Talk Titles and Abstracts
Zin Arai (Hokkaido University)
Rigorous verification of uniform hyperbolicity, subshifts of
finite type and the pruning front
In this talk, we propose a rigorous computational algorithm for
proving uniform hyperbolicity of dynamical systems. We avoid the
difficulty of constructing hyperbolic splittings by introducing
quasi-hyperbolicity, which is computationally much more tractable
than uniform hyperbolicity.
We also discuss how to describe the dynamics of hyperbolic invariant
sets. That is, we give some algorithms to rigorously construct
a conjugacy to subshifts of finite type. Using this algorithm,
we compute the exact value of the topological entropy of the real
Henon map for many regions of parameter values.
Furthermore, applications of the hyperbolicity algorithm include
rigorous computation of the monodromy action of the complex Henon
map, which turns out to be closely related to the pruning front
theory, a generalization of the kneading theory to higher dimensional
systems.
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Dror Bar-Natan (Toronto)
Dessert:
Hilbert's 13th Problem, in Full Colour
To end a week of deep thinking with a nice colourful light dessert,
we will present the Kolmogorov-Arnol'd solution of Hilbert's 13th
problem with lots of computer-generated rainbow-painted 3D pictures.
In short, Hilbert asked if a certain specific function of three
variables can be written as a multiple (yet finite) composition
of continuous functions of just two variables. Kolmogorov and
Arnol'd showed him silly (ok, it took about 60 years, so it was
a bit tricky) by showing that ANY continuous function f of any
finite number of variables is a finite composition of continuous
functions of a single variable and several instances of the binary
function "+" (addition). For f(x,y)=xy, this may be
xy=exp(log x + log y). For f(x,y,z)=x^y/z, this may be exp(exp(log
y + log log x) + (-log z)). What might it be for (say) the real
part of the Riemann zeta function?
The only original material in this talk will be the pictures;
the math was known since around 1957.
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Carlos Beltran (Universidad de Cantabria)
A difficult minimization problem: the distribution of points
in the sphere
Given N points in the 2-dimensional sphere, the logarithmic potential
is the sum of the logs of the inverses of the Euclidean distances
between the points. The 7th problem in Smale's list of problems
for the XXI Century asks, how to distribute N points in the 2-dimensional
sphere in such a way that the logarithmic potential is minimized
- or almost minimized? In the talk I will present an overview
of the history and progress of this problem, and present some
recent results which relate it to the distribution of the condition
number for polynomial solving. Home-made experiments with surprising
conclussions will also be presented. There may be more questions
than answers in this talk.
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Ilia Binder (Toronto)
Walk on Sphere algorithm and boundary geometry
Walk on Spheres algorithm is a standard algorithm for solving
Dirichlet problem. We discuss the relation of the rate of convergence
of this algorithm and certain fine geometric properties of the
domain boundary. This is a joint work with Mark Braverman (Microsoft
Research/University of Toronto)
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Mark Braverman (Microsoft Research)
Computability and Complexity of Julia Sets
In this talk we will present a survey on the computational properties
of quadratic Julia sets.
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Nathan Dunfield (Illinois)
Practical solutions to hard problems in 3-dimensional topology.
Many fundamental problems in 3-dimensional topology are known
to be algorithmically solvable, using tools ranging from normal
surface theory to hyperbolic geometry. However, most of these
algorithms are too slow to be of use in any interesting example,
and in fact some of these problems are known to be NP-hard or
worse. I will discuss other methods, based on various heurisitics,
randomization, and hyperbolic geometry, which can effectively
solve some of these problems in practice. Particular questions
of interest will be whether a 3-manifold fibers over the circle,
its Heegaard genus, and the rank of its fundamental group. I will
conclude with a brief demonstration of SnapPy, a modern Python
interface to Jeff Weeks SnapPea.
This is joint work with (separately) Dinakar Ramakrishnan, Helen
Wong, and Marc Culler.
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Herbert Edelsbrunner (Duke University)
Measuring with algebra
Persistent homology is an algebraic method for measuring the
scale of a homology class in a filtration. Taking the sublevel
sets of a real-valued function, we get a measure on the features
that are born and die as we increase the threshold. This bridge
between algebra and measure theory is solidified by results about
the stability of the diagram we use to express persistence. Using
the bottleneck distance for diagrams, we get stability for general,
tame functions. Using the Wasserstein distance, we get stability
for Lipschitz functions on a large class of spaces. Generalizing
the theory
from filtrations to zigzag modules, we extend the differential
notion of transversality to a measure and get stability for intersections.
These stability results have plenty of ramifications inside and
outside mathematics.
