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## Über dieses Buch

MATRIX is Australia’s international and residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each 1-4 weeks in duration. This book is a scientific record of the ten programs held at MATRIX in 2019 and the two programs held in January 2020:

· Topology of Manifolds: Interactions Between High and Low Dimensions

· Australian-German Workshop on Differential Geometry in the Large

· Aperiodic Order meets Number Theory

· Ergodic Theory, Diophantine Approximation and Related Topics

· Influencing Public Health Policy with Data-informed Mathematical Models of Infectious Diseases

· International Workshop on Spatial Statistics

· Mathematics of Physiological Rhythms

· Conservation Laws, Interfaces and Mixing

· Structural Graph Theory Downunder

· Tropical Geometry and Mirror Symmetry

· Early Career Researchers Workshop on Geometric Analysis and PDEs

· Harmonic Analysis and Dispersive PDEs: Problems and Progress

The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on a topic related to the MATRIX program; the remaining contributions are predominantly lecture notes or short articles based on talks or activities at MATRIX.

## Inhaltsverzeichnis

### Connected Sum Decompositions of High-Dimensional Manifolds

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.

Imre Bokor, Diarmuid Crowley, Stefan Friedl, Fabian Hebestreit, Daniel Kasproswki, Markus Land, Johnny Nicholson

### The Levine-Tristram Signature: A Survey

The Levine-Tristram signature associates to each oriented link L in S3 a function σL: S1 → $${\mathbb{z}}$$ z . This invariant can be defined in a variety of ways, and its numerous applications include the study of unlinking numbers and link concordance. In this survey, we recall the three and four dimensional definitions of σL, list its main properties and applications, and give comprehensive references for the proofs of these statements.

Anthony Conway

### PD4-Complexes and 2-Dimensional Duality Groups

This paper is a synthesis and extension of three earlier papers on PD4-complexes X such that π = π1(X) has one end and c.d.π = 2. The basic notion is that of strongly minimal PD4-complex, one for which the equivariant intersection pairing λX on π2(X) is null. The first main result is that two PD4-complexes with the same strongly minimal model are homotopy equivalent if and only if their intersection pairings are isometric. If c.d.π ≤ 2 every such complex has a strongly minimal model, and the second half of the paper focuses largely on determining the minimal models. In particular, if π is a surface group or is a semidirect product $$F(r) \rtimes {\mathbb{Z}}$$ F ( r ) ⋊ Z then the homotopy type of X is determined by π, the Stiefel-Whitney classes and λX. Although we expect that the strategy in the surface group case should extend to all π such that c.d.π = 2 and π has one end, we do not yet have a unified proof that covers the known cases. We conclude with an application to 2-knots and a short list of questions for further research.

Jonathan A. Hillman

### Topologically flat embedded 2-spheres in specific simply connected 4-manifolds

In this note we study whether specific elements in the second homology of specific simply connected closed 4-manifolds can be represented by smooth or topologically flat embedded spheres.

Daniel Kasprowski, Peter Lambert-Cole, Markus Land, Ana G. Lecuona

### Trisections of 5-Manifolds

Gay and Kirby introduced the notion of a trisection of a smooth 4-manifold, which is a decomposition of the 4-manifold into three ekementary pieces. Rubinstein and Tillmann later extended this idea to construct multisections of piecewise-linear manifolds in all dimensions. Given a PL manifold Y of dimension n, this is a decomposition of Y into $$\left\lfloor {{\text{n}}/{2}} \right\rfloor + {1}$$ n / 2 + 1 PL submanifolds. We show that every smooth, oriented, compact 5-manifold admits a smooth trisection. Furthermore, given a smooth cobordism W between trisected 4-manifolds, there is a smooth trisection of W extending the trisections on its boundary.

Peter Lambert-Cole, Maggie Miller

### The octonionic projective plane

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.

