%% Books
@book{fds166127,
Author = {edited and Viviana Ene},
Title = {Combinatorial aspects of commutative algebra},
Volume = {502},
Series = {Contemporary Mathematics},
Publisher = {AMS},
Year = {2009},
Key = {fds166127}
}
@book{fds166191,
Author = {edited and Victor Reiner and Bernd Sturmfels},
Title = {Geometric combinatorics (Lectures from the Graduate Summer
School held in Park City, UT, 2004)},
Volume = {13},
Series = {IAS/Park City Mathematics Series},
Pages = {xii+691},
Publisher = {American Mathematical Society},
Address = {Providence, RI},
Year = {2007},
ISBN = {978-0-8218-3736-8; 0-8218-3736-2},
MRCLASS = {05-01 (05-06 52-02)},
MRNUMBER = {2009i:05001},
url = {http://www.ams.org/mathscinet-getitem?mr=2009i:05001},
Key = {fds166191}
}
@book{fds166138,
Author = {First Miller and Srikanth Iyengar and Graham Leuschke and Anton Leykin and Claudia
Miller, Anurag Singh and Uli Walther},
Title = {Twenty-four hours of local cohomology},
Volume = {87},
Series = {Graduate Studies in Mathematics},
Pages = {xviii+282},
Publisher = {American Mathematical Society},
Address = {Providence, RI},
Year = {2007},
ISBN = {978-0-8218-4126-6},
MRCLASS = {13D45 (14B15 55N30)},
MRNUMBER = {2009a:13025},
url = {http://www.ams.org/mathscinet-getitem?mr=2009a:13025},
Key = {fds166138}
}
@book{fds166147,
Author = {First Miller and Bernd Sturmfels},
Title = {Combinatorial commutative algebra},
Volume = {227},
Series = {Graduate Texts in Mathematics},
Pages = {xiv+417},
Publisher = {Springer-Verlag},
Address = {New York},
Year = {2005},
ISBN = {0-387-22356-8},
MRCLASS = {13-01 (05-01 05E99 13D02 14M15 14M25)},
MRNUMBER = {2006d:13001},
url = {http://www.ams.org/mathscinet-getitem?mr=2006d:13001},
Key = {fds166147}
}
%% Papers Published
@article{fds374549,
Author = {Miller, E and Zhang, J},
Title = {Geodesic complexity of convex polyhedra},
Year = {2023},
Month = {March},
Key = {fds374549}
}
@article{fds360099,
Author = {Miller, E},
Title = {Stratifications of real vector spaces from constructible
sheaves with conical microsupport},
Journal = {Journal of Applied and Computational Topology},
Volume = {7},
Number = {3},
Pages = {473-489},
Publisher = {Springer},
Year = {2023},
url = {http://dx.doi.org/10.1007/s41468-023-00112-1},
Doi = {10.1007/s41468-023-00112-1},
Key = {fds360099}
}
@article{fds374550,
Author = {Miller, E and Geist, N},
Title = {Global dimension of real-exponent polynomial
rings},
Journal = {Algebra and Number Theory},
Volume = {17},
Number = {10},
Pages = {1779-1788},
Publisher = {Mathematical Sciences Publishers (MSP)},
Year = {2023},
url = {http://dx.doi.org/10.2140/ant.2023.17.1779},
Doi = {10.2140/ant.2023.17.1779},
Key = {fds374550}
}
@article{fds339830,
Author = {Katthän, L and Michałek, M and Miller, E},
Title = {When is a Polynomial Ideal Binomial After an Ambient
Automorphism?},
Journal = {Foundations of Computational Mathematics},
Volume = {19},
Number = {6},
Pages = {1363-1385},
Publisher = {Springer Nature America, Inc},
Year = {2019},
Month = {December},
url = {http://dx.doi.org/10.1007/s10208-018-9405-0},
Abstract = {Can an ideal I in a polynomial ring k[x] over a field be
moved by a change of coordinates into a position where it is
generated by binomials xA- λxb with λ∈ k, or by unital
binomials (i.e., with λ= 0 or 1)? Can a variety be moved
into a position where it is toric? By fibering the
G-translates of I over an algebraic group G acting on
affine space, these problems are special cases of questions
about a family I of ideals over an arbitrary base B. The
main results in this general setting are algorithms to find
the locus of points in B over which the fiber of Iis
contained in the fiber of a second family I′ of ideals
over B;defines a variety of dimension at least d;is
generated by binomials; oris generated by unital binomials.
A faster containment algorithm is also presented when the
fibers of I are prime. The big-fiber algorithm is
probabilistic but likely faster than known deterministic
ones. Applications include the setting where a second
group T acts on affine space, in addition to G, in which
case algorithms compute the set of G-translates of Iwhose
stabilizer subgroups in T have maximal dimension; orthat
admit a faithful multigrading by Zr of maximal rank r.
Even with no ambient group action given, the final
application is an algorithm todecide whether a normal
projective variety is abstractly toric. All of these loci
in B and subsets of G are constructible.},
Doi = {10.1007/s10208-018-9405-0},
Key = {fds339830}
}
@article{fds349179,
Author = {Ene, V and Miller, E},
Title = {Preface},
Journal = {Springer Proceedings in Mathematics and Statistics},
Volume = {238},
Pages = {v-viii},
Year = {2018},
Month = {January},
ISBN = {9783319904924},
Key = {fds349179}
}
@article{fds320533,
Author = {Berenstein, A and Braverman, M and Miller, E and Retakh, V and Weitsman,
J},
Title = {Andrei Zelevinsky, 1953–2013},
Journal = {Advances in Mathematics},
Volume = {300},
Pages = {1-4},
Publisher = {Elsevier BV},
Year = {2016},
Month = {September},
url = {http://dx.doi.org/10.1016/j.aim.2016.06.006},
Doi = {10.1016/j.aim.2016.06.006},
Key = {fds320533}
}
@article{fds303557,
Author = {Miller, E and Kahle, T and O'Neill, C},
Title = {Irreducible decomposition of binomial ideals},
Journal = {Compositio Mathematica},
Volume = {152},
Number = {6},
Pages = {15 pages},
Publisher = {Oxford University Press (OUP)},
Year = {2016},
Month = {June},
url = {http://arxiv.org/abs/1503.02607},
Abstract = {Building on coprincipal mesoprimary decomposition [Kahle and
Miller, 2014], we combinatorially construct an irreducible
decomposition of any given binomial ideal. In a parallel
manner, for congruences in commutative monoids we construct
decompositions that are direct combinatorial analogues of
binomial irreducible decompositions, and for binomial ideals
we construct decompositions into ideals that are as
irreducible as possible while remaining binomial. We provide
an example of a binomial ideal that is not an intersection
of irreducible binomial ideals, thus answering a question of
Eisenbud and Sturmfels [1996].},
Doi = {10.1112/S0010437X16007272},
Key = {fds303557}
}
@article{fds303556,
Author = {Bendich, P and Marron, JS and Miller, E and Pieloch, A and Skwerer,
S},
Title = {Persistent homology analysis of brain artery
trees},
Journal = {Annals of Applied Statistics},
Volume = {10},
Number = {1},
Pages = {198-218},
Year = {2016},
url = {http://arxiv.org/abs/1411.6652v1},
Abstract = {New representations of tree-structured data objects, using
ideas from topological data analysis, enable improved
statistical analyses of a population of brain artery trees.