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Joel Hass (University of California, Davis)
Unknot diagrams requiring a quadratic number of Reidemeister
moves to untangle
It is usually more difficult to obtain lower bounds on the computational
complexity of a problem than to obtain upper bounds. Several years
ago, in work with
Lagarias, we obtained exponential upper bounds on the number of
Reidemeister moves required to pass from an n crossing diagram
of the unknot to the trivial diagram.
In recent work with Tahl Nowik, we found a sequence of unknot
diagrams for which the minimum number of Reidemeister moves required
to pass to the trivial diagram is quadratic with respect to the
number of crossings. The result is proved by introducing a new
invariant of knot diagrams.
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Pat Hooper (Northwestern University)
Billiards in nearly isosceles triangles
It is a wide open question if polygons always have periodic billiard
paths. This question is unknown even for triangles. In this talk,
I will explain why nearly isosceles triangles have periodic billiard
paths.
While not proved using a computer, this result is ``computer
inspired.'' I will demonstrate McBilliards, a program used to
investigate periodic billiard paths in triangles. I will use this
program to illustrate the theorem alluded to above, and to guide
the audience through the main ideas in the proof.
Both the program McBilliards and the mathematics mentioned above
are joint work with R. Schwartz.
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Svetlana Katok (Pennsylvania State University)
Reduction theory and coding of geodesics on the modular surface.
In this talk I will describe two "computer inspired"
results related to coding of geodesics on the modular surface.
The first result deals with one-dimensional maps related to a
family of (a,b)-continued fractions. Numerical experiments led
us to formulate and eventually prove sufficient condition for
validity of the Reduction theory conjecture, proposed by Don Zagier,
that states that the associated natural extension maps have attractors
with finite rectangular structure where every point of the plane
is mapped after finitely many iterations. I will show that the
structure of these attractors can be ``computed" from the
data (a,b), and give a dynamical interpretation of the ``reduction
theory" that underlines these constructions.
The second result answers a question when arithmetic coding via
"minus" continued fractions (a=-1,b=0) coincides with
the geometric coding that consists of recording the successive
sides of the standard fundamental region cut by the geodesic.
This question is related to complexity of the geometric coding
and identification of a maximal 1- step topological Markov chain
of admissible geometric codes. Some of these results are joint
with Ilie Ugarcovici.
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Stefano Luzzatto (International Centre for
Theoretical Physics (ICTP))
Deciding the undecidable: Probabilistic arguments versus finite
resolution dynamics
Many families of dynamical systems exhibit so many bifurcations
that it is essentially impossible to determine the precise dynamical
properties corresponding to a specific parameter. In this talk
we will discuss two possible approaches to this issue. One is
to prove that certain dynamical phenomena occur for a positive
Lebesgue measure set of parameters, thus giving in some sense
a "probability" that the chosen parameter satisfies
the required conditions. This approach has been applied since
1981 to some families of one-dimensional and strongly dissipative
higher dimensional maps (though explicit quantitative estimates
have been obtained only in 2006 - Luzzatto & Takahasi, Nonlinearity),
but at the moment we are not even close to obtaining such results
for more general systems.
In this talk we will focus on an alternative approach suggested
recently by Luzzatto and Pilarczyk which applies to much more
general dissipative systems. Rather than obtaining a result about
the probability that a certain system satisfies certain conditions
we develop some notions which allow us to prove rigorous results
about a specific parameter value at the cost of describing the
dynamics of the systems only at a finite resolution. As an example
of the application of these ideas we prove rigorously by computer-assisted
calculations that the Henon map for Henon's classical parameter
values is "topologically mixing at all resolutions coarser
than 10^-5".
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Igor Pak, UCLA
Acute triangulations of polytopes
A simplex in R^d is called acute if all its dihedral angles are
< pi/2. An acute triangulation of a polytope is a triangulation
(as a CW-complex) into acute simplices. The problem of finding
acute triangulations is of interest in discrete geometry, the
finite element method, and other applications. In this talk I
will survey a number of known and recent results on acute triangulations.
In particular, we describe a complete solution for d-dim
cubes, which involve delicate computations (in 3-dim) and a topological
(negative) argument in higher dimensions. We conclude with
some open problems. Part of the talk is based on the recent paper
joint with Eryk Kopczynski and Piotr Przytycki.
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Roland Roeder (SUNY Stony Brook)
Computing arithmetic invariants for hyperbolic reflection groups
I will demonstrate a collection of computer scripts written in
PARI/GP to compute the commensurability invariants known as the
invariant trace field and invariant quaternion algebra for reflection
groups determined by compact polyhedra in H^3. These scripts also
allow one to determine arithmeticity of such groups and the isomorphism
class of the invariant quaternion algebra by analyzing its ramification.