Malte Lackmann

### Null-Homologous Twisting and the Algebraic Genus

The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman’s theorem that Alexander polynomial one knots are topologically slice. This paper develops null-homologous twisting operations as a tool for studying the algebraic genus and, consequently, for bounding the topological slice genus above. As applications we give new upper bounds on the algebraic genera of torus knots and satellite knots.

Duncan McCoy

### A Slicing Obstruction from the 10/8+4 Theorem

Using the 10/8+4 theorem of Hopkins, Lin, Shi, and Xu, we derive a smooth slicing obstruction for knots in the three-sphere using a spin 4-manifold whose boundary is 0–surgery on a knot. This improves upon the slicing obstruction bound by Vafaee and Donald that relies on Furuta’s 10/8 theorem. We give an example where our obstruction is able to detect the smooth non-sliceness of a knot by using a spin 4-manifold for which the Donald-Vafaee slice obstruction fails.

Linh Truong

### A Generalised Multidimensional Jarnìk-Besicovitch Theorem

In this short note we prove a general multidimensional Jarnìk-Besicovitch theorem which gives the Hausdorff dimension of simultaneously approximable set of points with error of approximations dependent on continuous functions in all dimensions. Consequently, the Hausdorff dimension of the set varies along continuous functions. This resolves a problem posed by Barral-Seuret (2011).

Mumtaz Hussain

### Model Structures and Structural Identifiability: What? Why? How?

We may attempt to encapsulate what we know about a physical system by a model structure, S. This collection of related models is defined by parametric relationships between system features; say observables (outputs), unobservable variables (states), and applied inputs. Each parameter vector in some parameter space is associated with a completely specified model in S. Before choosing a model in S to predict system behaviour, we must estimate its parameters from system observations. Inconveniently, multiple models (associated with distinct parameter estimates) may approximate data equally well. Yet, if these equally valid alternatives produce dissimilar predictions of unobserved quantities, then we cannot confidently make predictions. Thus, our study may not yield any useful result. We may anticipate the non-uniqueness of parameter estimates ahead of data collection by testing S for structural global identifiability (SGI). Here we will provide an overview of the importance of SGI, some essential theory and distinctions, and demonstrate these in testing some examples.

Jason M. Whyte

### Spatial Modelling of Linear Regression Coefficients for Gauge Measurements Against Satellite Estimates

Satellite imagery provides estimates for the amount of precipitation that has occurred in a region, these estimates are then used in models for predicting future precipitation trends. As these satellite images only provide an estimate for the amount of precipitation that has occurred, it is important that they be accurate estimates. If we assume that a rain gauge correctly measures the amount of precipitation that has occurred in some location over a specified time interval, then we can compare the satellite precipitation estimate to the gauge measurement for the same time interval. By expressing the relationship between the gauge measurement and the satellite precipitation estimate for the same time interval as a linear equation we can then spatially map the coefficients of this linear relationship to inspect the spatial trends of the regression coefficients. We then model the coefficients of the linear equations of each location by a spatial linear model and then use this model to predict the coefficients in location where there are no rain gauges available.

Benjamin Hines, Yuriy Kuleshov, Guoqi Qian

### The New Frontier of Network Physiology: Emerging Physiologic States in Health and Disease from Integrated Organ Network Interactions

An intriguing question in the new field of Network Physiology is how organ systems in the human body dynamically interact to coordinate functions, to maintain healthy homeostasis, and to generate distinct physiological states and behaviors at the organism level. Physiological systems exhibit complex dynamics, operate at different time scales and are regulated by multi-component mechanisms, which poses challenges to studying physiologic coupling and network interactions among systems with diverse dynamics. We present a conceptual framework and a method based on the concept of time delay stability to probe transient physiologic network interactions in a group of healthy subjects during sleep. We investigate the multi-layer network structure and dynamics of interactions among (i) physiologically relevant brain rhythms within and across cortical locations, (ii) brain rhythms and key peripheral organ systems, and (iii) the network structure and dynamics among peripheral organ systems across distinct physiological states. We demonstrate that each physiologic state (sleep stage) is characterized by a specific network structure and link strength distribution. The entire physiological network undergoes hierarchical reorganization across layers with the transition from one stage to another. Our findings are consistent across subjects and indicate a robust association of organ network structure and dynamics with physiologic state and function. The presented Network Physiology approach provides a new framework to explore physiologic states under health and disease through networks of organ interactions.

