History of Math #2
Describe how Beiberbach's political view shaped his role within mathematics
Beiberbach praised the Nazi's and because of this when he spoke against someone he forced tge German Mathematician's to take a side. He founded Deutsche Mathematik to promote german mathematics
What is the name and significance of Russell and Whitehead's famous book?
Principia Mathematica Attempted to describe a set of axioms in mathematical logic from which all of mathematics could be described
What is Hilbert's second problem?
Prove that the axioms of arithmetic are consistent
List three accomplishments of David Hilbert
Proved that E and Pi are transcendental Axiomized Geometry Completed the field of invariants Created Hilbert Space
Name three accomplishments of E.T. Bell
Published 15 novels Earned a Bocher Prize for his work in Paraphrases Published poetry
Describe visions of the future according to Doron Zeilberger
Semi-rigorous mathematics - it will be too expensive to get a complete proof and we will settle for getting .999999...
What is the significance of Nicolas Bourbaki?
Set out to kay a foundation that was broad enough to support the essential code of modern mathematics Wrote a series of books aiming to reestablish mathematics formally abstractly and rigorously
Who proved Fermat's Last Theorem?
Sir Andrew Wiles
List three accomplishments of John Forbes Nash
Solved the bargaining problem Game theory rests on the Nash equilibrium Earned PhD with 28 page dissertation on non-cooperative games
Which two mathematician's completed significant work on the fundamental lemma for the Langlands Program?
Thomas Callister Hales Ngo Bao-Chau
Name three combinatorial games
Tic-Tac-Toe Chess - assuming finite There are ten sticks on a table. Each player removes either 1 or 2 sticks per turn. The last player to moves wins.
Describe Von Neumann's impact on Economics and Computer Science
Tried to mathematize economics the way Newton mathematized Physics and Calculus Created the first real electronic computer with the inventors of ENIAC
What is the name and significance of Courant and Robbins famous book
What is Mathematics First time Courant's student was on the cover with him
Name accomplishment(s) of Terrence Tao
Won Bocher Prize Proved the Green-Tao Theorem Proved the Erdos Discrepancy Problem
Apply the Havel-Hakimi Algorithm to determine whether the dress sequence is graphic: A)54321 B)544221 C)54322222
Word Doc
List three accomplishments of Richard Courant
Wrote What is Mathematics with Herbert Hobbins Built up NYU's math department Built up Gottingen's math department Courant-Hilbert with Hilbert
List five key mathematicians in the 20th century and name their accomplishements
1) Stephen Hawking - theory of relativity and quantum mechanics 2) Pierre Deligne - Proof on one of Weil's conjecture, won fields medal and abel prize 3) David Blackwell - Roa - Blackwell Theorem 4) Jean - Pierre Serre - Helped link Fermat's Last theorem with the Taniyama-Shimura-Weil Conjecture, Youngest recipient of the Fields medal, first winner of the Abel prize 5) Paul Halmos - claims to have invented the abreviation iff for if and only if, originated the tombstone symbol to conclude proofs
State Godels incompleteness Theorems and discuss their significance
1. If a formal system is consistent then it cannot be complete 2. The consistency of axioms in a formal system cannot be proven within that system Therefore math is incomplete and inconsistent
Which of Hilbert's 23 problems have been solved?
5th problem on lie groups 10th problem on diophantine 19th problem on calculus variation
State one of Hilbert's problems and describe its resolution as of today
8th problem - the Riemann Hypothesis Unresolved
Which of Hilbert's 23 problems have not been solved?
9th problem on reimann hypothesis 12th problem The extension of the Kronecker-Weber theorem on abelian extensions of the rational numbers, to any base number field 16th problem Describe relative positions of ovals originating froma real algebraic curve and as limit cycles of a polynomial vector field on the plane
What is the significance of the polymath project?
A collaboration between mathematicians that aims to solve difficult and/or important problems
What is the significance of the IAS?
