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# Mathematics Seminars 2017 - 2018

## Mathematics Seminars 2017 - 2018

**Ralf Peeters (Maastricht University, 16 Jan)**

Department of Data Science and Knowledge Engineering, Maastricht University, The Netherlands

(this is joint work with my colleague Joel Karel)

Title: Data-driven design of orthogonal wavelets with compact support and vanishing moments

Abstract: We present a framework to design an orthogonal wavelet with compact support and vanishing moments, matched to a given application in a data-driven way. This is achieved by optimizing a criterion, such that a prototype signal, which is characteristic for the application and must be supplied by the user, becomes sparse in the wavelet domain. Such a sparsity approach is claimed to have benefits for compression and detection purposes.

Starting from a filter bank approach with lossless polyphase matrices, a parameterization is developed for which wavelet orthogonality and compact support are built in, and in terms of which we can express the vanishing moment conditions relatively conveniently. For filters of order 2n-1, the orthogonality and compact support requirements leave n degrees of freedom. Instead of using all this freedom to obtain as many vanishing moments as possible (which would lead to Daubechies wavelets), the idea is to impose p vanishing moments, where 0<p<n, still leaving n-p degrees of freedom for matching the wavelet to a prototype signal by optimization. For low values of p, these vanishing moment conditions can in fact be built into the parameterization; we show the topological structure this induces. For high values of p (and for high orders of the filters) this becomes hard; we then may have to resort to constrained optimization. The approach is developed for critically sampled wavelet transforms as well as for the stationary wavelet transform. Examples are given to illustrate the wavelets generated by these methods.

Reference:

Joël Karel, Ralf Peeters, (2017), Orthogonal Matched Wavelets with Vanishing Moments: A Sparsity Design Approach, *Circuits, Systems & Signal Processing (CSSP)*, published online Nov. 2017. https://doi.org/10.1007/s00034-017-0716-1.

**Steve Wills (University College Cork, 23 Jan)**

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Title: Construction of quantum stochastic cocycles

Abstract: Quantum stochastic cocycles generalise (semi)groups of automorphisms of operator algebras in two ways - the automorphism condition is weakened to a positivity preservation condition, and the time evolution law of a semigroup is suitably modified. These objects can be used to describe the evolution of an open quantum system interacting with its environment.

In this talk I will survey known results that show the correspondence between cocycles and solutions of the Evans-Hudson quantum stochastic differential equation for bounded generators. I will then outline the challenges of constructing such cocycles when the generator is unbounded, giving some techniques that can be used in a variety of settings to solve the QSDE/construct the cocycle.

**David Wraith (Maynooth University, 30 Jan)**

Title: When is a product not a product?

Abstract: The answer to the question in the title is: when the product is a Hatcher bundle. The first part of this talk will explore the construction and curious properties of Hatcher bundles. In the second part we will discuss the space of Ricci positive metrics on odd dimensional spheres, and indicate how Hatcher bundles can be used to uncover subtle aspects of the topology of this space. This is joint work with Boris Botvinnik and Mark Walsh.

**Andreea Nicoara (Trinity College Dublin, 5 Feb)**

Title: The Global Theory of Denjoy-Carleman Functions: Nullstellensatz Results and Beyond

Abstract: Denjoy-Carleman classes are intermediate classes between real-analytic and smooth defined by imposing certain bounds on the growth of their derivatives. Classically studied by analysts, they have more recently attracted the attention of model theorists and algebraic geometers due to their rather unique properties. I will describe recent progress on global Nullstellensatz results for the Denjoy-Carleman classes as well as challenges to figuring out their global algebraic geometric behaviour.

**Clifford Nolan (University of Limerick, 13 Feb)**

Title: Microlocal Analysis in Radar and Seismic Imaging

Abstract: In this talk we consider data acquisition geometries which are applicable to both radar and seismic imaging. What seems like a cosmetic change in how one acquires (or synthetically pre-processes) the data can have a dramatic effect on the image one obtains when performing standard back-projection, or almost any other imaging method that one cares to consider. Microlocal analysis is the perfect tool to analyse the (back-projected) image and identify whether or not one should expect artefacts to appear. We will use this approach to analyse the images obtained using some standard acquisition geometries used in radar and seismic data processing. No prior knowledge of microlocal analysis will be assumed.

