April 16 - 20, 2007
I will describe some results giving a single exponential upper bound on
the number of possible homotopy types of the fibres of a Pfaffian map, in terms
of the format of its graph.
In particular,
we show that if a semi-algebraic set S ⊂ ℝ^{m+n}
is defined by a Boolean formula with s polynomials of degrees less than d, and
π: (R)^{m+n} →
(R)^{n} is the projection
on a subspace, then the number of different homotopy types of fibres of
π does not exceed (2^{m} snd)^{O(nm)}.
All previously known bounds were doubly exponential.
As applications of our main results we prove single exponential bounds on the
number of homotopy types of semi-algebraic sets defined by
polynomials having a fixed number of monomials in their support
(we need to fix only the number of monomials, not the support set
itself), as well as by polynomials with bounded additive complexity.
We also prove single exponential
upper bounds on the radii of balls guaranteeing local contractibility
for semi-algebraic sets defined by polynomials with integer coefficients.
(Joint work with N. Vorobjov).
My goal in this talk is to advertise an algorithm
found by
James Van Buskirk, the first improvement in more than thirty
years in
the exact complexity of the discrete Fourier transform over the
reals.
The previous speed record was held by the split-radix FFT,
announced
by Yavne in 1968 and widely understood since the early 1980s.
The
split-radix FFT uses 4nlg n-6n+8 operations over the reals
for a
size-n complex DFT when n is a large power of 2, and
therefore
(12+o(1))nlg n operations for a complex cyclic convolution
of
length n. Bruun's real-factor FFT also uses (12+o(1))nlg n
operations. An analysis by Johnson and Frigo shows that Van
Buskirk's
new algorithm uses only (34/3+o(1))nlg n operations.
This is a joint work of the presenter and Ruud Pellikaan (Technical University of Eindhoven, Netherlands). We
investigate a question of decoding linear codes, in particular up to half the minimum distance. We propose a method based on solving systems of polynomial equations over a finite field. Such a method was already quite successfully applied for cyclic codes earlier, we move on to the general case. We concentrate on solving the systems that emerge with the use of Groebner bases. The poster reflects a theoretical framework, the main result (the system we want to solve has a unique solution, etc.), and also some results on complexity estimation and experimental results.
The running time of many iterative numerical algorithms is
dominated by the condition number of the input,
a quantity measuring the sensitivity of the solution with
regard to small perturbations of the input.
Examples are iterative methods of linear algebra,
interior-point methods of linear and convex optimization,
as well as homotopy methods for solving systems of polynomial
equations.
Spielman and Teng introduced in 2001 the seminal concept of
smoothed analysis, which
arguably blends the best of both worst-case and average-case
analysis of algorithms.
This led to a much better understanding of the success of the
simplex method in practice.
We present a general theorem providing smoothed analysis
estimates for conic condition numbers of problems
of numerical analysis. Our probability estimates depend only on
geometric invariants of the corresponding sets
of ill-posed inputs. Several applications to linear and
polynomial equation solving show that the estimates obtained
in this way are easy to derive and quite accurate. The main
theorem is based on a volume estimate of ε-tubular
neighborhoods around a real algebraic subvariety of a sphere,
intersected with a disk of radius σ. Besides
ε and σ, this bound depends only the dimension of the
sphere and on the degree of the defining equations.
This is joint work with Felipe Cucker and Martin Lotz.
List decoding Reed-Solomon codes via an interpolation and root-finding algorithm was pioneered by Madhu Sudan in the mid to late 1990's. In 2006, Kwankyu Lee and Michael O'Sullivan developed a Gröbner basis based implementation of the interpolation algorithm, which they showed was an efficient generalization of the Berlekamp-Massey Algorithm.
We propose to apply these ideas to list decoding BCH codes by
presenting BCH codes as subfield subcodes of generalized Reed-Solomon
codes and to explore the resulting algorithm. In particular, we seek ways to optimize the new algorithm and to compare it with the Berlekamp-Massey algorithm.
During the past decade, complex networks have become increasingly important in communication and information technology. Vast, self-engineered networks, like the Internet, the World Wide Web, and Instant Messaging Networks, have facilitated the flow of information, and served as media for social and economic interaction. In social networks, the ease of information flow goes by many names: the "small world" phenomenon, the "Kevin Bacon phenomenon," and "six degrees of separation"—the claim that any two people on earth can be connected through a chain of acquaintances with at most five intermediaries. Unfortunately, many of the properties that facilitate information transmission also facilitate the spread of viruses in both technological and social networks. Dr. Chayes uses simple mathematical models to explain these epidemics and to examine strategies for their containment.