A number of representations of each data tree arise from
persistence diagrams that quantify branching and looping of
vessels at multiple scales. Novel approaches to the
statistical analysis, through various summaries of the
persistence diagrams, lead to heightened correlations with
covariates such as age and sex, relative to earlier analyses
of this data set. The correlation with age continues to be
significant even after controlling for correlations from
earlier significant summaries},
Doi = {10.1214/15-AOAS886},
Key = {fds303556}
}
@article{fds290936,
Author = {Miller, E},
Title = {Fruit flies and moduli: Interactions between biology and
mathematics},
Journal = {Notices of the American Mathematical Society},
Volume = {62},
Number = {10},
Pages = {1178-1184},
Publisher = {American Mathematical Society (AMS)},
Year = {2015},
Month = {November},
ISSN = {0002-9920},
url = {http://dx.doi.org/10.1090/noti1290},
Doi = {10.1090/noti1290},
Key = {fds290936}
}
@article{fds243887,
Author = {Miller, E and Owen, M and Provan, JS},
Title = {Polyhedral computational geometry for averaging metric
phylogenetic trees},
Journal = {Advances in Applied Mathematics},
Volume = {68},
Pages = {51-91},
Publisher = {Elsevier BV},
Year = {2015},
Month = {July},
ISSN = {0196-8858},
url = {http://dx.doi.org/10.1016/j.aam.2015.04.002},
Abstract = {This paper investigates the computational geometry relevant
to calculations of the Fréchet mean and variance for
probability distributions on the phylogenetic tree space of
Billera, Holmes and Vogtmann, using the theory of
probability measures on spaces of nonpositive curvature
developed by Sturm. We show that the combinatorics of
geodesics with a specified fixed endpoint in tree space are
determined by the location of the varying endpoint in a
certain polyhedral subdivision of tree space. The variance
function associated to a finite subset of tree space has a
fixed C∞ algebraic formula within each cell of the
corresponding subdivision, and is continuously
differentiable in the interior of each orthant of tree
space. We use this subdivision to establish two iterative
methods for producing sequences that converge to the
Fréchet mean: one based on Sturm's Law of Large Numbers,
and another based on descent algorithms for finding optima
of smooth functions on convex polyhedra. We present
properties and biological applications of Fréchet means and
extend our main results to more general globally
nonpositively curved spaces composed of Euclidean
orthants.},
Doi = {10.1016/j.aam.2015.04.002},
Key = {fds243887}
}
@article{fds243885,
Author = {Zamaere, CB and Griffeth, S and Miller, E},
Title = {Systems of parameters and holonomicity of A-hypergeometric
systems},
Journal = {Pacific Journal of Mathematics},
Volume = {276},
Number = {2},
Pages = {281-286},
Publisher = {Mathematical Sciences Publishers},
Year = {2015},
Month = {January},
ISSN = {0030-8730},
url = {http://dx.doi.org/10.2140/pjm.2015.276.281},
Abstract = {We give an elementary proof of holonomicity for
A-hypergeometric systems, with no requirements on the
behavior of their singularities, a result originally due to
Adolphson (1994) after the regular singular case by Gelfand
and Gelfand (1986). Our method yields a direct de novo proof
that A-hypergeometric systems form holonomic families over
their parameter spaces, as shown by Matusevich, Miller, and
Walther (2005).},
Doi = {10.2140/pjm.2015.276.281},
Key = {fds243885}
}
@article{fds243886,
Author = {Huckemann, S and Mattingly, JC and Miller, E and Nolen,
J},
Title = {Sticky central limit theorems at isolated hyperbolic planar
singularities},
Journal = {Electronic Journal of Probability},
Volume = {20},
Pages = {1-34},
Publisher = {Institute of Mathematical Statistics},
Year = {2015},
url = {http://hdl.handle.net/10161/9516 Duke open
access},
Abstract = {We derive the limiting distribution of the barycenter bn of
an i.i.d. sample of n random points on a planar cone with
angular spread larger than 2π. There are three mutually
exclusive possibilities: (i) (fully sticky case) after a
finite random time the barycenter is almost surely at the
origin; (ii) (partly sticky case) the limiting distribution
of √nb<inf>n</inf> comprises a point mass at the origin,
an open sector of a Gaussian, and the projection of a
Gaussian to the sector’s bounding rays; or (iii)
(nonsticky case) the barycenter stays away from the origin
and the renormalized fluctuations have a fully supported
limit distribution—usually Gaussian but not always. We
conclude with an alternative, topological definition of
stickiness that generalizes readily to measures on general
metric spaces.},
Doi = {10.1214/EJP.v20-3887},
Key = {fds243886}
}
@article{fds243889,
Author = {First Miller and Gopalkrishnan, M and Miller, E and Shiu, A},
Title = {A geometric approach to the global attractor
conjecture},
Journal = {SIAM Journal on Applied Dynamical Systems},
Volume = {13},
Number = {2},
Pages = {758-797},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2014},
Month = {January},
url = {http://dx.doi.org/10.1137/130928170},
Abstract = {This paper introduces the class of strongly endotactic
networks, a subclass of the endotactic networks introduced
by Craciun, Nazarov, and Pantea. The main result states that
the global attractor conjecture holds for complex-balanced
systems that are strongly endotactic: every trajectory with
positive initial condition converges to the unique positive
equilibrium allowed by conservation laws. This extends a
recent result by Anderson for systems where the reaction
diagram has only one linkage class (connected component).
The results here are proved using differential inclusions, a
setting that includes power-law systems. The key ideas
include a perspective on reaction kinetics in terms of
combinatorial geometry of reaction diagrams, a projection
argument that enables analysis of a given system in terms of
systems with lower dimension, and an extension of Birch's
theorem, a well-known result about intersections of affine
subspaces with manifolds parameterized by
monomials.},
Doi = {10.1137/130928170},
Key = {fds243889}
}
@article{fds243890,
Author = {Skwerer, S and Bullitt, E and Huckemann, S and Miller, E and Oguz, I and Owen, M and Patrangenaru, V and Provan, S and Marron,
JS},
Title = {Tree-oriented analysis of brain artery structure},
Journal = {Journal of Mathematical Imaging and Vision},
Volume = {50},
Number = {1},
Pages = {126-143},
Year = {2014},
Month = {January},
ISSN = {0924-9907},
url = {http://dx.doi.org/10.1007/s10851-013-0473-0},
Abstract = {Statistical analysis of magnetic resonance angiography (MRA)
brain artery trees is performed using two methods for
mapping brain artery trees to points in phylogenetic
treespace: cortical landmark correspondence and descendant
correspondence. The differences in end-results based on
these mappings are highlighted to emphasize the importance
of correspondence in tree-oriented data analysis.
Representation of brain artery systems as points in
phylogenetic treespace, a mathematical space developed in
(Billera et al. Adv. Appl. Math 27:733–767, 2001),
facilitates this analysis. The phylogenetic treespace is a
rich setting for tree-oriented data analysis. The Fréchet
sample mean or an approximation is reported.
Multidimensional scaling is used to explore structure in the
data set based on pairwise distances between data points.
This analysis of MRA data shows a statistically significant
effect of age and sex on brain artery structure. Variation
in the proximity of brain arteries to the cortical surface
results in strong statistical difference between sexes and
statistically significant age effect. That particular
observation is possible with cortical correspondence but did
not show up in the descendant correspondence.},
Doi = {10.1007/s10851-013-0473-0},
Key = {fds243890}
}
@article{fds243902,
Author = {Miller, E},
Title = {Affine stratifications from finite misère
quotients},
Journal = {Journal of Algebraic Combinatorics},
Volume = {37},
Number = {1},
Pages = {1-9},
Publisher = {Springer Nature},
Year = {2013},
Month = {February},
ISSN = {0925-9899},
url = {http://dx.doi.org/10.1007/s10801-012-0355-3},
Abstract = {Given a morphism from an affine semigroup to an arbitrary
commutative monoid, it is shown that every fiber possesses
an affine stratification: a partition into a finite disjoint
union of translates of normal affine semigroups. The proof
rests on mesoprimary decomposition of monoid congruences and
a novel list of equivalent conditions characterizing the
existence of an affine stratification. The motivating
consequence of the main result is a special case of a
conjecture due to Guo and the author on the existence of
affine stratifications for (the set of winning positions of)
any lattice game. The special case proved here assumes that
the lattice game has finite misère quotient, in the sense
of Plambeck and Siegel. © 2012 Springer Science+Business
Media, LLC.},
Doi = {10.1007/s10801-012-0355-3},
Key = {fds243902}
}
@article{fds243900,
Author = {First Miller and Hotz, T and Huckemann, S and Le, H and Marron, JS and Mattingly, JC and Miller, E and Nolen, J and Owen, M and Patrangenaru, V and Skwerer,
S},
Title = {Sticky central limit theorems on open books},
Journal = {The Annals of Applied Probability},
Volume = {23},
Number = {6},
Pages = {2238-2258},
Publisher = {Institute of Mathematical Statistics},
Year = {2013},
ISSN = {1050-5164},
url = {http://dx.doi.org/10.1214/12-AAP899},
Abstract = {Given a probability distribution on an open book (a metric
space obtained by gluing a disjoint union of copies of a
half-space along their boundary hyperplanes), we define a
precise concept of when the Fréchet mean (barycenter)
is <i>sticky</i>. This non-classical phenomenon is
quantified by a law of large numbers (LLN) stating that the
empirical mean eventually almost surely lies on the
(codimension 1 and hence measure 0) <i>spine</i> that is the
glued hyperplane, and a central limit theorem (CLT) stating
that the limiting distribution is Gaussian and supported on
the spine. We also state versions of the LLN and CLT for the
cases where the mean is nonsticky (that is, not lying on the
spine) and partly sticky (that is, on the spine but not
sticky).},
Doi = {10.1214/12-AAP899},
Key = {fds243900}
}
@article{fds243898,
Author = {Guo, A and Miller, E},
Title = {Erratum: Lattice point methods for combinatorial games
(Advances in Applied Mathematics (2011) 46:1
(363-378))},
Journal = {Advances in Applied Mathematics},
Volume = {48},
Number = {1},
Pages = {269-271},
Publisher = {Elsevier BV},
Year = {2012},
Month = {January},
ISSN = {0196-8858},
url = {http://dx.doi.org/10.1016/j.aam.2011.09.001},
Doi = {10.1016/j.aam.2011.09.001},
Key = {fds243898}
}
@article{fds243899,
Author = {First Miller and Gopalkrishnan, M and Shiu, A},
Title = {A projection argument for differential inclusions, with
applications to persistence of mass-action
kinetics},
Journal = {SIGMA (Symmetry, Integrability, and Geometry: Methods and
Applications)},
Volume = {9},
Publisher = {SIGMA (Symmetry, Integrability and Geometry: Methods and
Application)},
Year = {2012},
url = {http://dx.doi.org/10.3842/SIGMA.2013.025},
Abstract = {Motivated by questions in mass-action kinetics, we introduce
the notion of vertexical family of differential inclusions.
Defined on open hypercubes, these families are characterized
by particular good behavior under projection maps. The
motivating examples are certain families of reaction
networks—including reversible, weakly reversible,
endotactic, and strongly endotactic reaction
networks—that give rise to vertexical families of
mass-action differential inclusions. We prove that
vertexical families are amenable to structural induction.
