Research Interests
Algebra:
I really appreciate tricks
in algebra, especially how mathematicians treat exact sequence. However, since
I haven’t learnt enough classical examples, many general theories in algebra
seems quite abstract for me. Some theories are not so abstract, like finite
simple groups, but they are too concrete and with too much details. I really
get lost in the sea of various algebras.
Plan to
study these days:
(i) Categories and homological algebra
(ii) Galois Cohomology
(iii) Representation of Lie Groups
Analysis:
I always believe I
have more talent in analysis than in algebra, maybe because of my solid
foundation on analyzing inequalities. I am quite interested in classical
functional analysis and its applications in PDE. Some topics in complex
analysis also interest me, like Oka’s work in several complex variables
Plan to
review contents in real/complex analysis this quarter.
Topology/Geometry:
I have to admit that
I forgot most topology I have learnt before. Now I can hardly remember how to
compute cohomology groups. Maybe it’s best for me to have a close study on Homology Theory by Bo-ju Jiang this
quarter. Together with course materials on Math 225A about differential
topology, I can have a more clear view on topology. For topology/geometry, it’s
too early for me to do any research, since I know almost nothing.
Plan to
study:
(i) Algebraic Topology (by Jiang’s book)
(ii) Differential Topology (textbook of Math 225A and Hirsch’s book)
Number Theory:
Number Theory is no
doubt the core of mathematics before, nowadays and in the future, especially
after Poincaré’s Conjecture has been solved. To study number theory means to
study every aspects of pure mathematics. A mathematician (unfortunately I
forgot his name) said mathematics can be divided into two parts: physics and
number theory. I think that’s quite true. My interest focuses on various
indefinite equations and Diophantine approximation in algebraic number field. I
have a conjecture that x^3+y^3-z^2=n has infinitely many non-trivial
integer solutions (x,y,z) for any fixed integer n in high school. Until now I
have no idea about how to prove it or construct a counterexample.
My current
plan is to read Cassels and Frohlich’s book in this quarter or next one. For
analytical number theory, I will try to understand Chen’s proof on “1+2” simplified
Goldbach’s conjecture. The proof can be found in Apostol’s book and I believe
it’s suitable as a change of taste.
Combinatorics:
I got to tons of combinatorics problems in high school. I am not good at
constructing examples (or counterexamples), except using the ordinary method
which can be done by a computer. I can’t give exact answers to many Ramsey-type
problems, but I am fond of guessing their orders.
Logic:
I never learn logic systematically. But I am always wondering one question:
most general theory relate two or more old theories together, point out their
intrinsic comparability. Can we get this relation by setting up a logic system
and find they are logically equivalent indeed? Maybe many people think this is
absurd…
Applied Mathematics:
I am fond of designing algorithms. I have experience in participating ACM/ICPC contest. I don’t like
those problems in segment trees or network flow. I prefer those which have
mathematical backgrounds. I used to solve problems on Judge Online. But as a
graduate student, I don’t think I can afford the time spending on it. I also
have some interests in operations research. However, getting used to different
languages in different mathematical softwares is annoying.