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The well-known Shamir secret sharing scheme uses polynomial
interpolation to recover a shared secret. The scheme and its
application to secure computation generalizes to algebraic
curve based schemes (Chen-Cramer 2006). For secure computation
against an active adversary a scheme needs to be strongly
multiplicative. We show that this can be achieved by using
what we call strongly self-orthogonal codes.
(Joint work with various authors)
Convolutional codes can be described by linear input-state-output systems. This gives rise to a state transition graph and an associated weight adjacency matrix. The latter counts in a detailed way the weights occurring in the code. After discussing some uniqueness issues we will present a MacWilliams identity theorem for convolutional codes and their duals in terms of the weight adjacency matrix. Furthermore, we will discuss isometries for convolutional codes and their effect on the weight adjacency matrix. It will be shown that for a particular class of codes the weight adjacency matrix forms a complete invariant under monomial equivalence, that is, under permutation and rescaling of the codeword coordinates.
The fundamental trade-off in coding theory is the one
between the rate of the code (a measure of amount of redundancy
introduced) and the amount of errors that can be corrected. In this
talk, I will describe an explicit construction of codes that achieves
the optimal trade-off between these parameters, for a worst-case noise model where the channel can corrrupt the codeword arbitrarily subject to a bound on the total number of errors.
Formally, for every 0 < R < 1 and eps > 0, we present an explicit
construction of codes of rate R (which encode R.n symbols into n
symbols) that can be list decoded in polynomial time from up to a
fraction (1-R-eps) of errors. Clearly, one needs at least Rn correct
symbols in the received word to have any hope of identifying the Rn
message symbols, so recovering from (R+eps)n correct symbols is
information-theoretically best possible. Our codes are simple to
describe: they are certain *folded* Reed-Solomon codes, which are just
Reed-Solomon codes, but viewed as a code over a larger alphabet by a
careful bundling of codeword symbols.
The talk will introduce the problem of list decoding, and
then give a peek into the algebraic ideas and constructions that lead
to the above result. Time permitting, I will also describe some challenging questions concerning algebraic-geometric codes that, if resolved, could improve the decoding complexity as one approaches the optimal trade-off.
Based on joint work with Atri Rudra (UW).
Let K =(K_{1…}K_{n}) be a n-tuple of
convex compact subsets in the Euclidean space
R^{n}, and let V (·) be the Euclidean volume in
R^{n} . The Minkowski polynomial
V_{K} is defined as V_{K}(x_{1},
…, x_{n})= V (λ_{1}K_{1}
+…+λ_{n}K_{n}) and the mixed volume
V(K_{1,…,}K_{n}) as
V(K_{1…}K_{n}) =
∂^{n} / ∂λ_{1…}∂λ_{n}
V_{K}(λ_{1}K_{1}
+…λ_{n}K_{n}).
The mixed volume is one of the backbones of convexity theory.
After BKH theorem, the mixed volume( and its generalizations)
had become crucially important in computational algebraic geometry.
We present in this talk randomized and deterministic algorithms to
approximate the mixed volume of well-presented convex compact sets. Our main result is
a poly-time randomized algorithm which approximates
V (K_{1,…,} K_{n}) with multiplicative
error e^{n} and with better rates if the affine dimensions of most of the sets
K_{i} are small.
Because of the famous Barany-Furedi lower bound, even such rate is not achievable by a poly-time deterministic oracle algorithm.
Our approach is based on the particular, geometric programming,
convex relaxation of log(V (K_{1,…,}
K_{n})). We prove the mixed volume analogues of the Van der
Waerden and the Schrijver/Valiant conjectures on the permanent. These results, interesting
on their own, allow to "justify" the above mentioned convex relaxation,
which is solved using the ellipsoid method and a randomized poly-time time
algorithm for the approximation of the volume of a convex set.
wo main categories of problems are studied in algebraic complexity theory:
evaluation problems and decision problems. A typical example of an evaluation
problem is the evaluation of the permanent of a matrix. Such problems can be
studied within Valiant's framework.
Deciding whether a multivariate polynomial has a real root is a typical example
of a decision problem. This problem is NP-complete in the Blum-Shub-Smale model
of computation over the real numbers.
In this talk I will present a transfer theorem which shows that if certain
evaluation problems are easy, then other decision problems (including the
above-mentioned NP-complete problem) are easy too.
Therefore, to show that that P is different from NP over the set of real
numbers, one should first be able show that certain evaluation problems are
hard.
Hodge gave basis for the homogeneous co-ordinate rings of the Grassmannian
and its Schubert varieties in terms of "standard monomials" in the Plucker
co-ordinates. This clasical work of Hodge was extended to the generalized flag variety
and its Schubert varieties by Lakshmibai-Littelmann-Musili-Seshadri. We shall give
a survey talk on this generalization and will also relate many important algebraic
varieties to Schubert varieties.
I'll give geometric formulations
and interpretations of some of the main results regarding the
complexity of matrix multiplication. I'll also explain the
relevance of this geometry for areas such as
algebraic statistics and quantum computing.
I will present new deterministic and probabilistic
recombination
algorithms to compute the irreducible factorization of
bivariate
polynomials via the classical Hensel lifting technique, and for
the
dense representation model. In bi-degree (d_{x},
d_{y}) with d_{y} ≤
d_{x}, and whatever the characteristic of the base field is,
these algorithms only require the precision of the lifting to be
d_{x}+1. The cost of the deterministic recombination algorithm
is sub-quadratic in d_{x} d_{y}, and the probabilistic version is
faster by
a factor of d_{y}. At the end, I will explain how these
algorithms can
be adapted to the computation of the absolute factorization,
and how
they extend to more than 2 variables.
Soon after Lee and O'Sullivan proposed a new interpolation algorithm for
algebraic soft-decision decoding of Reed-Solomon codes, there have been some
attempts to apply a coordinate transformation technique to the new
algorithm, with a remarkable complexity reducing effect. We propose a
conceptually simple way of applying the transformation technique to the
interpolation algorithm.
Systems of polynomial equations over the complex numbers can be used
to characterize NP-Complete graph-theoretic decision problems. From
the point of view of computer algebra and symbolic computation, these
are interesting polynomial systems because they are provably hard:
solving them is as hard as solving the underlying NP-Complete problem.
Furthermore, unless NP = coNP, there must exist infinite instances of
these infeasible systems whose Hilbert Nullstellensatz certificates
grow with respect to the underlying graphs.
Joint work with Martin Ziegler.
The word problem for discrete groups is well-known to be undecidable
by a Turing Machine; more precisely, it is reducible both to and from
and thus equivalent to the discrete Halting Problem.
The present work introduces and studies a real extension of
the word problem for a certain class of groups which are presented
as quotient groups of a free group and a normal subgroup.
The main result establishes the word problem for such groups to be
computationally equivalent to the Halting Problem for BSS machines
over the reals.
It thus provides the first non-trivial example of a concrete
problem that is computationally universal for the BSS model.
In this talk I will discuss several conceptual aspects leading a a probabilistic positive solution to the following problem proposed by S. Smale: "Can a zero of n complex polynomial equations in n unknowns be found approximately on the average, in polynomial time with a uniform algorithm?"
Just as the number of real roots of a real univariate quadratic
depends on the sign of the discriminant, the topological behavior of real
zero sets depends on (more general) A-discriminant variety complements.
More recently, in numerical linear algebra (and nonlinear work of Shub, Smale,
Beltran, Pardo, and other authors), the relationship between the numerical
behavior of zero sets and distance to the discriminant variety has been
clarified.
In this talk, we review some of the connections between
A-discriminants, the topology of real algebraic sets, and the
complexity of solving polynomial systems. In particular, we show
that outside a ball of sufficiently large radius (in the coefficient space),
one can assert the following with high probability:
(1) a new upper bound on the number of real roots of a fewnomial
system, significantly improving Khovanski's famous result
(2) the truth of a formerly broken conjecture of Itenberg and
Roy
Our main results are joint work in progress with Martin Avendano.
We also discuss a connection to a generalization of Smale's 17th Problem.
No background in algebraic geometry is assumed.
We discuss an extension of Descartes' Rule (for univariate
polynomials over the real numbers) to arbitrary k x n polynomial
systems over a field L which is either the real numbers or the
p-adic rationals. Over R, our extension fails for certain systems,
but we can characterize precise regions (in the coefficient space)
where our extension holds. Over the p-adic rationals, our extension
holds over even larger regions in the coefficient space. We then
conclude with a conjecture on what optimal fewnomial estimates should
like in general.
