09:20 to 09:40 Registration 09:40 to 09:50 Welcome from David Abrahams (Isaac Newton Institute) 09:50 to 10:40 Akram Aldroubi (Vanderbilt University); (Vanderbilt University)Dynamical sampling and frames generated from powers of exponential operators In this talk, I will give a brief review of the problem of frame generation from operator powers of exponentials acting on a set of vectors. I will discuss its relation to dynamical sampling, review some of the previous results and present several new ones. INI 1 10:40 to 11:10 Morning Coffee 11:10 to 12:00 Denka Kutzarova (University of Illinois at Urbana-Champaign); (Bulgarian Academy of Sciences)Transportation cost spaces on finite metric spaces Transportation cost spaces are studied by several groups of researchers, for different reasons and under different names. The term Lipschitz-free spaces is commonly used in Banach space theory. We prove that the transportation cost space on any finite metric space contains a large well-complemented subspace which is close to $\ell_1^n$. We show that transportation cost spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs. In the particular case of diamond graphs we prove that their cycle space is spanned by even levels of Haar functions. It is curious that the subspaces generated by all the even/odd levels of the Haar functions also appear in the study of quasi-greedy basic sequences in $L_1[0,1]$. This research is joint with Stephen Dilworth and Mikhail Ostrovskii. INI 1 12:00 to 13:30 Lunch at Churchill College 13:30 to 14:20 Albert Cohen (Université Pierre et Marie Curie Paris)Optimal sampling for approximation on general domains We consider the approximation of an arbirary function in any dimension from point samples. Approximants are picked from given or adaptively chosen finite dimensional spaces. Various recent works reveal that optimal approximations can be constructed at minimal sampling budget by least-squares methods with particular sampling measures. In this talk, we discuss strategies to construct these measures and their samples in the adaptive context and in general non-tensor-product multivariate domains. INI 1 14:20 to 15:10 Peter Binev (University of South Carolina)High Dimensional Approximation via Sparse Occupancy Trees Adaptive domain decomposition is often used in finite elements methods for solving partial differential equations in low space dimensions. The adaptive decisions are usually described by a tree. Assuming that can find the (approximate) error for approximating a function on each element of the partition, we have shown that a particular coarse-to-fine method provides a near-best approximation. This result can be extended to approximating point clouds any space dimension provided that we have relevant information about the errors and can organize properly the data. Of course, this is subject to the curse of dimensionality and nothing can be done in the general case. In case the intrinsic dimensionality of the data is much smaller than the space dimension, one can define algorithms that defy the curse. This is usually done by assuming that the data domain is close to a low dimensional manifold and first approximating this manifold and then the function defined by it. A few years ago, together with Philipp Lamby, Wolfgang Dahmen, and Ron DeVore, we proposed a direct method (without specifically identifying any low dimensional set) that we called "sparse occupancy trees". The method defines a piecewise constant or linear approximation on general simplicial partitions. This talk considers an extension of this method to find a similar approximation on conforming simplicial partitions following an idea from a recent result together with Francesca Fierro and Andreas Veeser about near-best approximation on conforming triangulations. INI 1 15:10 to 15:40 Afternoon Tea 15:40 to 16:30 Claire Boyer (Sorbonne Université); (ENS - Paris)Representer theorems and convex optimization We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and elements of the extreme rays of the regularizer level sets. As a side result, we characterize the minimizers of the total gradient variation. As an ongoing work, we will also study the geometry of the total gradient variation ball. This is a joint work with Antonin Chambolle, Yohann De Castro, Vincent Duval, Frédéric de Gournay, and Pierre Weiss. INI 1 16:30 to 17:30 Welcome Wine Reception at INI
