript is over 100 pages of complex
  math.
• The manuscript included the following
• Selmer groups
• Hecke algebras
• Elliptic curves
• Modular forms
• Euler systems
• Galois representations
• Number Theory and Geometry were used to solve
  this theorem.

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And the prize goes to

• A prize established in 1906 to reward the first
  valid proof of Fermat's Last Theorem
• Too late for the Field Medal
• International Mathematics Union honored Wiles
  with a special silver plaque in 1998

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Roman A.Polyak
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Mathematics to the Rescue!!!

• Dr. Roman Polyak found ways to treat cancer by
  using abstract algebra on March 5, 2008
• Polyak works in the field of mathematical
  optimization at George Mason University
• His work is accurate to the 10th digit.
• Application of optimization
• Image processing, medical diagnostics and
  finding the optimal distribution of power across
  a grid

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Dr. Roman Polyak continued

• Because he was a refusenik, he was unable to
  publish his research in the Soviet Union
• Family, friends, and math sustained him.
• He said that "If I couldn't do my mathematics, I
  would be spiritually dead."

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References

• Dr. Roman Polyak http//www.nsf.gov/discoveries/di
  sc_summ.jsp?cntn_id111193orgNSF
• Dr. Roman Polyakhttp//mason.gmu.edu/rpolyak/poly
  ak.html
• Fermats Theorem
• http//www.nsf.gov/discoveries/disc_summ.jsp?cntn_
  id100029orgNSF
• Problems solved
• http//en.wikipedia.org/wiki/Unsolved_problems_in_
  mathematicsProblems_solved_recently
• Four Color Theorem
• http//en.wikipedia.org/wiki/Four_color_theorem
• http//en.wikipedia.org/wiki/Four_color_theorem