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Christopher Hillar
NSA Young Investigator
NSF Postdoctoral fellow
Texas A&M University
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I am now in the Bay Area with new
coordinates:
Address:
MSRI:
Mathematical Sciences Research Institute
17 Gauss Way
Berkeley, CA 94120
Office: 316
Phone: (510) 643-6031
Email(s): chillar msri.org,
chillarmath.tamu.edu
Ph.D. U.C. Berkeley (Mathematics,
2001 - 2005)
B.S. Yale University (Mathematics,
Computer Science, 2000)
my CV
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Theoretical Neuroscience: Currently,
I am seeking a postdoc or similar position
that involves both mathematics and neuroscience.
The basic outline of my vision is contained in
the following preproposal
submitted to MSRI, which would include cross-disciplinary
research interaction between MSRI, UCB Neuroscience
(specifically the Redwood Center), and UCB Mathematics.
NSA Grant: I was awarded an
NSA
Young Investigators Grant. My proposal can
be found here. |
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I am presenting
a paper at the ISSAC
2008 conference in Austria. The pdf slides
from the talk can be found here.
See the section below on Finite Generation
of Symmetric Ideals for more information. |
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I was on leave
visiting University of Copenhagen Department
of Mathematical Sciences from April 6 - May
16. I gave a lecture series to the math department
on Groebner Basis and Applications (see below
for more details). Notes will be made available
soon. |
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Research Interests: Structured polynomial
systems, computational algebraic geometry, combinatorics,
matrix analysis, applications of mathematics to
neuroscience
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Lecture
Series on Groebner Bases and Applications
I am giving a lecture series at the University
of Copenhagen Department
of Mathematical Sciences.
Introduction
to Groebner Bases and Applications
Groebner
Basics
Applications
of Groebner Bases
Invariant
Theory and Symmetric Groebner Bases
Time permitting, I will try and put together a
set of notes for these lectures.
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Sums
of polynomial squares over totally real fields
are rational sums of squares
(Some slides)
Let K be a totally
real number field. We prove that if f
in Q[x1,...,xn]
is a sum of squares in K[x1,...,xn],
then f is a sum of squares in Q[x1,...,xn].
Moreover, our argument is constructive. This gives
a partial resolution to a question of Sturmfels
on the algebraic degree of certain semidefinite
programing problems. A practical application is
that searching numerically for sums of squares
can be done in exact arithmetic over the rationals
(if it can be done at all over K). Proceedings
of the American Mathematical Society, 137
(2009), 921-930. |
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Automorphisms
of Abelian Groups
We give a complete description of the automorphisms
of finite abelian groups. This leads to a formula
for their cardinalities. Although this characterization
has essentially been known for quite some time,
we provide a much needed modern and accessible
treatment. (Joint with D. Rhea). The American
Mathematical Monthly, 114
(2007), no. 10, 917-923. arXiv
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Algebraic
Characterization of Uniquely Vertex Colorable
Graphs
(slides
from a talk and code
for algorithms) The study of graph colorability
from an algebraic perspective has introduced novel
techniques and algorithms into the field. For
instance, k-colorability of a graph can
be characterized by whether its graph polynomial
is contained in a certain ideal. In this paper,
we interpret unique colorability in an analogous
manner and use Groebner basis techniques to prove
an algebraic characterization for uniquely k-colorable
graphs. Our result also gives algorithms for testing
unique colorability. As an application, we verify
a counterexample to a conjecture of Xu concerning
uniquely 3-colorable graphs without triangles.
(Joint with T. Windfeldt). Journal of Combinatorial
Theory Series B, 98 (2008),
400-414.
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Hilbert's
17th Problem for matrices
We
give an elementary and elegant proof of the following
generalization of a fundamental result of Artin:
Let A be a symmetric matrix with entries
in the polynomial ring R[x1,...,xn].
Theorem: If A is postive
semidefinite for all substitutions (x1,...,xn)
in Rn, then
A can be expressed as a sum of squares
of symmetric matrices with entries in R(x1,...,xn).
Moreover, our proof is constructive and gives
an explicit representation modulo the scalar case.
(Joint with J. Nie). Proceedings of the American
Mathematical Society, 136
(2008), 73-76. |
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Finite
Generation of Symmetric Ideals
(some slides
from a talk) We prove a finiteness theorem for
ideals in infinite dimensional polynomial rings
that are invariant under symmetric group action.
The proof involves introducing a certain well-quasi-ordering
on monomials and developing a theory of Groebner
bases and reduction in this setting. We also consider
the concept of an invariant chain of ideals for
finite-dimensional polynomial rings and relate
it to the finite generation result mentioned above.
Finally, a motivating question from chemistry
is presented. (Joint with Matthias
Aschenbrenner). Trans. Amer. Math. Soc.,
359 (2007), 5171-5192.
A follow up note (with Troels Windfeldt)
showing that invariant ideals are not always generated
by a fixed number of polynomials is here,
Proc. Amer. Math. Soc., to appear.
New: An Algorithm
for computing these Groebner bases | Preliminary
manuscript | arxiv |
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Advances
on the Bessis-Moussa-Villani Trace Conjecture
(A poster,
some talk slides,
and a brief
overview of what is known). We make some advances
on a long-standing BMV conjecture in mathematical
physics due to Bessis, Moussa, and Villani. The
statement is enticingly simple (thanks to a reformulation
by Elliot Lieb and Robert Seiringer): For every
positive integer m and every pair of
positive definite matrices A and B,
the polynomial
p(t) = Tr[(A+tB)m]
has nonnegative coefficients. One of the interesting
implications is that to prove the conjecture for
all m, it is enough to prove it for infinitely
many m. Moreover, if it is true for some
m, then it is true for all smaller
m. Linear Algebra and Applications,
426 (2007), 130-142.
A recent result of Daniel
Haegele proves the conjecture for m
= 7, although his technique does not generalize
to m = 6. However, by my results, this
implies the m = 6 case as well. |
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Solving
Symmetric Word Equations
(code that preforms the calculations in this
paper: maple
code 1 | maple
code 2). We continue a study of the solvability
of symmetric word equations in positive definite
letters. Let S(X,B)
be a palindromic word in two letters X
and B. A previous result
states that for each pair of positive definite
matrices B and P, there is a
positive definite solution X to the equation
S(X,B) = P.
The authors also conjectured that these solutions
are unique. In this paper, we resolve this conjecture
(negatively). Furthermore, we prove that, generically,
the number of solutions is odd (and thus finite)
in the real case. Our approach utilizes the theory
of Brouwer degree and also provides a second proof
of existence of such solutions in the real case.
(Joint with Scott
Armstrong). Journal of the London Mathematical
Society, 76 (2007), no.
3, 777-796. |
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