Plamen Ch Ivanov, Jilin W. J. L. Wang, Xiyun Zhang, Bolun Chen

### Modelling Oscillating Living Systems: Cell Energy Metabolism as Weighted Networks of Nonautonomous Oscillators

Oscillations are a common feature throughout life, forming a key mechanism by which living systems can regulate their internal processes and exchange information. To understand the functions and behaviours of these processes, we must understand the nature of their oscillations. Studying oscillations can be difficult within existing physical models that simulate the changes in a system’s masses through autonomous differential equations. We discuss an alternative approach that focuses on the phases of oscillating processes and incorporates time as a key consideration. We will also consider the application of these theories to the cell energy metabolic system, and present a novel model using weighted nonautonomous Kuramoto oscillator networks in this context.

Joe Rowland Adams, Aneta Stefanovska

### A Time-Series Approach to Assess Physiological and Biomechanical Regulatory Mechanisms

In various areas of Medicine there is interest to incorporate information on homeostasis and regulation to increase the predictive power of prognostic scales. This has proven to be difficult in practice because of an uncomplete understanding of how regulation works dynamically and because a common methodology does not exist to quantify the quality of regulation independent from the specific mechanism. In the present contribution, it is shown that time series of regulated and effector variables from different regulatory mechanisms show universal features that may be used to assess the underlying regulation.

Ruben Fossion, Ana Leonor Rivera, Lesli Alvarez-Milláan, Lorena García-Iglesias, Octavio Lecona, Adriana Robles-Cabrera, Bruno Esta˜nol

### InterCriteria Analysis Approach as a Tool for Promising Decision Making in Physiological Rhythms

Recently developed InterCriteria Analysis (ICrA) approach has being intensively gained popularity as quite promising approach to support decision making process in biomedical informatics studies, and in particular—in physiological rhythms. ICrA has been elaborated to discern possible similarities in the behaviour of pairs of criteria when multiple objects are considered. The approach is based on the theories of intuitionistic fuzzy sets and index matrices. Up to now, ICrA has been successfully applied in economics, different industry fields, ecology, artifficial intelligence, e-learning, etc. ICrA has been demonstrated as promising tool also in studies related to medicine and bioinformatics, which are in the focus of this investigation.

Krassimir Atanassov, Tania Pencheva

### “Ome” Sweet “ome”: From the Genome to the Conductome

The last few decades have seen science both changed and confronted by the appearance of enormous quantities of data, that have arisen from the development of multiple new technologies. The impact of this “data revolution” has been particularly acute in the biological sciences, where bioinformatics has made great strides in integrating such data into new theoretical frameworks and adopting new computational tools. One framework that has prospered is that of the “ome”, which adopts a more holistic view of the physical structures that make up a cell, tissue or organism and their mutual interactions. The structures associated with the principal “omes”—genome, proteome, transcriptome and metabolome—are all microscopic, being associated with different biological molecules. Recently, however, the omic approach has been applied to more “mesoscopic” structures, such as organs and tissues, with the resulting totality of structures conforming the physiolome. However, all these omes are associated with particular spatial and temporal scales, and are therefore inadequate for addressing the real complexity of living systems, which are both multi-scale and highly multi-factorial with respect to those scales. We argue that a “disease-ome”, for example, as the totality of factors associated with a given disease, requires the integration all the current omes, and more. Thus, a holistic description of an important disease, such as obesity, requires all micro, meso and macro factors, as well as an understanding of both their upstream and downstream causal relations. This is particularly challenging when the relevant factors are distant in scale. Thus, the causality between overeating and obesity at the individual level is clear. However, the link between a certain genotype and obesity or the link between food production and obesity is much less clear. In spite of this, all of these factors can, in principle, be collected and included in a prediction model, using present technology and computational tools. We argue that the fundamental concept that most naturally links the micro, meso and macro is that of behaviour, as it is influenced by both micro (nature) and macro (nurture) factors and, in turn, influences them. We discuss the concept of the Conductome—the totality of factors that influence behaviour, using as an example food consumption and obesity, and emphasise its utility as an unifying concept that allows for a truly systemic view of a living organism.