We use the study of bilinear forms over a finite field to give a decomposition of the linear codes similar to the one in [D. Ruano: On the structure of generalized toric codes, arXiv:cs.IT/0611010, 2006] for generalized toric codes. Such decomposition, called geometric decomposition of a linear code and which can be obtained in a constructive way, allows to express easily the dual of a linear code and gives a new paradigm for linear codes. The proofs for characteristic 2 are different, but they will be developed parallel.
Two orthonormal bases of a finite dimensional Hilbert space are
mutually unbiased if for every pair of vectors one from each basis,
the modulus of the standard inner product is the same.
The problem of determining bounds on the maximum number of
such bases (MUBs) for a fixed dimension is an important open problem
of 20 years that has been shown to arise in different guises in many
areas of mathematics.
The most common application of MUBs is found in quantum cryptography
where MUBs are the quantum states used in many protocols.
Here we consider 2 questions and provide preliminary results
and conjectures:
- When is a given set of MUBs non-extendible? I.e.,
characterize mutually unbiased sets
M1,M2,..,Mk of for which there is no
basis M unbiased to M1,..,Mk?
- Characterize families of bases where non-extendibility
of a set of MUBs imply maximality, and are there natural
families of bases with this property. I.e, within such a family of
bases, a greedy method for constructing MUBs would guarantee
a maximal cardinality MUB collection.
This is joint work with Oscar Boykin and Mohamad Tarifi
at the University of Florida.
We will outline a possible approach to outstanding computational
complexity theory (CCT) questions. The approach relies on a faithful
mapping of these questions to questions in geometric invariant (GIT)
theory and representation theory. We begin with a construction
of Valiant and show its connection to the Orbit Closure and
Membership problems in GIT. Proofs of non-existence of algorithms
translate to the existence of obstructions. This immediately leads
to the important notion of stability in invariant theory. For stable
points, we show that obstructions are easily constructed.
We then outline proofs of the stability of forms such as the
determinant and the permanent, which are important in CCT.
We next define our notion of partially stable points. We pose our
problems as (i) to understand the orbit closure for stable points
with distinctive stabilizers, and (ii) extension of these results
to partially stable points. We finish with an outline of our current
attempts at these teo, and also relate it to the wrok of other
researchers.
This is joint work with Ketan Mulmuley.
In 1980, Askold Khovanskii established his fewnomial bound
for the number of real solutions to a system of polynomials,
showing that the complexity of the set of real solutions to
a system of polynomials depends upon the number of monomials
and not on the degree. This fundamental finiteness result
in real algebraic geometry is believed to be unrealistically large.
I will Describe joint work with Frederic Bihan on a new
fewnomial bound which is substantially lower than Khovanskii's
bound and asymptotically optimal. This bound is obtained by
first reducing a given system to a Gale system, and then
bounding the number of solutions to a Gale system. Like
Khovanskii's bound, this bound is the product of an exponential
function and a polynomial in the dimension, with the exponents
in both terms depending upon the number of monomials. In our
bound, the exponents are smaller than in Khovanskii's.
We discuss the efficient computation of the zeta function of a
hypersurface X defined over a finite field k. The zeta function
Z(X,T) of X is the exponential generating series for the number of
points on X defined over k and its finite extensions; Z(X,T) is a
rational function in T. Zeta functions arise naturally in coding
theory, arithmetic geometry, complexity theory, and many other areas.
We consider the situation when X is affine, toric, or projective, when
its defining equation f is nondegenerate (a condition which implies
that X is smooth and which holds outside of a codimension 1 subset),
and when f is sparse (consisting of few monomials). In this
situation, one can use Dwork cohomology to give a very efficient
method of computing Z(X,T). We exhibit this method and provide
several examples.
The term "pseudocodeword" is used in relationship to decoding
linear block codes in at least three different ways. In linear
programming decoding (introduced by Feldman), any vertex of the
fundamental polytope is called a pseudocodeword. In a rigorous
formulation (as first proposed by Wiberg) of min-sum or sum-product
decoding, any valid configuration of any computation tree is called a
pseudocodeword. And, using a more intuitive interpretation of iterative
message-passing decoding, any valid configuration on any finite cover of
the Tanner graph corresponds to a pseudocodeword (this is made precise
through Graph Cover Decoding, proposed by Vontobel and Koetter). Though
some justification of the triple-use of the term has been given, these
three notions of pseudocodeword are indeed distinct. In this talk, we
will describe the connections and disconnections involved.