Baris Coskunuzer

committee: Paul Yang, Dave Gabai, Jordan Ellenberg

special topics: differential geometry, algebraic topology

algebraic topology: (Gabai)

van kampen theorem. proof. give example connectedness of intersection is 
essential. (take two circle and consider union with 2 intersection points.)
relation between first homology and fundamental group.
covering of genus 3 surface by genus 2 surface is possible? (no, compute euler 
char. of covering, and it should be multiple of base)
cover genus 2 surface by genus 3 surface.
what is pi_3(S^2) ? (use hopf fibration +induced long exact sequence)

differential geometry:(Yang)
prove by using diff. geo. that you cannot cover genus 3 surface with genus 2 
surface. ( use gauss-bonnet to prove euler chars. must be multiple)
prove gauss-bonnet.
define degree of a smooth function.
how do you know that the critical points of a smooth fuction is not too much? 
(sard's theorem)
define minimal surface.
prove bernstein theorem by using diff.geo. (S=f(R^2) is a minimal surface in 
R^3 then S is a plane: by definition, S has complex structure + consider 
gauss map, it stays in northern hemisphere + by liouville's theorem, it is 
constant then S is plane.)

complex analysis:(Yang)
prove entire function is bounded then constant. (liouville)
prove entire function less than a polynomial is constant (same 
argument with liouville)

algebra: (Ellenberg)
define Noetherian Ring.
give example: Z :)
show Z is PID. (division algorithm)
define lower central series, upper central series, nilpotent and solvable 
groups, prove SL(2,Z) is not solvable. 
give example of simple groups, are there infinitely many? (A_n, n>4)
is C[x,y] PID? (No, <x,y>) is <x,y> prime ideal? yes.
etc.

Real Analysis: (Yang)
is there any analog of Fundamental Theorem of Calculus in measure theory? 
(yes, Radon-Nikodym Theorem) Prove it. 
etc.