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, ) is 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.