A peaceful place where academics would work in the pursuit of knowledge
Name five individuals who attempted and failed to prove the four color theorem
Alfred Kempe Peter Tait Hermann Minkowski Philip Franklin J. Mayer
Describe Russell's paradox and discuss it's significance
An ordinary set is a set that does not contain itself An extraordinary set is a set that DOES contain itself Let S be the set containing all ordinary sets...is S ordinary or extraordinary? If S is ordinary it would have to be in S. However, this would make S extraordinary. If S is extraordinary it would not be in S. However, this would make S ordinary. Therefore S is neither ordinary or extraordinary
Who served as the de factor leader of the Bourbaki group?
Andre Weil
Name five members of the Bourbaki group
Andre Weil Jean Pierre-Serre Henri Cartan Claude Chevalley Laurent Schwartz
Prove that in any gathering of six people, there are either three people who are mutually acquainted of three people who are mutually unacquainted or both
Assume the six people are named 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, 𝐹 Focus in on vertex 𝐴 Assume 𝐴 connects to three (or more) other vertices in 𝐺 ... say 𝐵, 𝐶, 𝐷 If any one of 𝐵𝐶 , 𝐵𝐷 , {𝐶𝐷} appear in 𝐺, then we are done! If NOT, then ALL of 𝐵𝐶 , 𝐵𝐷 , {𝐶𝐷} appear in 𝐺3, and we are done! If 𝐴 connects to less than three other vertices in 𝐺, then it connects to three (or more) vertices in 𝐺3 If any one of 𝐵𝐶 , 𝐵𝐷 , {𝐶𝐷} appear in 𝐺3, then we are done! If NOT, then ALL of 𝐵𝐶 , 𝐵𝐷 , {𝐶𝐷} appear in 𝐺, and we are done!
Describe visions of the future according to Timothy Gowers
Automated Mathematics - Computers will surpass humans in being able to prove theorems
Which two mathematician's wrote Principia Mathematica?
Bertrand Russell and Whitehead
Describe visions of the future according to Joel E. Cohen
Biomathematics
Name the open Milennium Prize Problems. What is the significance of these problems?
Birch and Swinnerton Dyer Conjecture Hodge Conjecture P vs NP Reimann Hypothesis Yang-Mills Existence and Mass Gap Navier-Stokes Existence and Smoothness Will win $1 million if solved
Name three individuals with early computing machine models
Blaise Pascal Gottfried Leibniz Charles Babbage
Describe the significance of the Langlands Program
Brings conjectures together with the goal of drawing connections between Number Theory, Algebra, Geometry and Analysis
Name four mathematicians who won the fields medal in 2018
Caucher Birkar Alessio Figalli Peter Scholze Akshay Venkatesh
What is the significance of the treatise Elements de Mathematique?
Coined term Injective Bijective and Srujective Broken into core volumes of many mathematical topics Written by the Bourbaki Group
Describe the significance of the Fields Medal and name three 20th century winners
Considered the highest honor a mathematician can receive Awarded only to mathematicians under the age of 40 Given at the ICM Three prize winners: Lars Ahlfors Jesse Douglas Paul Cohen Jean-Pierre Serre
What is the name and significance of Courant and Hilbert's famous book?
Courant-Hilbert Book on mathematical physics
Who solved "Gordon's Problem" in Invariant Theory in 1888?
David Hilbert
Name accomplishment(s) of John H. Conway
Doomsday Algortihm Developed "The Game of Life" Established the Surreal Numbers
What is an Erdos number? Who has an Erdos number of 0? Name five individuals with an Erdos number of 1.
Erdos number is the collaborative distance between mathematician Paul Erdos and another person. Erdos himself has an Erdos number of 0. Fan Chung Samuel Wagstaff Mary Ellen Rudin Ronald Graham Steven Krantz
Name three accomplishments of George David Birkhoff
Ergodic Theorem Solved Poincare's "Last Geometric Theorem" Established his own set of axioms for Euclidean Geometry
Name five individuals who attempted but failed to prove Fermat's last theorem.
