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Nonparametric Bayesian Inference for Stochastic Processes with Piecewise Constant Priors
Belomestny D., van der Meulen F., Spreij P.
We present a survey of some of our recent results on Bayesian nonparametric inference for a multitude of stochastic processes. The common feature is that the prior distribution in the cases considered is on suitable sets of piecewise constant or piecewise linear functions, that differ for the specific situations at hand. Posterior consistency and in most cases contraction rates for the estimators are presented. Numerical studies on simulated and real data accompany the theoretical results.
Pachner’s Theorem
Friedl S., Hannes J.
Pachner’s Theorem is a purely combinatorial theorem which plays an important role in low-dimensional topology. We give a topology-free self-contained proof of Pachner’s Theorem and we outline why Pachner’s Theorem is such an essential tool in low-dimensional topology.
A Large Class of Conjecturally Stable Chromatic Symmetric Functions
Matherne J.P.
The theory of stable and Lorentzian polynomials has recently found a number of successes in a variety of research areas including combinatorics, engineering, and computer science; in particular, they have played a key role in solving long-standing open problems such as the Kadison–Singer problem and Mason’s log-concavity conjecture. More recently, the classes of stable polynomials and Lorentzian polynomials have appeared in representation theory, algebraic combinatorics, and even knot theory. We further highlight their ubiquity by introducing a large class of chromatic symmetric functions related to Hessenberg varieties and the Stanley–Stembridge conjecture that are conjecturally Lorentzian and stable.
Compactifying Torus Fibrations Over Integral Affine Manifolds with Singularities
Ruddat H., Zharkov I.
This is an announcement of the following construction: given an integral affine manifold B with singularities, we build a topological space X which is a torus fibration over B. The main new feature of the fibration X → B is that it has the discriminant in codimension 2.
Connected Sum Decompositions of High-Dimensional Manifolds
Bokor I., Crowley D., Friedl S., Hebestreit F., Kasprowski D., Land M., Nicholson J.
The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fails in higher dimensions.
Renormalisation for Inflation Tilings I:General Theory
G¨ahler F.
Inflation tilings are generated by iterating an inflation procedure r, which first expands a (partial) tiling linearly by a factor l, and then divides each expanded tile (called supertile) according to a fixed rule into a set of original tiles.
Topologically flat embedded 2-spheres in specific simply connected 4-manifolds
Kasprowski D., Lambert-Cole P., Land M., Lecuona A.G.
Aperiodic Order Meets Number Theory: Origin and Structure of the Field
Baake M., Coons M., Grimm U., Roberts J.A., Yassawi R.
Aperiodic order is a relatively young area of mathematics with connections to many other fields, including discrete geometry, harmonic analysis, dynamical systems, algebra, combinatorics and, above all, number theory. In fact, numbertheoretic methods and results are present in practically all of these connections. It was one aim of this workshop to review, strengthen and foster these connections.
The octonionic projective plane
Lackmann M.
As mathematicians found out in the last century, there are only four normed division algebras1 over ℝ: the real numbers themselves, the complex numbers, the quaternions and the octonions. Whereas the real and complex numbers are very well-known and most of their properties carry over to the quaternions (apart from the fact that these are not commutative), the octonions are very different and harder to handle since they are not even associative. However, they can be used for several interesting topological constructions, often paralleling constructions known for ℝ, ℂ or ℍ.
The Status of the QSSA Approximation in Stochastic Simulations of Reaction Networks
Bobadilla A.V., Bartmanski B.J., Grima R., Othmer H.G.
Stochastic models of chemical reactions are needed in many contexts in which the copy numbers of species are low, but only the simplest models can be treated analytically. However, direct simulation of computational models for systems with many components can be very time-consuming, and approximate methods are frequently used. One method that has been used in systems with multiple time scales is to approximate the fast dynamics, and in this note we study one such approach, in which the deterministic QSSH is used for the fast variables and the result used in the rate equations for the slow variables. We examine the classical Michaelis-Menten kinetics using this approach to determine when it is applicable.
On the Power of Restricted Monte Carlo Algorithms
Heinrich S.
We introduce a general notion of restricted Monte Carlo algorithms that generalizes previous notions in two ways: it includes full adaptivity and general (i.e. not only bit) restrictions. We show that for each such restricted setting there is a computational problem that can be solved in the general randomized setting but not under the restriction.
The Radiative Transport Equation with Heterogeneous Cross-Sections
Blake J.C., Graham I.G., Scheben F., Spence A.
We consider the classical integral equation reformulation of the radiative transport equation (RTE) in a heterogeneous medium, assuming isotropic scattering. We prove an estimate for the norm of the integral operator in this formulation which is explicit in the (variable) coefficients of the problem (also known as the cross-sections). This result uses only elementary properties of the transport operator and some classical functional analysis. As a corollary, we obtain a bound on the convergence rate of source iteration (a classical stationary iterative method for solving the RTE). We also obtain an estimate for the solution of the RTE which is explicit in its dependence on the cross-sections. The latter can be used to estimate the solution in certain Bochner norms when the cross-sections are random fields. Finally we use our results to give an elementary proof that the generalised eigenvalue problem arising in nuclear reactor safety has only real and positive eigenvalues.