 09:00 to 09:50 Aicke Hinrichs (Johannes Kepler Universität)How good is random information compared to optimal information? We study approximation and integration problems and compare the quality of optimal information with the quality of random information. For some problems random information is almost optimal and for other problems random information is much worse than optimal information. We give a survey about known and new results. Parts of the talk are based on joint work with D. Krieg, E. Novak, J. Prochno and M. Ullrich. INI 1 09:50 to 10:40 Karlheinz Groechenig (University of Vienna)Totally positive functions in sampling theory and time-frequency analysis Totally positive functions play an important role in approximation theory and statistics. In this talk I will present recent new applications of totally positive functions (TPFs) in sampling theory and time-frequency analysis.   (i) We study the sampling problem for shift-invariant spaces generated by a TPF. These spaces arise the span of the integer shifts of a TPF and are often used as a substitute for bandlimited functions.   We give a complete characterization of sampling sets for a shift-invariant space with a TPF generator of Gaussian type in the style of Beurling.   (ii) A related problem is the question of Gabor frames, i.e., the spanning properties of time-frequency shifts of a given function. It is conjectured that the lattice shifts of a TPF generate a frame, if and only if the density of the lattice  exceeds 1. At this time this conjecture has been proved  for two important subclasses of TPFs. For  rational lattices it is true for arbitrary TPFs.  So far, TPFs seem to be the only window functions for which the fine structure of the associated Gabor  frames is tractable.   (iii) Yet another question in time-frequency analysis is the existence of zeros of the Wigner distribution (or the radar ambiguity function). So far all examples of zero-free ambiguity functions are related to TPFs, e.g., the ambiguity function of the Gaussian is zero free. INI 1 10:40 to 11:10 Morning Coffee 11:10 to 12:00 Feng Dai (University of Alberta)Integral norm discretization and related problems In this talk, we will discuss the problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure. We study the problem for elements of finite dimensional spaces in a general setting, paying a special attention to the case of the multivariate trigonometric polynomials with frequencies from a finite set with fixed cardinality. Both new results and a survey of known results will be presented. This is a joint work with A. Prymak, V.N. Temlyakov and S. Tikhonov. INI 1 12:00 to 13:30 Lunch at Churchill College 13:30 to 14:20 Milana Gataric (University of Cambridge)Imaging through optical fibres In this talk, I'll present some recent results on reconstruction of optical vector-fields using measurements acquired by an optical fibre characterised by a non-unitary integral transform with an unknown spatially-variant kernel. A new imaging framework will be introduced, which through regularisation is able to recover an optical vector-field with respect to an arbitrary representation system potentially different from the one used for fibre calibration. In particular, this enables the recovery of an optical vector-field with respect to a Fourier basis, which is shown to yield indicative features of increased scattering associated with tissue abnormalities. The effectiveness of this framework is demonstrated using biological tissue samples in an experimental setting where measurements are acquired by a fibre endoscope, and it is observed that indeed the recovered Fourier coefficients are useful in distinguishing healthy tissues from lesions in early stages of cancer. If time permits, I'll also briefly present a new method that enables recovery of non-unitary fibre transmission matrices necessary for minimally invasive optical imaging in inaccessible areas of the body. INI 1 14:20 to 15:10 Geno Nikolov (Sofia University St. Kliment Ohridski)Markov-type inequalities and extreme zeros of orthogonal polynomials The talk is centered around the problem of finding (obtaining  tight two-sided bounds for)  the sharp constants in certain Markov-Bernstein type inequalities in weighted $L_2$ norms. It turns out that, under certain assumptions, this problem is equivalent to the estimation of the extreme zeros of orthogonal polynomials with respect to a measure supported on $R_{+}$. It will be shown how classical tools like the Euler-Rayleigh method and Gershgorin circle theorem produce surprisingly good bounds for the extreme zeros of the Jacobi, Gegenbauer and Laguerre polynomials. The sharp constants in the $L_2$  Markov inequalities with the Laguerre and Gegenbauer weight functions and in a discrete $\ell_2$ Markov-Bernstein inequality are investigated using the same tool. INI 1