Euler Sophie Germain Kummer Harry Vandiver Samuel Wagstaff
Describe Euler's role in Graph Theory
Eulerian Circuits - a circuit containing every edge of G
State and describe the significance of Ostrowski's Theorem
Every non-trivial norm on Q is equivalent to either |.|inf or |.|p for some prime p The finding is significant because it showed that the p-adics are the only alternative to the reals!
Describe visions of the future according to Paul Halmos
Experimental Mathematics - the use of computers will allow for experimentation and trial-and-error approaches to generate and prove conjectures, as well as develop and run advanced algorithms
List three accomplishments of John von Neumann
Famous for his contributions of pure math, applied math, physics, and computer science Helped develop quantum mechanics, establish game theory and economics as mathematical disciplines, and participated in inventing computers Was a part of the Manhattan Project creating the atomic bomb at Los Alamos
Which mathematician was so serious that he had one joke for the fall and one joke for the spring?
Felix Klein
Name Lars Ahlfors' accomplishements
First winner of Fields Medal Wrote Complex Analysis Worked with Riemann Surfaces
Name Jesse Douglas' accomplishments
First winner of fields medal Solved Lagrange's Problem of Plateau
Name three accomplishments of Maryam Mirzakhani
First women and Iranian to win a fields medal Contributed to Riemann surfaces Got a PHD from Harvard
Name three accomplishments of Julia Robinson
First women elected to National Academy of Sciences First female President of the American Mathematical Society Advanced Hilbert's 10th problem
Name three accomplishments of Karen Uhlenbeck
First women to win the Abel Prize Helped expand upon Einstein's Theory of Relativity She's an Emeritus professor at the University of Texas
State the axiom of choice and discuss it's significance
For a nonempty set composed of nonempty disjoint sets there exists a second set that has exactly one member of each of the sets in the original set Used in a proof of the Well-Ordering Theorem
State the axiom of choice
For a nonempty set composed of nonempty, disjoint sets (do not share members), there exists a second set that has exactly one member of each of the sets in the original set • Informally: Given a collection of jars, it is possible to choose exactly one item from each jar, even if the collection contains an infinite number of jars
What was the name and significance of David Hilbert's book?
Foundations of Geometry Hilbert established 21 axioms
Which mathematicians were associated with Gottingen?
Georg Cantor Karl Fredrich Gauss Sonya Kovalevskaya David Hilbert Richard Courant John Von Neumann Emmy Noether André Weil
Who published Begriffsschrift?
Gottlob Frege
Name 10 female mathematician's who lived/worked in the 20th century or 21st century
Grace Hopper Dorothy Voughan Katherine Johnson Mary Jackson Julia Robinson Cora Sadosky Mary Ellen Rudin Mary Gray Karen Uhlenbeck Lenore Blum (Judith Roitman) (Dusa Mcduff)
Why was George David Birkhoff a controversial figure?
He was accused of preventing Jews from obtaining jobs in the U.S. during the 1930's
List three accomplishments of Paul Erdos
Helped pioneer Ramsey theory Erdos with Alte Selberg proved The Prime Number Theorem Proved Bertrand's postulate more elegantly as a college freshman
Describe the ideas behind formalism and name it's leader
Hilbert A formal system is one that uses axioms and a set of rules to derive theorems (banish intuition)
Describe the Nash Equilibrium. What is its significance?
In a game with 2 or more players, each one may select a particular strategy Let's assume (for simplicity) that there are 2 players, A and B • Player A selects a strategy, knowing what player B's best option is • Player B selects a strategy, knowing what player A's best option is Assuming that neither player will change his/her strategy, this game is in a Nash Equilibrium if ... Both players are making the best decision based on their opponent's strategy In other words, if player A were to switch to any other strategy (while player B keeps his/her strategy), player A would NOT be more successful Nash showed that any finite game will always have at least one Nash Equilibrium
State the Fundamental Theorem of Combinatorial Games. Who proved it?
In any combinatorial game at least one player has a non-losing strategy Ernest Zermello
Who were the first two winners of the Fields Medal?
