Let $M$ be the Cantor space or an $n$-dimensional manifold with $C(M,M)$ the set of continuous self-maps of $M$. We analyse the behaviour of the generic $f$ in $C(M,M)$ in terms of attractors and some notions of chaos.
The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. We introduce the ASAP algorithm, for the graph realization problem in R^d, given a sparse and noisy set of distance measurements associated to the edges of a globally rigid graph. ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in R^3, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. The reflections and rotations are estimated using a recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. Furthermore, the algorithm successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose SP-ASAP, a faster version of ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that ASAP and SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms. Time permitting, we briefly discuss the analogy between the graph realization and the low-rank matrix completion problems, as well as an application of synchronization over Z_2 and its variations to bipartite multislice networks.
By recent work of Voevodsky and others, type theories are now considered as a candidate
for a homotopical foundations of mathematics. I will explain what are type theories using the language
of (essentially) algebraic theories. This shows that type theories are in the same "family" of algebraic
concepts such as groups and categories. I will also explain what is homotopic in (intensional) type theories.
Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the first talk.
Random geometric graphs have been well studied over the last 50 years or so. These are graphs that
are formed between points randomly allocated on a Euclidean space and any two of them are joined if
they are close enough. However, all this theory has been developed when the underlying space is
equipped with the Euclidean metric. But, what if the underlying space is curved?
The aim of this talk is to initiate the study of such random graphs and lead to the development of
their theory. Our focus will be on the case where the underlying space is a hyperbolic space. We
will discuss some typical structural features of these random graphs as well as some applications,
related to their potential as a model for networks that emerge in social life or in biological
sciences.
Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the
first talk.
Large-scale zonal jets are observed in a wide range of geophysical and astrophysical flows; most strikingly in the atmospheres of the Jovian gas giant planets. Jupiter's upper atmosphere is highly turbulent, with many small vortices, and strong westerly winds at the equator. We consider the thermal shallow water equations as a model for Jupiter's upper atmosphere. Originally proposed for the terrestrial atmosphere and tropical oceans, this model extends the conventional shallow water equations by allowing horizontal temperature variations with a modified Newtonian cooling for the temperature field. We perform numerical simulations that reproduce many of the key features of Jupiter’s upper atmosphere. However, the simulations take a long time to run because their time step is severely constrained by the inertia-gravity wave speed. We filter out the inertia-gravity waves by forming the quasigeostrophic limit, which describes the rapidly rotating (small Rossby number) regime. We also show that the quasigeostrophic energy equation is the quasigeostrophic limit of the thermal shallow water pseudo-energy equation, analogous to the derivation of the acoustic energy equation from gas dynamics. We perform numerical simulations of the quasigeostrophic equations, which again reproduce many of the key features of Jupiter’s upper atmosphere. We gain substantial performance increases by running these simulations on graphical processing units (GPUs).
We show how the reduction procedure for generalized Kahler
structures can be used to recover Hitchin's results about the
existence of a generalized Kahler structure on the moduli space of
instantons on bundle over a generalized Kahler manifold. In this setup
the proof follows closely the proof of the same claim for the Kahler
case and clarifies some of the stranger considerations from Hitchin's
proof.
We consider the stationary flow of Prandtl-Eyring fluids in two
dimensions. This model is a good approximation of perfect plasticity.
The corresponding potential is only slightly super linear. Thus, many
severe problems arise in the existence theory of weak solutions. These
problems are overcome by use of a divergence free Lipschitz
truncation. As a second application of this technique, we generalize
the concept of almost harmonic functions to the Stokes system.
The classical associahedra are cell complexes, in fact polytopes,
introduced by Stasheff to parametrize the multivariate operations
naturally occurring on loop spaces of connected spaces.
They form a topological operad $ Ass_\infty $ (which provides a resolution
of the operad $ Ass $ governing spaces-with-associative-multiplication)
and the complexes of cellular chains on the associahedra form a dg
operad governing $A_\infty$-algebras (that is, a resolution of the
operad governing associative algebras).
In classical applications it was not necessary to consider units for
multiplication, or it was assumed units were strict. The introduction
of non-strict units into the picture was considerably harder:
Fukaya-Ono-Oh-Ohta introduced homotopy units for $A_\infty$-algebras in
their work on Lagrangian intersection Floer theory, and equivalent
descriptions of the dg operad for homotopy unital $A_\infty$-algebras
have now been given, for example, by Lyubashenko and by Milles-Hirsch.
In this talk we present the "missing link": a cellular topological
operad $uAss_\infty$ of "unital associahedra", providing a resolution
for the operad governing topological monoids, such that the cellular
chains on $uAss_\infty$ is precisely the dg operad of
Fukaya-Ono-Oh-Ohta.
(joint work with Fernando Muro, arxiv:1110.1959, to appear Forum Math)
This is joint work with P. Caputo and D. Chafai. In this talk, we
will consider various probability distributions on the set of stochastic
matrices with n states and on the set of Laplacian/Kirchhoff
matrices on n states. They will arise naturally from the conductance model on
n states with i.i.d conductances. With the help of random matrix
theory, we will study the spectrum of these processes.
To study turbulence,B. Mandelbrot introduced a random fractal which is called
now Mandelbrot percolation or fractal percolation. The construction is as follows:
given an integer M _ 2 and a probability 0
The partition function of 3d N=2 superconformal theories on the
3-sphere can be computed exactly by localization methods. I will explain
some sublteties associated to that important result. As a by-product, this
analysis establishes the so-called F-maximization principle for N=2 SCFTs in
3d: the exact superconformal R-charge maximizes the 3-sphere free energy
F=-log Z.
A cactus product is much like a wedge product of pointed spaces, but instead of being uniquely defined there is a moduli space of possible cactus products. I will discuss how this space can be interpreted geometrically and how its combinatorics calculates the homology of the automorphism group of a free product with no free group factors. Then I will reinterpret the moduli space with Outer space in mind: the lobes of the cacti now behave like boundaries and our free products can now include free group factors.
What is a phase transition?
The first thing that comes to mind is boiling and freezing of water. The material clearly changes its behaviour without any chemical reaction. One way to arrive at a mathematical model is to associate different material behavior, ie., constitutive laws, to different phases. This is a continuum physics viewpoint, and when a law for the switching between phases is specified, we arrive at pde problems. The oldest paper on such a problem by Clapeyron and Lame is nearly 200 years old; it is basically on what has later been called the Stefan problem for the heat equation.
The law for switching is given e.g. by the melting temperature. This can be taken to be a phenomenological law or thermodynamically justified as an equilibrium condition.
The theory does not explain delayed switching (undercooling) and it does not give insight in structural differences between the phases.
To some extent the first can be explained with the help of a free energy associated with the interface between different phases. This was proposed by Gibbs, is relevant on small space scales, and leads to mean curvature equations for the interface – the so-called Gibbs Thompson condition.
The equations do not by themselves lead to a unique evolution. Indeed to close the resulting pde’s with a reasonable switching or nucleation law is an open problem.
Based on atomistic concepts, making use of surface energy in a purely phenomenological way, Becker and Döring developed a model for nucleation as a kinetic theory for size distributions of nuclei. The internal structure of each phase is still not considered in this ansatz.
An easier problem concerns solid-solid phase transitions. The theory is furthest developped in the context of equilibrium statistical mechanics on lattices, starting with the Ising model for ferromagnets. In this context phases correspond to (extremal) equilibrium Gibbs measures in infinite volume. Interfacial free energy appears as a finite volume correction to free energy.
The drawback is that the theory is still basically equilibrium and isothermal. There is no satisfactory theory of metastable states and of local kinetic energy in this framework.
The use of gain-loss ratio as a measure of attractiveness has been
introduced by Bernardo and Ledoit. In their well-known paper, they
show that gain-loss ratio restrictions have a dual representation in
terms of restricted pricing kernels.
In spite of its clear financial significance, gain-loss ratio has
been largely ignored in the mathematical finance literature, with few
exceptions (Cherny and Madan, Pinar). The main reason is intrinsic
lack of good mathematical properties. This paper aims to be a
rigorous study of gain-loss ratio and its dual representations
in a continuous-time market setting, placing it in the context of
risk measures and acceptability indexes. We also point out (and
correctly reformulate) an erroneous statement made by Bernardo and
Ledoit in their main result. This is joint work with M. Pinar.
*Please note that this is a joint seminar with the William Dunn School of Pathology and will take place in EPA Seminar Room which is located inside the Sir William Dunn School of Pathology and must be entered from the main entrance on South Parks Road. Link http://g.co/maps/8cbbx
'Pattern-of-life' is a current buzz-word in sensor systems. One aspect to this is the automatic estimation of traffic flow patterns, perhaps where existing road maps are not available. For example, a sensor might measure the position of a number of vehicles in 2D, with a finite time interval between each observation of the scene. It is desired to estimate the time-average spatial density, current density, sources and sinks etc. Are there practical methods to do this without tracking individual vehicles, given that there may also be false 'clutter' detections, the density of vehicles may be high, and each vehicle may not be detected in every timestep? And what if the traffic flow has periodicity, e.g. variations on the timescale of a day?
I will give an overview of the description of imaginaries in algebraically closed (and some other) valued fields, and then discuss the related issue for valued fields with analytic structure (in the sense of Lipshitz-Robinson, and Denef – van Den Dries). In particular, I will describe joint work with Haskell and Hrushovski showing that in characteristic 0, elimination of imaginaries in the `geometric sorts’ of ACVF no longer holds if restricted exponentiation is definable.