Freshman-Sophomore Seminar


Check regularly the following ever-growing list of ever-changing  pdf  files
(the files grew out of lectures in various semesters; hence the occasional overlaps):


Algebra

some basic algebraic structures

groups and fields

try these simple algebra problems

a summary about groups

cyclic groups

isomorphism of groups

cosets in groups

polynomials, field extensions, Euclidean constructions


Linear Algebra

vector spaces

spanning sets, independent sets, bases, dimension

Here is an interesting generalization
of the notion of subspaces, spanning sets, independent sets, and bases.

linear transformations

orthogonal transformations


Number Theory

groups and number theory

integers that can be written as the sum of two squares

Wilson short

Wilson's theorem and Fermat's ``little theorem"


Miscellanea

isometries of Rn

additive functions - the Cauchy equation

images and inverse images

symbols and notations

finite projective planes


Some further topics for talks (hopefully with writeups later):

The Bolyai-Gerwien theorem
Hilbert's third problem; Dehn's solution
Cantor's world: Cardinalities; The Bernstein-Schröder theorem; The Banach-Tarski paradox
Lagrange's Four Squares Theorem


Names:

Banach, Stefan (Polish; 1892-1945) [born in Kraków, then Austria-Hungary now Poland]
Bernstein, Felix (German; 1878-1956)
Bolyai, Farkas (Wolfgang) (Hungarian; 1775-1856) [father of János Bolyai 1802-1860]

[born in Bolya, Transylvania, Austrian Empire, now Sibiu, Romania]
Cantor, Georg (born in St. Petersburg, Russia but lived in Germany; 1845-1918) [Georg Ferdinand Ludwig Philipp Cantor]
Cauchy, Augustin (French; 1789-1857) [Augustin Louis Cauchy]
Dehn, Max (German; 1878-1952) [Max Wilhelm Dehn]
Euler, Leonhard (Swiss; 1707-1783) [He mostly worked in St. Petersburg, Russia, and in Berlin, Prussia]
[He introduced the notations e, pi and i.]
Fermat, Pierre de (French; 1601-1665)
Gerwien, P???. (Prussian; 18??-18??)
Hilbert, David (German; 1862-1943) [born in Königsberg, Prussia, now Kaliningrad, Russia] [His lecture "The Problems of Mathematics" on 2nd ICM, 1900 Paris, contains his famous 23 problems for the twentieth century; see Professor David Joyce's webpage.]
Laczkovich, Miklós (Hungarian; 1948-)
Lagrange, Joseph-Louis (Italian; 1736-1813) [born as Giuseppe Lodovico Lagrangia]
Schröder, Ernst (German; 1841-1902) [Friedrich Wilhelm Karl Ernst Schröder]
Tarski, Alfred (Polish; 1902-1983) [born in Warsaw, then Russia now Poland]
Wilson, John (English; 1741-1793)

For more names referenced by these pages, see   names.htm   or the original source:   http://www-history.mcs.st-andrews.ac.uk/history/BiogIndex.html

 


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Thanks jeaig

Last modified on Monday, February 4, 2008.