3. Postdoctoral Program - MSRI

Christopher J. Hillar
University of California, Berkeley, 2005, Ph.D.
Ph.D. advisor: Bernd Sturmfels
Christopher Hillar Grant Report
2010-2011 Academic Year
Institution prior to obtaining the MSRI PD fellowship: Texas A&M University
Visiting Assistant Professor and NSF Postdoctoral Fellow
Postdoc Advisor: Frank Sottile
Institution where you held your MSRI PD fellowship: University of California, Berkeley
Mentor: Friedrich Sommer (Redwood Center for Theoretical Neuroscience)
Institution (or company) where you are going next year:
Redwood Center for Theoretical Neuroscience
University of California, Berkeley
Position: Postdoc
Anticipated length: 1 Year
My work falls into two sections: (1) Pure Mathematics and (2) Theoretical Neuroscience. As I
am transitioning fields, it is important to honor my obligations to collaborators (both old and
new), so I have tried to be as diligent as possible in pushing out old projects and folding in new
ones.
(1) Pure Mathematics
With Seth Sullivant, we have proved the Independent Set Conjecture in Algebraic Statistics. The
proof involved unifying and generalizing several ideas concerning polynomial rings with infinite
numbers of variables. We also introduced the notion of monoidal Groebner Bases, which are
useful tools for proving finiteness of invariant ideals in semigroup rings that have a monoidal
action. The work has recently been accepted:
C. Hillar and S. Sullivant, Finite Groebner bases in infinite dimensional polynomial rings and
applications, Advances in Mathematics, to appear.
http://arxiv.org/abs/0908.1777
Related to this work is a paper in preparation with Abraham Martin del Campo on invariant
chains of toric ideals:
C. Hillar and A. Martin del Campo, Symmetric stabilization of toric ideals, in preparation.
www.msri.org/people/members/chillar/files/HM-InvarChainStabDec10.pdf
With Lek-heng Lim, we have proved that most tensor problems are NP-hard (tensors are the
natural generalization of matrices to higher dimensions). We are preparing the article for
publication, but a preliminary manuscript is here: