Tell us your research

[Yi]

Hi,

I am glad to receive a post from Bi, Shuchao in Berkeley telling us
his research interests.

Please describe your interests or research areas to us.

Best,

Yi


============================================
Name: Bi, Shuchao
Institute: UC Berkeley
Year: second year grad
Interests: as below
===============================================
Dear all,

All the comments are awesome! But those are all about the "classical"
algebraic geometry. There is a big amazing world outside. I will list
some
of these in the following (ordered by the distance to classical AG).

Toric Varieties: This subject was developed in the 1990s. There is a
classical book "Introduction to toric varieties" by William Fulton and
a
uncompleted online book by David A. Cox, John B. Little, and Hal
Schenck
which I like a lot. Quote W. Fulton, "toric variety have provided a
remarkably fertile testing ground for general theories".

Algebraic Statistics: This is a rather new research area, and there is
only one book on this subject by Bernd Sturmfels which is published
this
year I think. Basically, algebraic statistics is about using algebraic
geometry to study statistical models. We know that, traditionally
people
treat these models as manifold and use analysis to study them, on
drawback
is that people don't know how to deal with singularities in the
analysis
setting, but resolution of singularities in AG provides some hope.
Quote
B. Sturmfels, "Algebra is as powerful as analysis in statistics".

Tropical Geometry: This is a piece-wise linear version of AG. For a
classical varieties, there is a so-called evaluation map project them
to
piece-wise linear things (so called polyhedral fans or polyhedral
complexes), hope that we can get information of the original varieties
from their projections. There is no standard book on these subject
yet, I
know Sturmfels are working on this, and I could email you a very draft
version of their "book" if anyone is interested.

I really liked our Cocktail Seminar back to undergraduate days at
Hangzhou, thanks to Prof. Shiu-chun Wong!

Cheers,
Shuchao

[Shuchao Bi]

Dear all,

Right now, I am considering the following two open problems:

1. Irreducibility of certain matrix varieties, the motivation comes from
the so call Salmon Conjecture in algebraic statistics. In particular, I'd
like to know the irreducibility of the variety {(A,B,C)| Badj(A)C=Cadj(A)B
} where A,B,C are n by n matrices and adj(A) stands for the adjugate. I
proved the subset where A is nonsingular is irreducible, so the
irreducibility of this variety is equivalent to the statement that this
subset is dense.

2. The Cohen-Macaulay property and minimal free resolution of the
commuting ideals. More explicitly, let X=(x_ij), Y=(y_ij) be n by n
matrices with variable entries. The ideal generated by the entries of the
matrix XY-YX is called the commuting ideal. The variety cut out by this
ideal is called the commuting variety. It is still open that whether the
commuting ideal is the defining ideal of the commuting variety, i.e.,
whether the commuting ideal is radical. There are a lot of conjectures
about this ideal, people (I think by people, I mean M. Artin) conjectured
that this ideal is Cohen-Macaulay), also there are a lot of guesses of
what the free resolution should be look like.


There is a long story between me and the commuting ideal ("who" entertains
me all the time) back to when I was a freshman. I remembered that I wrote
a
write-up trying to find a new criterion when two matrices commute. It
turns out that all the question I had at that time are clear now and these
techniques turn out to be very important.

It is a really long email already, I'd better stop.

Best wishes to all and our teacher Shiu-chun Wong.