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Catalin Zara
Text Only Version
Last updated:
November 11, 2007
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Research
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Published papers
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Abstract:
Let $M$ be a compact Hamiltonian $T-$space, with finite fixed point
set, $M^T$. An equivariant class
is determined by its restriction to $M^T$, and to each fixed point
$p \in M^T$ and generic component
of the moment map, there corresponds a canonical class $\tau_p$.
For a special class of Hamiltonian
$T-$spaces, the value $\tau_{p,q}$ of $\tau_p$ at a fixed point
$q$ can be determined through an
iterated interpolation procedure, and we obtained a formula for
$\tau_{p,q}$ as a sum over ascending
chains from $p$ to $q$. In general the number of such chains is huge,
and the main result of this paper
is a procedure to reduce the number of relevant chains, through a
systematic degeneration of the interpolation
direction. The resulting formula for $\tau_{p,q}$ resembles,
via the localization formula, an integral
over a space of chains, and we prove that, for complex
Grassmannians, $\tau_{p,q}$ can indeed be expressed
as the integral of an equivariant form over a smooth Schubert
variety.
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Complete Padovan Sequences in Finite Fields, The Fibonacci Quarterly.
Vol. 45 (2007) , No. 1, 64--75. (with J. Gil and M. Weiner). [PDF]
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Abstract:
Given a prime $p \geq 5$, and given
$1 < k < p-1$, we call a sequence
$(a_n)_{n}$ in $\mathbb{F}_p$ a
$\Phi_{k}$-sequence if it is periodic with period $p-1$,
and if it satisfies the linear recurrence
$a_n+a_{n+1}=a_{n+k}$ with $a_0=1$. Such a sequence is said
to be a complete
$\Phi_{k}$-sequence if in addition
$\{a_0,a_1,...,a_{p-2}\}=\{1,...,p-1\}$. For instance,
every primitive root $b$ mod $p$ generates a complete
$\Phi_{k}$-sequence
$a_n=b^n$ for some (unique) $k$. A natural question is whether every
complete $\Phi_{k}$-sequence
is necessarily defined by a primitive root. For $k=2$ the answer is
known to be
positive. In this paper we reexamine that case and investigate the case
$k=3$
together with the associated cases $k=p-2$ and $k=p-3$.
One of the sequences introduced in this paper is included in
The On-Line Encyclopedia of Integer Sequences
(sequence A134573)
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A GKM description of the equivariant cohomology ring of a homogeneous space,
J. Algebraic Combin. 23 (2006), 21--41 (with V. Guillemin and T. Holm).
[PDF]
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Abstract:
Let T be a torus of dimension n >1 and M a compact
T-manifold. M is a GKM manifold if the set of zero
dimensional orbits in the orbit space M/T is zero dimensional and
the set of one dimensional orbits in M/T is one dimensional. For
such a manifold these sets of orbits have the structure of a labelled
graph and it is known that a lot of topological information about M
is encoded in this graph. In this paper we prove that every compact
homogeneous space M of non-zero Euler characteristic is of GKM type
and show that the graph associated with M encodes geometric
information about M as well as topological information. For
example, from this graph one can detect whether M admits an
invariant complex structure or an invariant almost complex structure.
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Chains, Subwords, and Fillings: Strong Equivalence of Three Definitions of the Bruhat Order,
Electron. J. Combin. 13 (1) (2006), #N5.
[PDF]
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Abstract:
Let Sn be the group of permutations of [n]={1,...,n}. The
Bruhat order on Sn is a partial order relation, for which there
are several equivalent definitions. Three well-known conditions are based on
ascending chains, subwords, and comparison of matrices, respectively. We
express the last using fillings of tableaux, and prove that the three equivalent
conditions are satisfied in the same number of ways.
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Parking Functions, Stack-Sortable Permutations, and Spaces of Paths in the Johnson Graph,
Electron. J. Combin. 9 (2) (2003), #R11.