Consequently, a trajectory of a vertexical family approaches
the boundary if and only if either the trajectory approaches
a vertex of the hypercube, or a trajectory in a
lower-dimensional member of the family approaches the
boundary. With this technology, we make progress on the
global attractor conjecture, a central open problem
concerning mass-action kinetics systems. Additionally, we
phrase mass-action kinetics as a functor on reaction
networks with variable rates.},
Doi = {10.3842/SIGMA.2013.025},
Key = {fds243899}
}
@article{fds243901,
Author = {First Miller and Thomas Kahle},
Title = {Decompositions of commutative monoid congruences and
binomial ideals},
Journal = {Algebra & Number Theory},
Volume = {8},
Number = {6},
Pages = {1297-1364},
Publisher = {Mathematical Sciences Publishers},
Year = {2012},
url = {http://dx.doi.org/10.2140/ant.2014.8.1297},
Abstract = {We demonstrate how primary decomposition of commutative
monoid congruences fails to capture the essence of primary
decomposition in commutative rings by exhibiting a more
sensitive theory of mesoprimary decomposition of
congruences, complete with witnesses, associated prime
objects, and an analogue of irreducible decomposition called
coprincipal decomposition. We lift the combinatorial theory
of mesoprimary decomposition to arbitrary binomial ideals in
monoid algebras. The resulting binomial mesoprimary
decomposition is a new type of intersection decomposition
for binomial ideals that enjoys computational efficiency and
independence from ground field hypotheses. Furthermore,
binomial primary decomposition is easily recovered from
mesoprimary decomposition, as is binomial irreducible
decomposition -- which was previously not known to exist --
from binomial coprincipal decomposition.},
Doi = {10.2140/ant.2014.8.1297},
Key = {fds243901}
}
@article{fds243892,
Author = {Miller, E},
Title = {Theory and applications of lattice point methods for
binomial ideals},
Journal = {Combinatorial Aspects of Commutative Algebra and Algebraic
Geometry: The Abel Symposium 2009},
Volume = {6},
Series = {Abel Symposia},
Pages = {99-154},
Booktitle = {Combinatorial aspects of commutative algebra and algebraic
geometry, Proceedings of Abel Symposium held at Voss,
Norway, 1--4 June 2009},
Publisher = {Springer Berlin Heidelberg},
Address = {Berlin-Heidelberg},
Year = {2011},
Month = {December},
ISBN = {9783642194917},
url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000303079200008&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
Abstract = {This survey of methods surrounding lattice point methods for
binomial ideals begins with a leisurely treatment of the
geometric combinatorics of binomial primary decomposition.
It then proceeds to three independent applications whose
motivations come from outside of commutative algebra:
hypergeometric systems, combinatorial game theory, and
chemical dynamics. The exposition is aimed at students and
researchers in algebra; it includes many examples, open
problems, and elementary introductions to the motivations
and background from outside of algebra. © Springer-Verlag
Berlin Heidelberg 2011.},
Doi = {10.1007/978-3-642-19492-4_8},
Key = {fds243892}
}
@article{fds243904,
Author = {First Miller and Miller, E and Novik, I and Swartz, E},
Title = {Face rings of simplicial complexes with singularities},
Journal = {Mathematische Annalen},
Volume = {351},
Number = {4},
Pages = {857-875},
Publisher = {Springer Nature},
Year = {2011},
Month = {December},
ISSN = {0025-5831},
url = {http://dx.doi.org/10.1007/s00208-010-0620-5},
Abstract = {The face ring of a simplicial complex modulo m generic
linear forms is shown to have finite local cohomology if and
only if the link of every face of dimension m or more is
nonsingular, i.e., has the homology of a wedge of spheres of
the expected dimension. This is derived from an enumerative
result for local cohomology of face rings modulo generic
linear forms, as compared with local cohomology of the face
ring itself. The enumerative result is generalized to
squarefree modules. A concept of Cohen-Macaulay in
codimension c is defined and characterized for arbitrary
finitely generated modules and coherent sheaves. For the
face ring of an r-dimensional complex Δ, it is equivalent
to nonsingularity of Δ in dimension r-c; for a coherent
sheaf on projective space, this condition is shown to be
equivalent to the same condition on any single generic
hyperplane section. The characterization of nonsingularity
in dimension m via finite local cohomology thus generalizes
from face rings to arbitrary graded modules. © 2010
Springer-Verlag.},
Doi = {10.1007/s00208-010-0620-5},
Key = {fds243904}
}
@article{fds243905,
Author = {First Miller and Anderson, D and Griffeth, S and Miller, E},
Title = {Positivity and Kleiman transversality in equivariant
K-theory of homogeneous spaces},
Journal = {Journal of the European Mathematical Society},
Volume = {13},
Number = {1},
Pages = {57-84},
Publisher = {European Mathematical Publishing House},
Year = {2011},
Month = {January},
ISSN = {1435-9855},
url = {http://dx.doi.org/10.4171/JEMS/244},
Abstract = {We prove the conjectures of Graham-Kumar [GrKu08] and
Griffeth-Ram [GrRa04] concerning the alternation of signs in
the structure constants for torus-equivariant K-theory of
generalized flag varieties G/P. These results are immediate
consequences of an equivariant homological Kleiman
transversality principle for the Borel mixing spaces of
homogeneous spaces, and their subvarieties, under a natural
group action with finitely many orbits. The computation of
the coefficients in the expansion of the equivariant K-class
of a subvariety in terms of Schubert classes is reduced to
an Euler characteristic using the homological transversality
theorem for nontransitive group actions due to S. Sierra. A
vanishing theorem, when the subvariety has rational
singularities, shows that the Euler characteristic is a sum
of at most one term-the top one-with a well-defined sign.
The vanishing is proved by suitably modifying a geometric
argument due to M. Brion in ordinary K-theory that brings
Kawamata-Viehweg vanishing to bear. © European Mathematical
Society 2011.},
Doi = {10.4171/JEMS/244},
Key = {fds243905}
}
@article{fds243907,
Author = {First Miller and Guo, A and Miller, E},
Title = {Lattice point methods for combinatorial games},
Journal = {Advances in Applied Mathematics},
Volume = {46},
Number = {1-4},
Pages = {363-378},
Publisher = {Elsevier BV},
Year = {2011},
Month = {January},
ISSN = {0196-8858},
url = {http://hdl.handle.net/10161/3749 Duke open
access},
Abstract = {We encode arbitrary finite impartial combinatorial games in
terms of lattice points in rational convex polyhedra.
Encodings provided by these lattice games can be made
particularly efficient for octal games, which we generalize
to squarefree games. These encompass all heap games in a
natural setting where the Sprague-Grundy theorem for normal
play manifests itself geometrically. We provide an algorithm
to compute normal play strategies. The setting of lattice
games naturally allows for misère play, where 0 is declared
a losing position. Lattice games also allow situations where
larger finite sets of positions are declared losing.
Generating functions for sets of winning positions provide
data structures for strategies of lattice games. We
conjecture that every lattice game has a rational strategy:
a rational generating function for its winning positions.
Additionally, we conjecture that every lattice game has an
affine stratification: a partition of its set of winning
positions into a finite disjoint union of finitely generated
modules for affine semigroups. This conjecture is true for
normal-play squarefree games and every lattice game with
finite misère quotient. © 2010 Elsevier Inc. All rights
reserved.},
Doi = {10.1016/j.aam.2010.10.004},
Key = {fds243907}
}
@article{fds243908,
Author = {First Miller and Dickenstein, A and Matusevich, LF and Miller, E},
Title = {Combinatorics of binomial primary decomposition},
Journal = {Mathematische Zeitschrift},
Volume = {264},
Number = {4},
Pages = {745-763},
Publisher = {Springer Nature},
Year = {2010},
Month = {April},
ISSN = {0025-5874},
url = {http://dx.doi.org/10.1007/s00209-009-0487-x},
Abstract = {An explicit lattice point realization is provided for the
primary components of an arbitrary binomial ideal in
characteristic zero. This decomposition is derived from a
characteristic-free combinatorial description of certain
primary components of binomial ideals in affine semigroup
rings, namely those that are associated to faces of the
semigroup. These results are intimately connected to
hypergeometric differential equations in several variables.
© Springer-Verlag 2009.},
Doi = {10.1007/s00209-009-0487-x},
Key = {fds243908}
}
@article{fds243909,
Author = {First Miller and Dickenstein, A and Matusevich, LF and Miller, E},
Title = {Binomial D-modules},
Journal = {Duke Mathematical Journal},
Volume = {151},
Number = {3},
Pages = {1-13},
Publisher = {Duke University Press},
Year = {2010},
Month = {January},
url = {http://projecteuclid.org/euclid.dmj/1265637658},
Abstract = {We study quotients of the Weyl algebra by left ideals whose
generators consist of an arbitrary Zd-graded binomial ideal
I in C[∂1∂n] along with Euler operators defined by the
grading and a parameter βεCd 2 Cd. We determine the
parameters β for which these D-modules (i) are holonomic
(equivalently, regular holonomic, when I is
standard-graded); (ii) decompose as direct sums indexed by
the primary components of I; and (iii) have holonomic rank
greater than the rank for generic β. In each of these three
cases, the parameters in question are precisely those
outside of a certain explicitly described affine subspace
arrangement in Cd. In the special case of Horn
hypergeometric D-modules, when I is a lattice basis ideal,
we furthermore compute the generic holonomic rank
combinatorially and write down a basis of solutions in terms
of associated A-hypergeometric functions. Fundamental in
this study is an explicit lattice point description of the
primary components of an arbitrary binomial ideal in
characteristic zero, which we derive from a
characteristic-free combinatorial result on binomial ideals
in affine semigroup rings. Effective methods can be derived
for the computation of primary components of arbitrary
binomial ideals and series solutions to classical Horn
systems.},
Doi = {10.1215/00127094-2010-002},
Key = {fds243909}
}
@article{fds304495,
Author = {Knutson, A and Miller, E and Yong, A},
Title = {Gröbner geometry of vertex decompositions and of flagged
tableaux},
Journal = {Journal fur die Reine und Angewandte Mathematik},
Volume = {2009},
Number = {630},
Pages = {1-31},
Publisher = {WALTER DE GRUYTER GMBH},
Year = {2009},
Month = {May},
ISSN = {0075-4102},
url = {http://dx.doi.org/10.1515/CRELLE.2009.033},
Abstract = {We relate a classic algebro-geometric degeneration
technique, dating at least to Hodge 1941 (J. London Math.