**Paolo Guasoni (Dublin City University, 20 Feb)**

Title: Asset Prices in Segmented and Integrated Markets

Abstract: Agents with equal preferences live in two regions that yield two respective dividend streams, cointegrated with each other, but with uncorrelated fluctuations. We find equilibrium asset prices and welfare both in segmentation, when each region holds its own asset and consumes its dividend, and in integration, when both regions trade both assets and consume both dividends. Integration always increases welfare. Asset prices may increase or decrease, depending on the time of integration, but decrease on average. Correlation in assets' returns is negligible before integration, but significantly positive afterwards, partially explaining financialization effects.

**Madalin Guta (University of Nottingham, 27 Feb)**

Title: An introduction to quantum statistics

Abstract: Statistical inference plays an increasingly important role in quantum information and technology. In this talk I will give an overview of some of tools and research topics in “quantum statistics” centred around the problem of quantum state tomography and system identification. In particular I will show how key statistical concepts such as Fisher information and local asymptotic normality can be extended to the quantum domain, and how thresholding and compressed sensing techniques can be used for estimating high dimensional, low rank quantum states.

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**James Wright (University of Edinburgh, 6 Mar)**

Title: A Sharp L2 Fourier Restriction Theorem

Abstract: We give a variant of the classical Stein-Tomas argument proving an L2 restriction

estimate. Interestingly the result is sharp for a class of homogeneous varieties of any dimension.

**Hiroaki Aikawa (Hokkaido University, 20 Mar)**

Title: Global integrability of superharmonic functions and supertemperatures

Abstract: Ever since Armitage showed, in 1972, that a nonnegative superharmonic function on a bounded smooth domain in $R^n$ is $L^p$-integrable up to the boundary, provided $0<p<n/(n-1)$, the global integrability of nonnegative supersolutions has attracted many mathematicians.

In this talk we consider a parabolic counterpart. We study the global integrability of nonnegative supertemperatures on the cylinder $D\times(0,T)$. We show that the integrability depends on the lower estimate of the Green function for the Dirichlet Laplacian on $D$.

In particular, if $D$ is a bounded $C^1$-domain, then every nonnegative supertemperature on $D\times(0,T)$ is $L^p$-integrable over $D\times(0,T')$ for any $0<T'<T$, provided $0<p<(n+2)/(n+1)$. The bound $(n+2)/(n+1)$ is sharp.

Joint work with Hara and Hirata.

**Dan Petersen (Stockholm University, 10 Apr)**

Title: Moduli spaces and operads

Abstract: The moduli space of Riemann surfaces parametrizes all complex structures on a two-dimensional surface. After compactifying the moduli space one finds an interesting structure corresponding to how surfaces can be glued together. Specifically, these gluings give rise to something called an operad, and this leads to an interesting interplay between algebra and geometry.

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**Neil O’Connell (University College Dublin, 17 Apr)**

Title: Birational RSK correspondence and Whittaker functions

Abstract: The Robinson-Schensted-Knuth (RSK) correspondence is a combinatorial bijection which plays an important role in the theory of Young tableaux and symmetric functions, particularly in understanding combinatorial aspects of Schur polynomials (Cauchy-Littlewood identity, Littlewood-Richardson rule, etc.). I will give some background on this and then explain how a birational version of the RSK correspondence provides a similar `combinatorial’ framework for the study of GL(n,R)-Whittaker functions. These functions arise in the context of automorphic forms associated with GL(n,R), and reduce to the classical Whittaker functions in the case n=2.

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**Eoin** **Ó ****Colgáin (Asia Pacific Center for Theoretical Physics, 24 Apr)**

Title: Classical Yang-Baxter Equation from Gravity

Abstract: Solving the equations of motion of a gravitational system, one expects to encounter differential equations. We present a theory where given a solution, there exists a deformation where the equations of motion are equivalent to the Classical Yang-Baxter Equation, a hallmark of integrability (exact solvability) across many subjects in physics. I will introduce the theory and present some examples and time permitting explain initial attempts at a proof.

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