Bounding the Spectral Gap for an Elliptic Eigenvalue Problem with Uniformly Bounded Stochastic Coefficients
Gilbert A.D., Graham I.G., Scheichl R., Sloan I.H.
A key quantity that occurs in the error analysis of several numerical methods for eigenvalue problems is the distance between the eigenvalue of interest and the next nearest eigenvalue. When we are interested in the smallest or fundamental eigenvalue, we call this the spectral or fundamental gap. In a recent manuscript [Gilbert et al.,
https://arxiv.org/abs/1808.02639
], the current authors, together with Frances Kuo, studied an elliptic eigenvalue problem with homogeneous Dirichlet boundary conditions, and with coefficients that depend on an infinite number of uniformly distributed stochastic parameters. In this setting, the eigen-values, and in turn the eigenvalue gap, also depend on the stochastic parameters. Hence, for a robust error analysis one needs to be able to bound the gap over all possible realisations of the parameters, and because the gap depends on infinitely- many random parameters, this is not trivial. This short note presents, in a simplified setting, an important result that was shown in the paper above. Namely, that, under certain decay assumptions on the coefficient, the spectral gap of such a random elliptic eigenvalue problem can be bounded away from 0, uniformly over the entire infinite-dimensional parameter space.
New Preasymptotic Estimates for Approximation of Periodic Sobolev Functions
Kühn T.
Approximation of Sobolev embeddings is a well-studied subject in high-dimensional approximation, with many application to different branches of mathematics. E.g., for isotropic Sobolev spaces $$ H^{s} \left( {{\mathbb{T}}^{d} } \right) $$ of fractional smoothness s > 0 on the d-dimensional torus it is known that the approximation numbers an of the embedding $$ H^{s} \left( {{\mathbb{T}}^{d} } \right) \hookrightarrow L_{2} \left( {{\mathbb{T}}^{d} } \right) $$ behave like $$ a_{n} \sim n^{{{{ - s} \mathord{\left/ {\vphantom {{ - s} d}} \right. \kern-0pt} d}}} $$ as $$ n \to \infty $$, where the (weak) equivalence $$ \sim $$ holds only up to multiplicative constants which are not known explicitly. However, for practical purposes it is more relevant to know the preasymptotic behaviour of the an, i.e. for small n, say $$ n \le 2^{d} $$. In this range the dependence on n is only logarithmic. The main results in this note are sharp two-sided preasymptotic estimates for approximation of isotropic Sobolev functions on $$ {\mathbb{T}}^{d} $$. In particular we give explicit constants, which show the exact dependence on the dimension d, the smoothness s, and further parameters of the norm. This improves the known results in the literature. Moreover, we prove a new preasymptotic estimate for approximation of Sobolev functions of dominating mixed smoothness.
A Set Optimization Technique for Domain Reconstruction from Single-Measurement Electrical Impedance Tomography Data
Harrach B., Rieger J.
We propose and test a numerical method for the computation of the convex source support from single-measurement electrical impedance tomography data. Our technique is based on the observation that the convex source support is the unique minimum of an optimization problem in the space of all convex and compact subsets of the imaged body.
CY-Operators and L-Functions
Straten D.V.
This a write up of a talk given at the MATRIX conference at Creswick in 2017 (to be precise, on Friday, January 20, 2017). It reports on work in progress with P. Candelas and X. de la Ossa. The aim of that work is to determine, under certain conditions, the local Euler factors of the L-functions of the fibres of a family of varieties without recourse to the equations of the varieties in question, but solely from the associated Picard–Fuchs equation.
Unravelling the Dodecahedral Spaces
Spreer J., Tillmann S.
The hyperbolic dodecahedral space of Weber and Seifert has a natural non-positively curved cubulation obtained by subdividing the dodecahedron into cubes. We show that the hyperbolic dodecahedral space has a 6-sheeted irregular cover with the property that the canonical hypersurfaces made up of the mid-cubes give a very short hierarchy. Moreover, we describe a 60-sheeted cover in which the associated cubulation is special. We also describe the natural cubulation and covers of the spherical dodecahedral space (aka Poincaré homology sphere).
Spectral Triples on O N
Goffeng M., Mesland B.
We give a construction of an odd spectral triple on the Cuntz algebra O
N
, whose K-homology class generates the odd K-homology group K
1(O
N
). Using a metric measure space structure on the Cuntz-Renault groupoid, we introduce a singular integral operator which is the formal analogue of the logarithm of the Laplacian on a Riemannian manifold. Assembling this operator with the infinitesimal generator of the gauge action on O
N
yields a θ-summable spectral triple whose phase is finitely summable. The relation to previous constructions of Fredholm modules and spectral triples on O
N
is discussed.
Application of Semifinite Index Theory to Weak Topological Phases
Bourne C., Schulz-Baldes H.
Recent work by Prodan and the second author showed that weak invariants of topological insulators can be described using Kasparov’s KK-theory. In this note, a complementary description using semifinite index theory is given. This provides an alternative proof of the index formulae for weak complex topological phases using the semifinite local index formula. Real invariants and the bulk-boundary correspondence are also briefly considered.
Endomorphisms of Lie Groups over Local Fields
Glöckner H.
Lie groups over totally disconnected local fields furnish prime examples of totally disconnected, locally compact groups. We discuss the scale, tidy subgroups and further subgroups (like contraction subgroups) for analytic endomorphisms of such groups.