Soc. 16: 245-255), to the notion of vertex decompositions of
simplicial complexes. The good case is when the degeneration
is reduced, and we call this a geometric vertex
decomposition. Our main example in this paper is the family
of vexillary matrix Schubert varieties, whose ideals are
also known as (one-sided) ladder determinantal ideals. Using
a diagonal term order to specify the (Gröbner)
degeneration, we show that these have geometric vertex
decompositions into simpler varieties of the same type. From
this, together with the combinatorics of the pipe dreams of
Fomin-Kirillov 1996 (Discr. Math. 153: 123-143), we derive a
new formula for the numerators of their multigraded Hilbert
series, the double Grothendieck polynomials, in terms of
flagged set-valued tableaux. This unifies work of Wachs 1985
(J. Combin. Th. (A) 40: 276-289) on flagged tableaux, and
Buch 2002 (Acta. Math. 189: 37-78) on set-valued tableaux,
giving geometric meaning to both. This work focuses on
diagonal term orders, giving results complementary to those
of Knutson-Miller 2005 (Ann. Math. 161: 1245-1318), where it
was shown that the generating minors form a Gröbner basis
for any antidiagonal term order and any matrix Schubert
variety. We show here that under a diagonal term order, the
only matrix Schubert varieties for which these minors form
Gröbner bases are the vexillary ones, reaching an end
toward which the ladder determinantal literature had been
building. © 2009 Walter de Gruyter Berlin, New
York.},
Doi = {10.1515/CRELLE.2009.033},
Key = {fds304495}
}
@article{fds299952,
Author = {Miller, E},
Title = {Topological Cohen-Macaulay criteria for monomial
ideals},
Journal = {COMBINATORIAL ASPECTS OF COMMUTATIVE ALGEBRA},
Volume = {502},
Series = {Contemporary Mathematics},
Pages = {137-155},
Booktitle = {Combinatorial aspects of commutative algebra},
Publisher = {AMER MATHEMATICAL SOC},
Editor = {Ene, V and Miller, E},
Year = {2009},
Month = {January},
ISBN = {978-0-8218-4758-9},
ISSN = {0271-4132},
url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000279748700010&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
Key = {fds299952}
}
@article{fds243934,
Author = {Miller, E},
Title = {What is... a toric variety?},
Journal = {Notices of the American Mathematical Society},
Volume = {55},
Number = {5},
Pages = {586-587},
Year = {2008},
Month = {May},
ISSN = {0002-9920},
MRCLASS = {14M25},
MRNUMBER = {MR2404030},
url = {http://www.ams.org/mathscinet-getitem?mr=2404030},
Key = {fds243934}
}
@article{fds243930,
Author = {First Miller and Ezra, M and Speyer, DE},
Title = {A kleiman-bertini theorem for sheaf tensor
products},
Journal = {Journal of Algebraic Geometry},
Volume = {17},
Number = {2},
Pages = {335-340},
Publisher = {American Mathematical Society (AMS)},
Year = {2008},
Month = {January},
ISSN = {1056-3911},
MRCLASS = {14F05 (14L30)},
MRNUMBER = {2008k:14044},
url = {http://dx.doi.org/10.1090/s1056-3911-07-00479-1},
Abstract = {Fix a variety X with a transitive (left) action by an
algebraic group G. Let ε and ℱ be coherent sheaves on X.
We prove that for elements g in a dense open subset of G,
the sheaf Tor¡X- (ε, gℱ) vanishes for all i > 0. When ε
and ℱ are structure sheaves of smooth subschemes of X in
characteristic zero, this follows from the Kleiman-Bertini
theorem; our result has no smoothness hypotheses on the
supports of ε or ℱ, or hypotheses on the characteristic
of the ground field.},
Doi = {10.1090/s1056-3911-07-00479-1},
Key = {fds243930}
}
@article{fds243931,
Author = {First Miller and Miller, E and Pak, I},
Title = {Metric combinatorics of convex polyhedra: Cut loci and
nonoverlapping unfoldings},
Journal = {Discrete and Computational Geometry},
Volume = {39},
Number = {1-3},
Pages = {339-388},
Publisher = {Springer Nature},
Year = {2008},
Month = {January},
ISSN = {0179-5376},
MRCLASS = {52B11 (52B70)},
MRNUMBER = {2008m:52027},
url = {http://dx.doi.org/10.1007/s00454-008-9052-3},
Abstract = {Let S be the boundary of a convex polytope of dimension d+1,
or more generally let S be a convex polyhedral
pseudomanifold. We prove that S has a polyhedral
nonoverlapping unfolding into ℝd, so the metric space S is
obtained from a closed (usually nonconvex) polyhedral ball
in ℝd by identifying pairs of boundary faces
isometrically. Our existence proof exploits geodesic flow
away from a source point v S, which is the exponential map
to S from the tangent space at v. We characterize the cut
locus (the closure of the set of points in S with more than
one shortest path to v) as a polyhedral complex in terms of
Voronoi diagrams on facets. Analyzing infinitesimal
expansion of the wavefront consisting of points at constant
distance from v on S produces an algorithmic method for
constructing Voronoi diagrams in each facet, and hence the
unfolding of S. The algorithm, for which we provide
pseudocode, solves the discrete geodesic problem. Its main
construction generalizes the source unfolding for boundaries
of three-polytopes into ℝ2. We present conjectures
concerning the number of shortest paths on the boundaries of
convex polyhedra, and concerning continuous unfolding of
convex polyhedra. We also comment on the intrinsic
nonpolynomial complexity of nonconvex manifolds. © 2008
Springer Science+Business Media, LLC.},
Doi = {10.1007/s00454-008-9052-3},
Key = {fds243931}
}
@article{fds243933,
Author = {First Miller and Knutson, A and Miller, E and Yong, A},
Title = {Tableau complexes},
Journal = {Israel Journal of Mathematics},
Volume = {163},
Number = {1},
Pages = {317-343},
Publisher = {Springer Nature},
Year = {2008},
Month = {January},
ISSN = {0021-2172},
MRCLASS = {05E10 (06A07 14M15)},
MRNUMBER = {2009b:05264},
url = {http://dx.doi.org/10.1007/s11856-008-0014-5},
Abstract = {Let X, Y be finite sets and T a set of functions X → Y
which we will call " tableaux". We define a simplicial
complex whose facets, all of the same dimension, correspond
to these tableaux. Such tableau complexes have many nice
properties, and are frequently homeomorphic to balls, which
we prove using vertex decompositions [BP79]. In our
motivating example, the facets are labeled by semistandard
Young tableaux, and the more general interior faces are
labeled by Buch's set-valued semistandard tableaux. One
vertex decomposition of this "Young tableau complex"
parallels Lascoux's transition formula for vexillary double
Grothendieck polynomials [La01, La03]. Consequently, we
obtain formulae (both old and new) for these polynomials. In
particular, we present a common generalization of the
formulae of Wachs [Wa85] and Buch [Bu02], each of which
implies the classical tableau formula for Schur polynomials.
© 2008 The Hebrew University of Jerusalem.},
Doi = {10.1007/s11856-008-0014-5},
Key = {fds243933}
}
@article{fds376730,
Author = {Jow, SY and Miller, E},
Title = {Multiplier ideals of sums via cellular resolutions},
Journal = {Mathematical Research Letters},
Volume = {15},
Number = {2-3},
Pages = {359-373},
Year = {2008},
Month = {January},
url = {http://dx.doi.org/10.4310/MRL.2008.v15.n2.a13},
Abstract = {Fix nonzero ideal sheaves a1, . . . ., ar and b on a normal
ℚ-Gorenstein complex variety X. For any positive real
numbers α and β, we construct a resolution of the
multiplier ideal script T((a1 + . . . + ar)αbβ) by sheaves
that are direct sums of multiplier ideals script T(a1λ1 . .
. arλrbβ) for various λ ε ℝ≥0r satisfying Σi=1r λi
= α. The resolution is cellular, in the sense that its
boundary maps are encoded by the algebraic chain complex of
a regular CW-complex. The CW-complex is naturally expressed
as a triangulation Δ of the simplex of nonnegative real
vectors λ ε ℝr with Σi=1r λi = α. The acyclicity of
our resolution reduces to that of a cellular free
resolution, supported on Δ, of a related monomial ideal.