(They are similar to the program SNAP that does the same thing
for hyperbolic manifolds of finite volume.)
I present many computed examples of these invariants. This is
enough to show that most of the groups that we consider are pairwise
incommensurable. For pairs of groups with identical invariants,
not all is lost: when both groups are arithmetic, having identical
invariants guarantees commensurability. We discover some "unexpected"
commensurable pairs this way.
I will conclude by describing a notion of mutation for polyhedra
which enables one to construct many (often non-arithmetic) pairs
of polyhedra having the same invariant trace field and quaternion
algebras. For these mutant pairs of polyhedra it is typically
an open question whether the reflection groups are commensurable.
This is joint work with Omar Antolin-Camarena and Gregg Maloney.
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Saul Schleimer (University of Warwick)
Twister: building triangulations of surface bundles
I will explain the main ideas behind Twister, a program written
by Tracy Hall and myself, that produces triangulations based on
Dehn twist data. We have used the program to produce a census
of bundles in genus two. Twister is also useful for producing
counterexamples. At the end of the talk I will demo the program.
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Richard Schwartz (Brown University)
Polygonal Outer Billiards
Outer billiards is a dynamical system consisting of a discrete
sequence of moves done on the outside of a convex shape in the
plane. I will demonstrate some graphical user interfaces I wrote
which help explore the geometry and dymamics of outer billiards
when it is defined relative to a polygon. Along the way, I'll
sketch my solution of the Moser-Neumann problem, which asks if
one can have unbounded orbits in an outer billiards system. The
main result is that outer billiards relative to a kite has unbounded
orbits if and only if the kite is irrational.
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Carles Simó (Universidad de Barcelona)
The role of Dynamical Systems in Celestial Mechanics. Applications
to Astronomy and Astrodynamics
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Sergei Tabachnikov (Penn State University)
Pentagrama Myrificum, Old wine into new wineskins
The Pentagram map is a projectively natural iteration on plane
polygons introduced by R. Schwartz 17 years ago. Computer experiments
show that the Pentagram map exhibits quasi-periodic behavior.
Based on joint work with R. Schwartz and V. Ovsienko, I shall
describe the Pentagram map as a completely integrable system.
A by-product of this research is a collections of new configuration
theorems of classical projective geometry, discovered in computer
experiments. A number of open problems will be formulated.
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Morwen Thistlethwaite (Tennessee)
The exact computation of some representation varieties
A method is described for the exact computation of representation
varieties of groups having small presentations, for example triangle
groups or fundamental groups of hyperbolic 3-manifolds with small
volume. We are particularly interested in deforming the canonical
SO(n,1)-representation of the group of a hyperbolic n-manifold
or orbifold into SL(n+1,R) or into SU(n,1), this last group bringing
us to the intriguing world of complex hyperbolic geometry.
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Dylan Thurston (Barnard College, Columbia)
Rigidity of Graphs
When do the lengths of the edges of a straight-edge framework
determine the positions of the vertices? The problem comes up
all the time in applications ranging molecular biology to sensor
networks to computer vision. But it also turns out that the problem
is NP-hard in general. It becomes easier if you require the initial
position to be generic; then there is a polynomial algorithm based
on the \emph{stress matrix} of the graph. But even in this case,
actually reconstructing the positions from the edge-length is
difficult. There is a good algorithm in case the framework is
\emph{universally} rigid: the edge lengths determine the framework
independently of the embedding dimension. There is again a characterization
of such frameworks in terms of the stress matrix.
This is joint work with Steven Gortler and Alex Healy.
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Warwick Tucker (Uppsala)
A rigorous lower bound for the stability regions of the quadratic
map
We establish a lower bound on the measure of the set of stable
parameters a for the quadratic map $Q_a(x) = ax(1?x2)$. For these
parameters, we prove that $Q_a$
either has a single stable periodic orbit or a period-doubling
bifurcation. From this result, we also obtain a non-trivial upper
bound on the set of stochastic parameters for $Q_a$.
This is joint work with Daniel Wilczak.
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Shmuel Weinberger (Chicago)
Persistent homology of data, loop spaces, and landscapes.
Persistent homology is a useful tool for dealing with "errors",
whether noise in data, or geometrical perturbations to topological
invariants. In this talk, I will explain some formalism and recast
some old theorems in these terms, giving a new proof or so, explain
the meaning of long and short "persistence intervals"
and speculative on some further directions.