[PDF]
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The Existence of Generating Families for the Cohomology Ring of a Graph,
Advances in Mathematics 174 (2003) No. 1, p. 115-153 (with V. Guillemin).
[PDF]
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Abstract:
Let Γ be a finite d-valent graph and G an n-dimensional torus. An action of G on Γ
is defined by a map that assigns to each oriented edge e of Γ a 1-dimensional representation
of G (or, alternatively, a weight αe in the weight lattice of G. For the assignment
e → αe to be a schematic description of a G-action, these weights have to satisfy
certain compatibility conditions). We attach to (Γ, α) an equivariant cohomology ring, H(Γ,α).
By definition, this ring contains the equivariant cohomology ring of a point, HG(pt)=S(g*), as a
subring, and in this paper we use graphical versions of standard Morse theoretical techniques to analyze the
structure of H(Γ,α) as an S(g*)-module.
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Abstract:
Let G be a torus of dimension n > 1 and M be a compact Hamiltonian G-manifold with MG finite.
A circle K in G is generic if MG = MK. For such a circle the moment map associated with
its action on M is a perfect Morse function. Let { Wp+; p in MG} be the
Morse--Whitney stratification of M associated with this function and let τp+ be the
equivariant Thom class dual to Wp+. These classes form a basis of HG*(M) as a
module over S(g*) and, in particular, τp+ τq+ = ∑
cpqr τr+ with cpqr in S(g*).
For a large class of manifolds of this type we obtain a combinatorial description of
these τp+'s and, from this description, a combinatorial formula for
cpqr.
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Abstract:
Let G be an n-dimensional torus and τ a Hamiltonian action of G on a compact symplectic manifold, M. If M is
pre-quantizable one can associate with τ a representation of G on a virtual vector space, Q (M), by
spinC-quantization. If M is a symplectic GKM manifold, we will show that several well-known
theorems about this "quantum action" of G: for example, the convexity theorem, the Kostant
multiplicity theorem and the "quantization commutes with reduction"
theorem for circle subgroups of G, are basically just theorems about G-actions on graphs.
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1-Skeletons, Betti Numbers and Equivariant Cohomology,
Duke Math. Journal 107 (2001), No.2, pp. 283-349 (with V. Guillemin).
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Abstract:
The one-skeleton of a G-manifold M is the set of points p in M
where dim Gp ≥ dim G -1; and M is a GKM manifold if the
dimension of this one-skeleton is 2. Goresky, Kottwitz and MacPherson
show that for such a manifold this one-skeleton has the structure of a
"labeled" graph, (Γ, α), and that the equivariant cohomology
ring of M is isomorphic to the "cohomology ring" of this graph. Hence,
if M is symplectic, one can show that this ring is a free module
over the symmetric algebra S(g*), with b2i(Γ) generators
in dimension 2i, b2i(Γ) being the "combinatorial" 2i-th
Betti number of Γ. In this article we show that this
"topological" result is , in fact, a combinatorial result about graphs.
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Equivariant de Rham Theory and Graphs,
Asian J. of Math 3 (1999), no. 1, pp.49-76. Reprinted in
Surveys in Differential Geometry 7 (2000) , pp. 221-257 (with V. Guillemin).
Abstract:
Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant
cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up
the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection,
turn out simply to be theorems about graphs. In this paper we show that for some familiar
theorems, this is indeed the case.
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A Characterization of Bi-Invariant Pseudo-Riemannian Metrics on Lie Groups,
Stud.Cerc.Mat. 50 (1998). no. 1-2, pp. 111-115.
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Theorems and Problems on Lie Groups (book, in Romanian),
Ed.Univ.Bucharest 1997 (with L. Nicolescu and G. Pripoae).
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On a theorem of D. Muller (Romanian),
Stud.Cerc.Mat. 47 (1995), no. 3-4, pp. 359--363.
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