Our resolution implies the multiplier ideal sum formula
generalizing Takagi's formula for two summands [Tak05], and
recovering Howald's multiplier ideal formula for monomial
ideals [How01] as a special case. Our resolution also yields
a new exactness proof for the Skoda complex [Laz04, Section
9.6.C]. © International Press 2008.},
Doi = {10.4310/MRL.2008.v15.n2.a13},
Key = {fds376730}
}
@article{fds243932,
Author = {First Miller and Shin Yao Jow},
Title = {Multiplier ideals of sums via cellular resolutions},
Journal = {Mathematical Research Letters},
Volume = {15},
Number = {2},
Pages = {359-373},
Year = {2008},
ISSN = {1073-2780},
MRCLASS = {14B05},
MRNUMBER = {2009b:14004},
url = {http://dx.doi.org/10.4310/MRL.2008.v15.n2.a13},
Abstract = {Fix nonzero ideal sheaves a1, . . . ., ar and b on a normal
ℚ-Gorenstein complex variety X. For any positive real
numbers α and β, we construct a resolution of the
multiplier ideal script T((a1 + . . . + ar)αbβ) by sheaves
that are direct sums of multiplier ideals script T(a1λ1 . .
. arλrbβ) for various λ ε ℝ≥0r satisfying Σi=1r λi
= α. The resolution is cellular, in the sense that its
boundary maps are encoded by the algebraic chain complex of
a regular CW-complex. The CW-complex is naturally expressed
as a triangulation Δ of the simplex of nonnegative real
vectors λ ε ℝr with Σi=1r λi = α. The acyclicity of
our resolution reduces to that of a cellular free
resolution, supported on Δ, of a related monomial ideal.
Our resolution implies the multiplier ideal sum formula
generalizing Takagi's formula for two summands [Tak05], and
recovering Howald's multiplier ideal formula for monomial
ideals [How01] as a special case. Our resolution also yields
a new exactness proof for the Skoda complex [Laz04, Section
9.6.C]. © International Press 2008.},
Doi = {10.4310/MRL.2008.v15.n2.a13},
Key = {fds243932}
}
@article{fds304494,
Author = {Jow, SY and Miller, E},
Title = {Multiplier ideals of sums via cellular resolutions},
Journal = {Mathematical Research Letters},
Volume = {15},
Number = {2-3},
Pages = {359-373},
Year = {2008},
ISSN = {1073-2780},
url = {http://dx.doi.org/10.4310/MRL.2008.v15.n2.a13},
Abstract = {Fix nonzero ideal sheaves a1, . . . ., ar and b on a normal
ℚ-Gorenstein complex variety X. For any positive real
numbers α and β, we construct a resolution of the
multiplier ideal script T((a1 + . . . + ar)αbβ) by sheaves
that are direct sums of multiplier ideals script T(a1λ1 . .
. arλrbβ) for various λ ε ℝ≥0r satisfying Σi=1r λi
= α. The resolution is cellular, in the sense that its
boundary maps are encoded by the algebraic chain complex of
a regular CW-complex. The CW-complex is naturally expressed
as a triangulation Δ of the simplex of nonnegative real
vectors λ ε ℝr with Σi=1r λi = α. The acyclicity of
our resolution reduces to that of a cellular free
resolution, supported on Δ, of a related monomial ideal.
Our resolution implies the multiplier ideal sum formula
generalizing Takagi's formula for two summands [Tak05], and
recovering Howald's multiplier ideal formula for monomial
ideals [How01] as a special case. Our resolution also yields
a new exactness proof for the Skoda complex [Laz04, Section
9.6.C]. © International Press 2008.},
Doi = {10.4310/MRL.2008.v15.n2.a13},
Key = {fds304494}
}
@article{fds166137,
Author = {First Miller and Victor Reiner},
Title = {What is geometric combinatorics?---An overview of the
graduate summer school},
Volume = {13},
Series = {IAS/Park City Math. Ser.},
Pages = {1-17},
Booktitle = {Geometric combinatorics},
Publisher = {Amer. Math. Soc.},
Address = {Providence, RI},
Year = {2007},
MRCLASS = {52-01 (05-01 05Bxx)},
MRNUMBER = {MR2383124},
url = {http://www.ams.org/mathscinet-getitem?mr=2383124},
Key = {fds166137}
}
@article{fds243911,
Author = {First Miller and Ning Jia},
Title = {Duality of antidiagonals and pipe dreams},
Journal = {Séminaire Lotharingien de Combinatoire},
Volume = {58},
Publisher = {Art. B58e (6 pages)},
Year = {2007},
MRCLASS = {05E15 (05A05 14M15)},
MRNUMBER = {2009m:05196},
url = {http://arxiv.org/abs/math/0706.3031},
Key = {fds243911}
}
@article{fds243929,
Author = {First Miller and Knutson, A and Miller, E and Shimozono, M},
Title = {Four positive formulae for type A quiver
polynomials},
Journal = {Inventiones Mathematicae},
Volume = {166},
Number = {2},
Pages = {229-325},
Publisher = {Springer Nature},
Year = {2006},
Month = {November},
ISSN = {0020-9910},
MRCLASS = {14M15 (05E15)},
MRNUMBER = {2007k:14098},
url = {http://dx.doi.org/10.1007/s00222-006-0505-0},
Abstract = {We give four positive formulae for the (equioriented type A)
quiver polynomials of Buch and Fulton [BF99 ]. All four
formulae are combinatorial, in the sense that they are
expressed in terms of combinatorial objects of certain
types: Zelevinsky permutations, lacing diagrams, Young
tableaux, and pipe dreams (also known as rc-graphs). Three
of our formulae are multiplicity-free and geometric, meaning
that their summands have coefficient 1 and correspond
bijectively to components of a torus-invariant scheme. The
remaining (presently non-geometric) formula is a variant of
the conjecture of Buch and Fulton in terms of factor
sequences of Young tableaux [BF99 ]; our proof of it
proceeds by way of a new characterization of the tableaux
counted by quiver constants. All four formulae come
naturally in "doubled" versions, two for double quiver
polynomials, and the other two for their stable limits, the
double quiver functions, where setting half the variables
equal to the other half specializes to the ordinary case.
Our method begins by identifying quiver polynomials as
multidegrees [BB82 , Jos84 , BB85 , Ros89 ] via equivariant
Chow groups [EG98 ]. Then we make use of Zelevinsky's map
from quiver loci to open subvarieties of Schubert varieties
in partial flag manifolds [Zel85 ]. Interpreted in
equivariant cohomology, this lets us write double quiver
polynomials as ratios of double Schubert polynomials [LS82 ]
associated to Zelevinsky permutations; this is our first
formula. In the process, we provide a simple argument that
Zelevinsky maps are scheme-theoretic isomorphisms
(originally proved in [LM98 ]). Writing double Schubert
polynomials in terms of pipe dreams [FK96 ] then provides
another geometric formula for double quiver polynomials, via
[KM05 ]. The combinatorics of pipe dreams for Zelevinsky
permutations implies an expression for limits of double
quiver polynomials in terms of products of Stanley symmetric
functions [Sta84 ]. A degeneration of quiver loci (orbit
closures of GL on quiver representations) to unions of
products of matrix Schubert varieties [Ful92 , KM05 ]
identifies the summands in our Stanley function formula
combinatorially, as lacing diagrams that we construct based
on the strands of Abeasis and Del Fra in the representation
theory of quivers [AD80 ]. Finally, we apply the
combinatorial theory of key polynomials to pass from our
lacing diagram formula to a double Schur function formula in
terms of peelable tableaux [RS95a , RS98 ], and from there
to our formula of Buch-Fulton type. © Springer-Verlag
2006.},
Doi = {10.1007/s00222-006-0505-0},
Key = {fds243929}
}
@article{fds243927,
Author = {First Miller and Matusevich, LF and Miller, E},
Title = {Combinatorics of rank jumps in simplicial hypergeometric
systems},
Journal = {Proceedings of the American Mathematical
Society},
Volume = {134},
Number = {5},
Pages = {1375-1381},
Publisher = {American Mathematical Society (AMS)},
Year = {2006},
Month = {May},
ISSN = {0002-9939},
MRCLASS = {33C70 (13D45 14M25 52B20)},
MRNUMBER = {2006j:33016},
url = {http://dx.doi.org/10.1090/S0002-9939-05-08245-6},
Abstract = {Let A be an integer d × n matrix, and assume that the
convex hull conv(A) of its columns is a simplex of dimension
d - 1 not containing the origin. It is known that the
semigroup ring ℂ[Ndbl;A] is Cohen-Macaulay if and only if
the rank of the GKZ hypergeometric system H A(β) equals the
normalized volume of conv(A) for all complex parameters β
ε ℂ d (Saito, 2002). Our refinement here shows that H
A(β) has rank strictly larger than the volume of conv(A) if
and only if β lies in the Zariski closure (in ℂ d) of all
Zdbl; d-graded degrees where the local cohomology ⊕ i<d H
mi(ℂ[ℕA]) is nonzero. We conjecture that the same
statement holds even when conv(A) is not a simplex. © 2005
American Mathematical Society.},
Doi = {10.1090/S0002-9939-05-08245-6},
Key = {fds243927}
}
@article{fds243928,
Author = {First Miller and Miller, E and Reiner, V},
Title = {Stanley's simplicial poset conjecture, after M.
