77,815 research outputs found
Combinatorial Yamabe Flow on Surfaces
In this paper we develop an approach to conformal geometry of piecewise flat
metrics on manifolds. In particular, we formulate the combinatorial Yamabe
problem for piecewise flat metrics. In the case of surfaces, we define the
combinatorial Yamabe flow on the space of all piecewise flat metrics associated
to a triangulated surface. We show that the flow either develops removable
singularities or converges exponentially fast to a constant combinatorial
curvature metric. If the singularity develops, we show that the singularity is
always removable by a surgery procedure on the triangulation.
We conjecture that after finitely many such surgery changes on the
triangulation, the flow converges to the constant combinatorial curvature
metric as time approaches infinity
Sensor Array Design Through Submodular Optimization
We consider the problem of far-field sensing by means of a sensor array.
Traditional array geometry design techniques are agnostic to prior information
about the far-field scene. However, in many applications such priors are
available and may be utilized to design more efficient array topologies. We
formulate the problem of array geometry design with scene prior as one of
finding a sampling configuration that enables efficient inference, which turns
out to be a combinatorial optimization problem. While generic combinatorial
optimization problems are NP-hard and resist efficient solvers, we show how for
array design problems the theory of submodular optimization may be utilized to
obtain efficient algorithms that are guaranteed to achieve solutions within a
constant approximation factor from the optimum. We leverage the connection
between array design problems and submodular optimization and port several
results of interest. We demonstrate efficient methods for designing arrays with
constraints on the sensing aperture, as well as arrays respecting combinatorial
placement constraints. This novel connection between array design and
submodularity suggests the possibility for utilizing other insights and
techniques from the growing body of literature on submodular optimization in
the field of array design
A survey of Elekes-R\'onyai-type problems
We give an overview of recent progress around a problem introduced by Elekes
and R\'onyai. The prototype problem is to show that a polynomial has a large image on a Cartesian product , unless has a group-related special form. We discuss a number
of variants and generalizations. This includes the Elekes-Szab\'o problem,
which generalizes the Elekes-R\'onyai problem to a question about an upper
bound on the intersection of an algebraic surface with a Cartesian product, and
curve variants, where we ask the same questions for Cartesian products of
finite subsets of algebraic curves. These problems lie at the crossroads of
combinatorics, algebra, and geometry: They ask combinatorial questions about
algebraic objects, whose answers turn out to have applications to geometric
questions involving basic objects like distances, lines, and circles, as well
as to sum-product-type questions from additive combinatorics. As part of a
recent surge of algebraic techniques in combinatorial geometry, a number of
quantitative and qualitative steps have been made within this framework.
Nevertheless, many tantalizing open questions remain
A geometric method of sector decomposition
We propose a new geometric method of IR factorization in sector
decomposition. The problem is converted into a set of problems in convex
geometry. The latter problems are solved using algorithms in combinatorial
geometry. This method provides a deterministic algorithm and never falls into
an infinite loop. The number of resulting sectors depends on the algorithm of
triangulation. Our test implementation shows smaller number of sectors
comparing with other existing methods with iterations.Comment: 17 pages, 2 eps figure
Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory
Arithmetic combinatorics is often concerned with the problem of bounding the
behaviour of arbitrary finite sets in a group or ring with respect to
arithmetic operations such as addition or multiplication. Similarly,
combinatorial geometry is often concerned with the problem of bounding the
behaviour of arbitrary finite collections of geometric objects such as points,
lines, or circles with respect to geometric operations such as incidence or
distance. Given the presence of arbitrary finite sets in these problems, the
methods used to attack these problems have primarily been combinatorial in
nature. In recent years, however, many outstanding problems in these questions
have been solved by algebraic means (and more specifically, using tools from
algebraic geometry and/or algebraic topology), giving rise to an emerging set
of techniques which is now known as the polynomial method.