Jesse Douglas and Lars Ahlfors
Name three female mathematicians employed by NACA starting in the 1950s
Katherine Johnson Mary Jackson Dorothy Vaughan
Who proved the four color theorem and what is the significance of their proof?
Kenneth Appel Wolfgang Haken Firs computer assisted proof
Who was the first to discover the P-Adic numbers?
Kurt Hensel
Describe the ideas behind Intuitionism and name its leader
L.E.J. Brouwer They believed that an object exists is only proven if there exists a method to enable the object to be found or constructed
Which mathematician did not believe that irrational numbers even existed?
Leopold Kronecker
State the four color theorem
Let G be a planar graph. There exists a proper 4-coloring of G
Describe a winning strategy in the following game: There are eleven sticks on a table. Each player removes 1 or 2 sticks per turn. The last player to move wins
Let 𝑛 be a positive integer and 𝑘 an integer such that 1 ≤ 𝑘 < 𝑛. •Starting with a pile of 𝑛 sticks, each player removes 1,2,3,...𝑘 sticks. • Winning Strategy? • I: If 𝑘 + 1 does not divide 𝑛, Player A wins .• A takes an amount leaving a multiple of 𝑘+1. When B takes "𝑥", A takes 𝑘+1−𝑥 •II:If 𝑘+1 does divide 𝑛, Player B wins: When A takes "𝑥", B takes 𝑘+1−𝑥
What is the significance of Hilberts speech at the ICM in paris in 1900?
Looked towards the future of mathematics and proposed 23 unsolved problems that mathematicians should solve for the field to develop
What is the name and significance of E.T. Bell's famous book which influenced Nash?
Men of Mathematics It was his first glimpse of real mathematics
What is the name and significance of Abraham Robinson's famous book?
Non-Standard Analysis Introduced the field and it's relevant ideas
Describe visions of the future according to Steven Kranz
Subject Division- believes that the distinction between 'physicist', 'mathematician' and 'engineer' will merge into one
Name accomplishment(s) of Yitang Zang
Successfully proved that there are infinitely many pairs of primes that differ by 70 million or less
Describe three mathematical contributions of Andre Wiles
Taniyama - Shimura - Weil Conjecture = Modularity Theorem (played a role in the proof of Fermat's Last Theorem) Helped Establish the foundations of Algebraic Geometry and its connection to Number Theory Weil Conjectures - 4 conjectures regarding local zeta functions
Prove that rationals are countable and reals are uncountable
The Reals are Uncountable: • Suppose the Reals on 1 < 𝑥 < 2 countable • Then elements can be written as a sequence, each with the unique decimal expansion seen here (assume no infinite chain of 9's) • Assume this list contains all the real numbers on the interval • Define a new 𝑥 = 1. 𝑎33𝑎44𝑎55𝑎66 ... in the following way: • Choose𝑎33 ≠𝑑33,𝑎44 ≠𝑑44,𝑎55 ≠𝑑55,𝑎66 ≠𝑑66, etc .• This 𝑥 therefore differs from all others already in the sequence •It differs from the first number because 𝑎33 ≠ 𝑑33, from the second number because 𝑎44≠𝑑44, etc. •Therefore, 𝑥 is a real number that was not initially on this list • Since 1 < 𝑥 < 2 still holds, we have a CONTRADICTION! • And ... If the real #s on 1 < 𝑥 < 2 are uncountable, so too is R in general
What is the name and significance of von Neumann and Morgenstern's famous book?
The Theory of Games and Economic Behavior Wanted to establish economics on a rigorous mathematical foundation
List three accomplishments of Emmy Noether
Theory of Ideals in ring domains (establishing the term noethering) Guided the establishment of modern physics with her work on differential equations and the calculus of variations (Noether's Theorem)
State and describe the significance of the continuum hypothesis. Who proved a key result related to this hypothesis
There is no set whose cardinality is strictly between that of the integers and the real numbers Paul Cohen proved that the continuum hypothesis cannot be proven nor disproven, it is independent of all established axioms