Masuda},
Journal = {Communications in Algebra},
Volume = {34},
Number = {3},
Pages = {1049-1053},
Publisher = {Informa UK Limited},
Year = {2006},
Month = {February},
ISSN = {0092-7872},
MRCLASS = {13F55 (05E99 13F50 55U10)},
MRNUMBER = {2006m:13023},
url = {http://dx.doi.org/10.1080/00927870500442005},
Abstract = {M. Masuda recently provided the missing piece proving a
conjecture of R.P. Stanley on the characterization of
f-vectors for Gorenstein *simplicial posets. We propose a
slight simplification of Masuda's proof.},
Doi = {10.1080/00927870500442005},
Key = {fds243928}
}
@article{fds243925,
Author = {First Miller and Matusevich, LF and Miller, E and Walther, U},
Title = {Homological methods for hypergeometric families},
Journal = {Journal of the American Mathematical Society},
Volume = {18},
Number = {4},
Pages = {919-941},
Year = {2005},
Month = {October},
ISSN = {0894-0347},
MRCLASS = {13D45 (13H10 13N10 14M25 33C70)},
MRNUMBER = {2007d:13027},
url = {http://dx.doi.org/10.1090/S0894-0347-05-00488-1},
Doi = {10.1090/S0894-0347-05-00488-1},
Key = {fds243925}
}
@article{fds243924,
Author = {First Miller and Miller, E},
Title = {Alternating formulas for K-theoretic quiver
polynomials},
Journal = {Duke Mathematical Journal},
Volume = {128},
Number = {1},
Pages = {1-17},
Publisher = {Duke University Press},
Year = {2005},
Month = {May},
MRCLASS = {05E05 (14C17)},
MRNUMBER = {2006e:05181},
url = {http://dx.doi.org/10.1215/S0012-7094-04-12811-8},
Abstract = {The main theorem here is the K-theoretic analogue of the
cohomological "stable double component formula" for quiver
polynomials in [KMS]. This K-theoretic version is still in
terms of lacing diagrams, but nonminimal diagrams contribute
terms of higher degree. The motivating consequence is a
conjecture of Buch [B1] on the sign alternation of the
coefficients appearing in his expansion of quiver
K-polynomials in terms of stable Grothendieck polynomials
for partitions.},
Doi = {10.1215/S0012-7094-04-12811-8},
Key = {fds243924}
}
@article{fds243923,
Author = {First Miller and Kogan, M and Miller, E},
Title = {Toric degeneration of Schubert varieties and Gelfand-Tsetlin
polytopes},
Journal = {Advances in Mathematics},
Volume = {193},
Number = {1},
Pages = {1-17},
Publisher = {Elsevier BV},
Year = {2005},
Month = {May},
MRCLASS = {14M15},
MRNUMBER = {2006d:14054},
url = {http://dx.doi.org/10.1016/j.aim.2004.03.017},
Abstract = {This note constructs the flat toric degeneration of the
manifold ℱℓn of flags in ℂn due to Gonciulea and
Lakshmibai (Transform. Groups 1(3) (1996) 215) as an
explicit GIT quotient of the Gröbner degeneration due to
Knutson and Miller (Gröbner geometry of Schubert
polynomials, Ann. Math. (2) to appear). This implies that
Schubert varieties degenerate to reduced unions of toric
varieties, associated to faces indexed by rc-graphs (reduced
pipe dreams) in the Gelfand-Tsetlin polytope. Our explicit
description of the toric degeneration of ℱℓn provides a
simple explanation of how Gelfand-Tsetlin decompositions for
irreducible polynomial representations of GLn arise via
geometric quantization. © 2004 Elsevier Inc. All rights
reserved.},
Doi = {10.1016/j.aim.2004.03.017},
Key = {fds243923}
}
@article{fds304492,
Author = {Helm, D and Miller, E},
Title = {Algorithms for graded injective resolutions and local
cohomology over semigroup rings},
Journal = {Journal of Symbolic Computation},
Volume = {39},
Number = {3-4 SPEC. ISS.},
Pages = {373-395},
Publisher = {Elsevier BV},
Year = {2005},
Month = {March},
url = {http://dx.doi.org/10.1016/j.jsc.2004.11.009},
Abstract = {Let Q be an affine semigroup generating ℤd, and fix a
finitely generated ℤd-graded module M over the semigroup
algebra k[Q] for a field k. We provide an algorithm to
compute a minimal ℤd-graded injective resolution of M up
to any desired cohomological degree. As an application, we
derive an algorithm computing the local cohomology modules
HIi supported on any monomial (that is, ℤd-graded) ideal
I. Since these local cohomology modules are neither finitely
generated nor finitely cogenerated, part of this task is
defining a finite data structure to encode them. © 2005
Elsevier Ltd. All rights reserved.},
Doi = {10.1016/j.jsc.2004.11.009},
Key = {fds304492}
}
@article{fds243894,
Author = {Miller, E and Reiner, V},
Title = {Reciprocal domains and Cohen-Macaulay d-complexes in
ℝd},
Journal = {Electronic Journal of Combinatorics},
Volume = {11},
Number = {2 N},
Pages = {1-9},
Year = {2005},
Month = {January},
ISSN = {1077-8926},
Abstract = {We extend a reciprocity theorem of Stanley about enumeration
of integer points in polyhedral cones when one exchanges
strict and weak inequalities. The proof highlights the roles
played by Cohen-Macaulayness and canonical modules. The
extension raises the issue of whether a Cohen-Macaulay
complex of dimension d embedded piecewise-linearly in ℝd
is necessarily a d-ball. This is observed to be true for d
≤ 3, but false for d = 4.},
Key = {fds243894}
}
@article{fds304493,
Author = {Knutson, A and Miller, E},
Title = {Gröbner geometry of Schubert polynomials},
Journal = {Annals of Mathematics},
Volume = {161},
Number = {3},
Pages = {1245-1318},
Publisher = {Annals of Mathematics, Princeton U},
Year = {2005},
Month = {January},
ISSN = {0003-486X},
url = {http://dx.doi.org/10.4007/annals.2005.161.1245},
Abstract = {Given a permutation w ∈ Sn, we consider a determinantal
ideal Iw whose generators are certain minors in the generic
n × n matrix (filled with independent variables). Using
'multidegrees' as simple algebraic substitutes for
torus-equivariant cohomology classes on vector spaces, our
main theorems describe, for each ideal Iw: variously graded
multidegrees and Hubert series in terms of ordinary and
double Schubert and Grothendieck polynomials; a Gröbner
basis consisting of minors in the generic n × n matrix; the
Stanley-Reisner simplicial complex of the initial ideal in
terms of known combinatorial diagrams [FK96], [BB93]
associated to permutations in Sn; and a procedure inductive
on weak Bruhat order for listing the facets of this complex.
We show that the initial ideal is Cohen-Macaulay, by
identifying the Stanley-Reisner complex as a special kind of
"subword complex in Sn;", which we define generally for
arbitrary Coxeter groups, and prove to be shellable by
giving an explicit vertex decomposition. We also prove
geometrically a general positivity statement for
multidegrees of subschemes. Our main theorems provide a
geometric explanation for the naturality of Schubert
polynomials and their associated combinatorics. More
precisely, we apply these theorems to: define a single
geometric setting in which polynomial representatives for
Schubert classes in the integral cohomology ring of the flag
manifold are determined uniquely, and have positive
coefficients for geometric reasons; rederive from a
topological perspective Fulton's Schubert polynomial formula
for universal cohomology classes of degeneracy loci of maps
between flagged vector bundles; supply new proofs that
Schubert and Grothendieck polynomials represent cohomology
and K-theory classes on the flag manifold; and provide
determinantal formulae for the multidegrees of ladder
determinantal rings. The proofs of the main theorems
introduce the technique of "Bruhat induction", consisting of
a collection of geometric, algebraic, and combinatorial
tools, based on divided and isobaric divided differences,
that allow one to prove statements about determinantal
ideals by induction on weak Bruhat order.},
Doi = {10.4007/annals.2005.161.1245},
Key = {fds304493}
}
@article{fds243921,
Author = {First Miller and Knutson, A and Miller, E},
Title = {Subword complexes in Coxeter groups},
Journal = {Advances in Mathematics},
Volume = {184},
Number = {1},
Pages = {161-176},
Publisher = {Elsevier BV},
Year = {2004},
Month = {May},
MRCLASS = {20F55 (05E05 05E15 05E25 13F55)},
MRNUMBER = {2005c:20066},
url = {http://dx.doi.org/10.1016/S0001-8708(03)00142-7},
Abstract = {Let (Π, Σ) be a Coxeter system. An ordered list of
elements in Σ and an element in Π determine a subword
complex, as introduced in Knutson and Miller (Ann. of Math.