While various instances of the polynomial method have been known for decades
(e.g. Stepanov's method, the combinatorial nullstellensatz, or Baker's
theorem), the general theory of this method is still in the process of
maturing; in particular, the limitations of the polynomial method are not well
understood, and there is still considerable scope to apply deeper results from
algebraic geometry or algebraic topology to strengthen the method further. In
this survey we present several of the known applications of these methods,
focusing on the simplest cases to illustrate the techniques. We will assume as
little prior knowledge of algebraic geometry as possible.Comment: 44 pages, no figures. Final revision, incorporating several minor
correction
L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry
We survey connections between the theory of bi-Lipschitz embeddings and the
Sparsest Cut Problem in combinatorial optimization. The story of the Sparsest
Cut Problem is a striking example of the deep interplay between analysis,
geometry, and probability on the one hand, and computational issues in discrete
mathematics on the other. We explain how the key ideas evolved over the past 20
years, emphasizing the interactions with Banach space theory, geometric measure
theory, and geometric group theory. As an important illustrative example, we
shall examine recently established connections to the the structure of the
Heisenberg group, and the incompatibility of its Carnot-Carath\'eodory geometry
with the geometry of the Lebesgue space .Comment: To appear in Proceedings of the International Congress of
Mathematicians, Hyderabad India, 201
Schur functions for approximation problems
In this paper we propose a new approach to least squares approximation
problems. This approach is based on partitioning and Schur function. The nature
of this approach is combinatorial, while most existing approaches are based on
algebra and algebraic geometry. This problem has several practical
applications. One of them is curve clustering. We use this application to
illustrate the results
Toric Degenerations and tropical geometry of branching algebras
We construct polyhedral families of valuations on the branching algebra of a
morphism of reductive groups. This establishes a connection between the
combinatorial rules for studying a branching problem and the tropical geometry
of the branching algebra. In the special case when the branching problem comes
from the inclusion of a Levi subgroup or a diagonal subgroup, we use the dual
canonical basis of Lusztig and Kashiwara to build toric deformations of the
branching algebra.Comment: 11 figures; shortene
Limits of Order Types
We apply ideas from the theory of limits of dense combinatorial structures to
study order types, which are combinatorial encodings of finite point sets.
Using flag algebras we obtain new numerical results on the Erd\H{o}s problem of
finding the minimal density of 5-or 6-tuples in convex position in an arbitrary
point set, and also an inequality expressing the difficulty of sampling order
types uniformly. Next we establish results on the analytic representation of
limits of order types by planar measures. Our main result is a rigidity
theorem: we show that if sampling two measures induce the same probability
distribution on order types, then these measures are projectively equivalent
provided the support of at least one of them has non-empty interior. We also
show that some condition on the Hausdorff dimension of the support is necessary
to obtain projective rigidity and we construct limits of order types that
cannot be represented by a planar measure. Returning to combinatorial geometry
we relate the regularity of this analytic representation to the aforementioned
problem of Erd\H{o}s on the density of k-tuples in convex position, for large
k
Hodge theory in combinatorics
George Birkhoff proved in 1912 that the number of proper colorings of a
finite graph G with n colors is a polynomial in n, called the chromatic
polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of
absolute values of coefficients of the chromatic polynomial is unimodal: it
goes up, hits a peak, and then goes down. Read's conjecture was proved by June
Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh's
result was subsequently refined and generalized by Huh and Katz, again using
substantial doses of algebraic geometry. Both papers in fact establish
log-concavity of the coefficients, which is stronger than unimodality.
The breakthroughs of Huh and Huh-Katz left open the more general Rota-Welsh
conjecture where graphs are generalized to (not necessarily representable)
matroids and the chromatic polynomial of a graph is replaced by the
characteristic polynomial of a matroid. The Huh and Huh-Katz techniques are not
applicable in this level of generality, since there is no underlying algebraic
geometry to which to relate the problem. But in 2015 Adiprasito, Huh, and Katz
announced a proof of the Rota-Welsh conjecture based on a novel approach
motivated by but not making use of any results from algebraic geometry. The
authors first prove that the Rota-Welsh conjecture would follow from
combinatorial analogues of the Hard Lefschetz Theorem and Hodge-Riemann
relations in algebraic geometry. They then implement an elaborate inductive
procedure to prove the combinatorial Hard Lefschetz Theorem and Hodge-Riemann
relations using purely combinatorial arguments.
We will survey these developments.Comment: 22 pages. This is an expository paper to accompany my lecture at the
2017 AMS Current Events Bulletin. v2: Numerous minor correction
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