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Daniel Wilczak (University of Bergen)
Rigorous numerics for homoclinic tangencies
We propose a method for computer-assisted verification of the
existence of homoclinic tangencies for maps. The method is geometric
in the spirit and assumes that the map is of class C^2 for which
we can compute enclosures for values and derivatives up to the
second order. Having a rigorous solver for variational equations
one can apply the method to Poincare maps.
The method has been successfully applied to the Henon map and
the forced-damped pendulum equation.
The main numerical result states that a suitable Poincare map
for the
forced-damped pendulum
equation
x'' + bx' + sin(x) = cos(t)
admits a homoclinic tangency unfolding generically for some parameter
value beta close to 0.2471.
This is a joint work with Piotr Zgliczynski
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Yosef Yomdin (The Weizmann Institute of
Science)
I. Center-Focus Problem for Abel equation, Moment vanishing,
Compositions, and Mathieu and Zhao conjectures
Center-Focus problem for Abel equation (1) y'=p(x)y^2+q(x)y^3
on [a,b] is to give conditions on p and q for all the solutions
of (1) to satisfy y(a)=y(b). This problem turns out to be closely
related with the vanishing problem for the moments of the form
\int_a^b P^k(x)Q(x) dP(x), with P=\int p, Q=\int q. Recently F.
Pakovich and M. Muzychuk completely solved the Moments vanishing
problem for p,q - polynomials. The answer is given in terms of
certain composition relations between P and Q. For Laurent polynomials
situation is more complicated. In a very recent work F. Pakovich
has achieved a serious progress in understanding vanishing conditions
for rational functions and, in particular, for Laurent polynomials.
In particular, new relations with the Mathieu conjecture in representations
of compact Lie groups have appeared, and (through the recent work
of Wenhua Zhao) to certain questions closely related to the Jacobian
conjecture.
Moment vanishing problem is closely related to the "moment
inversion problem": reconstruct the functions P and Q from
their known moments. The corresponding question for Abel equation
(1) provides a generalization of the Center-Focus problem. It
also appears naturally in Signal Processing. We plan to present
this part in a separate talk at this workshop.
Yosef Yomdin (The Weizmann Institute of Science)
II. "Algebraic" reconstruction of Signals and Images
from Fourier Data
We discuss the problem of reconstruction of signals with singularities
from integral measurements (like Fourier coefficients or moments).
The "Algebraic Reconstruction" assumes rather detailed
a priori information on the form of the signals: we need their
parametric models. But assuming such models are known, Algebraic
Reconstruction promises a very high resolution, significantly
better than with the "Compressed Sensing" approach.
The mathematics involved strongly resembles "the moment inversion
problem" considered in the first talk at this workshop, in
connection with the Center-Focus problem for Abel differential
equation. We give an overview of some essential mathematical ingredients
in Algebraic Reconstruction approach.
However, in order to apply Algebraic Reconstruction method to
images, we need their "parametric representation". Is
it possible at all? This is a well known and very difficult problem
(sometimes called "vectorization of photo-realistic images"):
Develop a representation scheme for images, based on "geometric
models", and satisfying the following two requirements:
- The representation preserves a full visual quality of high
resolution photo-realistic images.
- The total data volume in the representation is significantly
lower than in the original pixel image ("compression").
Here "geometric models" are assumed to be analytic
parametric expressions, capturing faithfully a local visual content
of the image. In order to satisfy the "compression"
requirement their scale must be of order at least of a few pixels.
This problem is well known to be extremely difficult. On the other
hand, even its partial solution could provide strong tools in
a wide variety of imaging problems, in particular, in image analysis
and processing. We plan to discuss the "state of the art"
in the geometric image modelization.
Finally, we plan to shortly present a problem of capturing image
objects with high-level models. As an application, we shall discuss
a "Photo-Animator" tool allowing one to easily produce
complicated video-clips from a single photo.
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Piotr Zgliczynski (Jagiellonian University)
Heteroclinic connections between fixed points for Kuramoto-Sivashinki
PDE, a computer assisted proof
We will discuss the method of self-consistent bounds for dissipitaive
PDEs. This methods allows for a direct application of tools from
dynamical system theory (finite dimensional) to dissipative PDE.
This includes both: abstract theorems and rigours algorithms for
integration of PDEs. As an example we will discuss a computer
assisted proof of the existence of some heteroclinic connections
between fixed points for Kuramoto-Sivashinski PDE on the line
with odd and periodic boundary conditions.
The proof consists of the following stages:
1) the proof of the existence of two fixed points, "the source"
and "the target"
2) rigorous estimates for the attracting region around the target
point
3) rigorous estimates for one dimensional unstable manifold of
the source point
4) rigorous integration of PDE - the propagation of the unstable
manifold of the source until it enters the basin of attraction
of the target point.
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