(2) (2003), to appear). Subword complexes are demonstrated
here to be homeomorphic to balls or spheres, and their
Hilbert series are shown to reflect combinatorial properties
of reduced expressions in Coxeter groups. Two formulae for
double Grothendieck polynomials, one of which appeared in
Fomin and Kirillov (Proceedings of the Sixth Conference in
Formal Power Series and Algebraic Combinatorics, DIMACS,
1994, pp. 183-190), are recovered in the context of
simplicial topology for subword complexes. Some open
questions related to subword complexes are presented. ©
2003 Elsevier Inc. All rights reserved.},
Doi = {10.1016/S0001-8708(03)00142-7},
Key = {fds243921}
}
@article{fds166192,
Author = {Appendix to: Mark Haiman},
Title = {Commutative algebra of n points in the plane},
Volume = {51},
Series = {Math. Sci. Res. Inst. Publ.},
Pages = {153-180},
Booktitle = {Trends in commutative algebra},
Publisher = {Cambridge Univ. Press},
Address = {Cambridge},
Editor = {Luchezar Avramov and Mark Green and Craig Huneke and Karen Smith and Bernd Sturmfels},
Year = {2004},
MRCLASS = {14C05 (05E05 13P10)},
MRNUMBER = {2006c:14005},
url = {http://www.ams.org/mathscinet-getitem?mr=2006c:14005},
Key = {fds166192}
}
@article{fds243922,
Author = {First Miller and Victor Reiner},
Title = {Reciprocal domains and Cohen-Macaulay d-complexes in
R^d},
Journal = {Electronic Journal of Combinatorics},
Volume = {11},
Number = {2},
Pages = {Note 1, 9 pp.},
Year = {2004},
MRCLASS = {52B20 (05A15 13H10)},
MRNUMBER = {2005k:52024},
url = {http://arxiv.org/abs/math/0408169},
Key = {fds243922}
}
@article{fds243919,
Author = {First Miller and Helm, D and Miller, E},
Title = {Bass numbers of semigroup-graded local cohomology},
Journal = {Pacific Journal of Mathematics},
Volume = {209},
Number = {1},
Pages = {41-66},
Publisher = {Mathematical Sciences Publishers},
Year = {2003},
Month = {January},
MRCLASS = {13D45},
MRNUMBER = {2004c:13028},
url = {http://dx.doi.org/10.2140/pjm.2003.209.41},
Abstract = {Given a module M over a ring R that has a grading by a
semigroup Q, we present a spectral sequence that computes
the local cohomology HIi(M) at any graded ideal I in terms
of Ext modules. We use this method to obtain flniteness
results for the local cohomology of graded modules over
semigroup rings. In particular we prove that for a semigroup
Q whose saturation Qsat is simplicial, and a finitely
generated module M over k[Q] that is graded by Qgp, the Bass
numbers of HIi(M) are finite for any Q-graded ideal I of
k[Q]. Conversely, if Qsat is not simplicial, we find a
graded ideal I and graded k[Q]-module M such that the local
cohomology module HIi(M) has infinite-dimensional socle. We
introduce and exploit the combinatorially defined essential
set of a semigroup.},
Doi = {10.2140/pjm.2003.209.41},
Key = {fds243919}
}
@article{fds243920,
Author = {Miller, E},
Title = {Mitosis recursion for coefficients of Schubert
polynomials},
Journal = {Journal of Combinatorial Theory. Series A},
Volume = {103},
Number = {2},
Pages = {223-235},
Publisher = {Elsevier BV},
Year = {2003},
Month = {January},
MRCLASS = {05E15 (14M15)},
MRNUMBER = {2004i:05157},
url = {http://dx.doi.org/10.1016/S0097-3165(03)00020-7},
Abstract = {Mitosis is a rule introduced by Knutson and Miller for
manipulating subsets of the n × n grid. It provides an
algorithm that lists the reduced pipe dreams (also known as
rc-graphs) of Fomin and Kirillov for a permutation w ∈ Sn
by downward induction on weak Bruhat order, thereby
generating the coefficients of Schubert polynomials of
Lascoux and Schützenberger inductively. This note provides
a short and purely combinatorial proof of these properties
of mitosis. © 2003 Published by Elsevier
Inc.},
Doi = {10.1016/S0097-3165(03)00020-7},
Key = {fds243920}
}
@article{fds243915,
Author = {Miller, E},
Title = {Planar graphs as minimal resolutions of trivariate monomial
ideals},
Journal = {Documenta Mathematica},
Volume = {7},
Number = {1},
Pages = {43-90},
Year = {2002},
Month = {January},
MRCLASS = {05C10 (13D02)},
MRNUMBER = {2003d:05055},
url = {http://www.math.uiuc.edu/documenta/vol-07/03.html},
Abstract = {We introduce the notion of rigid embedding in a grid
surface, a new kind of plane drawing for simple triconnected
planar graphs. Rigid embeddings provide methods to (1) find
well-structured (cellular, here) minimal free resolutions
for arbitrary monomial ideals in three variables: (2)
strengthen the Brightwell-Trotter bound on the order
dimension of triconnected planar maps by giving a geometric
reformulation: and (3) generalize Schnyder's angle coloring
of planar triangulations to arbitrary triconnected planar
maps via geometry. The notion of rigid embedding is stable
under duality for planar maps, and has certain uniqueness
properties.},
Key = {fds243915}
}
@article{fds243918,
Author = {Miller, E},
Title = {Cohen-Macaulay quotients of normal semioroup rings via
irreducible resolutions},
Journal = {Mathematical Research Letters},
Volume = {9},
Number = {1},
Pages = {117-128},
Publisher = {International Press of Boston},
Year = {2002},
Month = {January},
MRCLASS = {13D02 (13D25 13H10)},
MRNUMBER = {2003a:13015},
url = {http://dx.doi.org/10.4310/MRL.2002.v9.n1.a9},
Abstract = {For a radical monomial ideal I in a normal semigroup ring
κ[Q], there is a unique minimal irreducible resolution 0
→ κ[Q]/I → W̄0 → W̄1 ... by modules W̄i of the
form ⊕jk[Fij], where the Fij are (not necessarily
distinct) faces of Q. That is, W̄i is a direct sum of
quotients of κ[Q] by prime ideals. This paper characterizes
Cohen-Macaulay quotients κ[Q]/I as those whose rainimal
irreducible resolutions are linear, meaning that W̄i is
pure of dimension dim(κ[Q]/I) - i for i ≥ 0. The proof
exploits a graded ring-theoretic analogue of the Zeeman
spectral sequence [Zee63], thereby also providing a
combinatorial topological version involving no commutative
algebra. The characterization via linear irreducible
resolutions reduces to the Eagon-Reiner theorem [ER98] by
Alexander duality when Q = Nd.},
Doi = {10.4310/MRL.2002.v9.n1.a9},
Key = {fds243918}
}
@article{fds166154,
Title = {Graded Greenlees-May duality and the Čech
hull},
Volume = {226},
Series = {Lecture Notes in Pure and Appl. Math.},
Pages = {233-253},
Booktitle = {Local cohomology and its applications (Guanajuato,
1999)},
Publisher = {Dekker},
Address = {New York},
Year = {2002},
MRCLASS = {13D45 (13D02)},
MRNUMBER = {2004b:13019},
url = {http://www.ams.org/mathscinet-getitem?mr=2004b:13019},
Key = {fds166154}
}
@article{fds339831,
Author = {Miller, E},
Title = {Graded greenlees-may duality and the cech
hull},
Pages = {233-253},
Booktitle = {Local Cohomology and its Applications},
Year = {2001},
Month = {January},
ISBN = {9781138402133},
Abstract = {The duality theorem of Greenlees and May relating local
cohomology with support on an ideal I and the left derived
functors of J-adic completion [GM92) holds for rather
general ideals in commutative rings. Here, simple formulas
are provided for both local cohomology and derived functors
of zn-graded completion, when I is a monomial ideal in the
Zn-graded polynomial ring k[xl,…, xn] Greenlees-May
duality for this case is a consequence. A key construction
is the combinatorially defined Cech hull operation on
Zn-graded modules [Mil98, MilOO, YanOO]. A simple
self-contained proof of GM duality in the derived category
is presented for arbitrarily graded noetherian rings, using
methods motivated by the Čech hull.},
Key = {fds339831}
}
@article{fds243916,
Author = {First Miller and David Perkinson},
Title = {Eight lectures on monomial ideals},
Journal = {Queen's papers in pure and applied mathematics},
Volume = {120},
Pages = {3-105},
Year = {2001},
Key = {fds243916}
}
@article{fds243914,
Author = {Miller, E},
Title = {The Alexander duality functors and local duality with
monomial support},
Journal = {Journal of Algebra},
Volume = {231},
Number = {1},
Pages = {180-234},
Publisher = {Elsevier BV},
Year = {2000},
Month = {September},
MRCLASS = {13D45 (13F55)},
MRNUMBER = {2001k:13028},
url = {http://www.sciencedirect.com/science?_ob=ArticleURL&_aset=V-WA-A-W-BE-MsSAYWW-UUW-U-AAWZEEBCVD-AAWVCDVBVD-WEACYAWCA-BE-U&_rdoc=5&_fmt=summary&_udi=B6WH2-45F4PW6-52&_coverDate=09%2F01%2F2000&_cdi=6838&_orig=search&_st=13&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=a45cf03a490d09f2fff59d5813682703},
Abstract = {Alexander duality is made into a functor which extends the
notion for monomial ideals to any finitely generated
Nn-graded module. The functors associated with Alexander
duality provide a duality on the level of free and injective
resolutions, and numerous Bass and Betti number relations
result as corollaries. A minimal injective resolution of a
module M is equivalent to the injective resolution of its
Alexander dual and contains all of the maps in the minimal
free resolution of M over every Zn-graded localization.
Results are obtained on the interaction of duality for
resolutions with cellular resolutions and lcm-lattices.
Using injective resolutions, theorems of Eagon, Reiner, and
Terai are generalized to all Nn-graded modules: the
projective dimension of M equals the support-regularity of
its Alexander dual, and M is Cohen-Macaulay if and only if
its Alexander dual has a support-linear free resolution.
Alexander duality is applied in the context of the Zn-graded
local cohomology functors HiI(-) for squarefree monomial
ideals I in the polynomial ring S, proving a duality
directly generalizing local duality, which is the case when
I=m is maximal. In the process, a new flat complex for
calculating local cohomology at monomial ideals is
introduced, showing, as a consequence, that Terai's formula
for the Hilbert series of HiI(S) is equivalent to Hochster's
for Hn-im(S/I). © 2000 Academic Press.},
Doi = {10.1006/jabr.2000.8359},
Key = {fds243914}
}
@article{fds243910,
Author = {Miller, EN},
Title = {Icosahedra constructed from congruent triangles},
Journal = {Discrete and Computational Geometry},
Volume = {24},
Number = {2-3},
Pages = {437-451},
Year = {2000},
Month = {January},
MRCLASS = {52B12 (52B15)},
MRNUMBER = {2001d:52021},
url = {http://dx.doi.org/10.1007/s004540010047},
Abstract = {It is possible to construct a figure in three dimensions
which is combinatorially equivalent to a regular
icosahedron, and whose faces are all congruent but not
equilateral. Such icosamonohedra can be convex or nonconvex,
and can be deformed continuously. A scalene triangle can
construct precisely zero, one, or two convex icosamonohedra,
and each occurs. Demonstrated here are two explicit convex
examples, the first of which is the unique such object
constructed from scalene right triangles, proving a
conjecture of Banchoff and Strauss.},
Doi = {10.1007/s004540010047},
Key = {fds243910}
}
@article{fds243917,
Author = {First Miller and Miller, E and Sturmfels, B and Yanagawa, K},
Title = {Generic and cogeneric monomial ideals},
Journal = {Journal of Symbolic Computation},
Volume = {29},
Number = {4-5},
Pages = {691-708},
Publisher = {Elsevier BV},
Year = {2000},
Month = {January},
MRCLASS = {13P10 (13F20 13F55)},
MRNUMBER = {2001m:13051},
url = {http://dx.doi.org/10.1006/jsco.1999.0290},
Abstract = {Monomial ideals which are generic with respect to either
their generators or irreducible components have minimal free
resolutions encoded by simplicial complexes. There are
numerous equivalent ways to say that a monomial ideal is
generic or cogeneric. For a generic monomial ideal, the
associated primes satisfy a saturated chain condition, and
the Cohen-Macaulay property implies shellability for both
the Scarf complex and the Stanley-Reisner complex. Reverse
lexicographic initial ideals of generic lattice ideals are
generic. Cohen-Macaulayness for cogeneric ideals is
characterized combinatorially; in the cogeneric case, the
Cohen-Macaulay type is greater than or equal to the number
of irreducible components. Methods of proof include
Alexander duality and Stanley's theory of local h -vectors.
© 2000 Academic Press.},
Doi = {10.1006/jsco.1999.0290},
Key = {fds243917}
}
@article{fds243912,
Author = {Ezra Miller},
Title = {Multiplicities of ideals in Noetherian rings},
Journal = {Beiträge zur Algebra und Geometrie},
Volume = {39},
Number = {1},
Pages = {47-51},
Year = {1998},
MRCLASS = {13H10},
MRNUMBER = {99c:13044},
url = {http://www.ams.org/mathscinet-getitem?mr=99c:13044},
Key = {fds243912}
}
@article{fds320535,
Author = {First Miller and Miller, E and Sturmfels, B},
Title = {Monomial ideals and planar graphs},
Journal = {Applied Algebra, Algebraic Algorithms and Error-Correcting
Codes},
Volume = {1719},
Series = {Lecture Notes in Computer Science},
Pages = {19-28},
Booktitle = {Applied algebra, algebraic algorithms and error-correcting
codes (Honolulu, HI, 1999)},
Publisher = {Springer Berlin Heidelberg},
Address = {Berlin},
Editor = {Fossorier, M and Imai, H and Lin, S and Poli, A},
Year = {1998},
ISBN = {3540667237},
MRCLASS = {13P10 (13D02 13F20)},
MRNUMBER = {2002h:13041},
url = {http://dx.doi.org/10.1007/3-540-46796-3_3},
Abstract = {Gröbner basis theory reduces questions about systems of
polynomial equations to the combinatorial study of monomial
ideals, or staircases. This article gives an elementary
introduction to current research in this area. After
reviewing the bivariate case, a new correspondence is
established between planar graphs and minimal resolutions of
monomial ideals in three variables. A brief guide is given
to the literature on complexity issues and monomial ideals
in four or more variables.},
Doi = {10.1007/3-540-46796-3_3},
Key = {fds320535}
}
%% Papers Submitted
@article{fds212344,
Author = {First Miller and Megan Owen and Scott Provan},
Title = {Polyhedral computational geometry for averaging metric
phylogenetic trees},
Year = {2012},
url = {http://arxiv.org/abs/math/1211.7046},
Abstract = {This paper investigates the computational geometry relevant
to calculations of the Fréchet mean and variance for
probability distributions on the phylogenetic tree space of
Billera, Holmes and Vogtmann, using the theory of
probability measures on spaces of nonpositive curvature
developed by Sturm. We show that the combinatorics of
geodesics with a specified fixed endpoint in tree space are
determined by the location of the varying endpoint in a
certain polyhedral subdivision of tree space. The variance
function associated to a finite subset of tree space is
continuously differentiable within each cell of the
corresponding subdivision. We use this subdivision to
establish two iterative methods for producing sequences that
converge to the Fréchet mean: one based on Sturm's
Law of Large Numbers, and another based on descent
algorithms for finding optima of smooth functions on convex
polyhedra. We present properties and biological applications
of Frechet means and extend our main results to more general
globally nonpositively curved spaces composed of Euclidean
orthants.},
Key = {fds212344}
}
@article{fds212345,
Author = {First Miller and Christine Berkesch and Stephen Griffeth},
Title = {Systems of parameters and holonomicity of hypergeometric
systems},
Year = {2012},
Abstract = {The main result is an elementary proof of holonomicity for
<var>A</var>-hypergeometric systems, with no requirements on
the behavior of their singularities, originally due to
Adolphson (1994) after the regular singular case by Gelfand
and Gelfand (1986). Our method yields a direct de novo proof
that <var>A</var>-hypergeometric systems form holonomic
families over their parameter spaces, as shown by
Matusevich, Miller, and Walther (2005).},
Key = {fds212345}
}
%% Other
@misc{fds166158,
Author = {First Miller and Alan Guo and Mike Weimerskirch},
Title = {Potential applications of commutative algebra to
combinatorial game theory},
Volume = {22},
Series = {Oberwolfach Reports},
Pages = {23-26},
Booktitle = {Kommutative Algebra},
Editor = {Winfried Bruns and Hubert Flenner and Craig
Huneke},
Year = {2009},
Key = {fds166158}
}
@misc{fds166130,
Title = {Alexander duality for monomial ideals and their
resolutions},
Journal = {Rejecta Mathematica},
Volume = {1},
Pages = {18-57},
Year = {2009},
url = {http://math.rejecta.org/vol1-num1/18-57},
Key = {fds166130}
}
@misc{fds166159,
Author = {First Miller and Alicia Dickenstein and Laura Matusevich},
Title = {Extended abstract: Binomial D-modules},
Booktitle = {Proceedings MEGA (Effective Methods in Algebraic Geometry),
Strobl, Austria},
Publisher = {13 pages},
Year = {2007},
url = {http://www.ricam.oeaw.ac.at/mega2007/electronic/17.pdf},
Key = {fds166159}
}
@misc{fds166160,
Author = {First Miller and Shin-Yao Jow},
Title = {Extended abstract: Cellular resolutions of multiplier ideals
of sums},
Volume = {4},
Series = {Oberwolfach Reports},
Number = {1},
Booktitle = {Topological and geometric combinatorics, abstracts from the
Jan. 28-Feb. 3, 2007 workshop},
Publisher = {3 pages},
Editor = {Anders Björner and Gil Kalai and Günter
Ziegler},
Year = {2007},
Key = {fds166160}
}
@misc{fds166161,
Author = {First Miller and Laura Matusevich and Uli Walther},
Title = {Extended abstract: Homological methods for hypergeometric
families},
Volume = {3},
Series = {Oberwolfach Reports},
Number = {1},
Booktitle = {Convex and algebraic geometry, abstracts from the workshop
held Jan. 29-Feb. 4, 2006},
Publisher = {3 pages},
Editor = {Klaus Altmann and Victory Batyrev and Bernard
Teissier},
Year = {2006},
Key = {fds166161}
}
@misc{fds166188,
Author = {First Miller and David Helm},
Title = {Extended abstract: Algorithms for graded injective
resolutions and local cohomology over semigroup
rings},
Booktitle = {Proceedings MEGA (Effective Methods in Algebraic Geometry),
Kaiserslautern, Germany},
Publisher = {5 pages},
Year = {2003},
Key = {fds166188}
}
@misc{fds166189,
Author = {First Miller and Allen Knutson},
Title = {Extended abstract: Gröbner geometry of Schubert
polynomials},
Booktitle = {Proceedings FPSAC (Formal Power Series and Algebraic
Combinatorics), Melbourne},
Publisher = {10 pages},
Year = {2002},
Key = {fds166189}
}
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Mathematics Department