Christopher R. Stephens

### Delay-Differential Equations for Glucose-Insulin Regulation

In this paper, a model based on a system of delay differential equations, describing a process of glucose-insulin regulation in the human body, is studied numerically. For simplicity, the system is based on a single delay due to the practical importance of one of the two delays appearing in more complex models. The stability of the system is investigated numerically. The regions, where the solutions demonstrate periodicity and asymptotic stability, are explicitly calculated. The sensitivity of the solutions to the parameters of the model, which describes the insulin production in the system, is analysed.

Maia Angelova, Sergiy Shelyag

### Effect of Adiabatic Index on Richtmyer-Meshkov Flows Induced by Strong Shocks

Richtmyer-Meshkov Instability (RMI) is an instability that develops at the interface between fluids of contrasting densities when impacted by a shock wave. Its applications include inertial confinement fusion, supernovae explosions, and the evolution of blast waves. We systematically study the effect of the adiabatic index of the fluids on the dynamics of strong-shock driven flows, particularly the amount of shock energy available for interfacial mixing. Only limited information is currently available about the dynamic properties of matter at these extreme regimes. Smooth Particle Hydrodynamics simulations are employed to ensure accurate shock capturing and interface tracking. A range of adiabatic indexes is considered, approaching limits which, to the best of the author’s knowledge, have never been considered before. We analyse the effect of the adiabatic indexes on the interface speed and growth-rate immediately after the shock passage. The simulation results are compared, wherever possible, with rigorous theories and with experiments, achieving good quantitative and qualitative agreement. We find that the more challenging cases for simulations arise where the adiabatic indexes are further apart, and that the initial growth rate is a non-monotone function of the initial perturbation amplitude, which holds across all adiabatic indexes of the fluids considered. The applications of these findings on experiment design are discussed.

Cameron E. Wright, Snezhana I. Abarzhi

### Compressibility Effect on Markstein Number for a Flame Front in Long-Wavelength Approximation

The effect of compressibility on the Markstein number for a planar front of a premixed flame is examined, at small Mach numbers, in the form of M2-expansions. The method of matched asymptotic expansions is used to analyze the solution in the preheat zone in a power series in two small parameters, the relative thickness of the preheat zone and the Mach number. We employ a specific form of perturbations, valid at long wavelengths, for the thermodynamic variables, which produces the correction term, to the Markstein number, of second order in the Mach number in drastically simple form. Our analysis accounts for the pressure variation as a source term in the heat-conduction equation and calls for the Navier–Stokes equation. The suppression effect of the front curvature on the Darrieus-Landau instability is enhanced by the viscous effect if Pr > 4/3, but is weakened if otherwise.

Keigo Wada, Yasuhide Fukumoto

### Computational fluid dynamics modelling of a transient solids concentration in a lagoon

Investigation of slurry flows is important for the mineral industry, biomass processing and waste processing. In the design of slurry handling systems such as channel flows, separators where solids concentrates are separated from clear liquid streams, knowledge of physics underlying slurry flows is required. In this study,slurry flows in tanks have been investigated. The transient profiles of the solids concentration along the length have been modelled using computational fluid dynamics (CFD). This investigation examines multiphase flows with settling solids in a non-Newtonian flow. The dynamical model gives guidance in determining formation accumulation of solids as a sludge blanket. In addition the clear liquid solids interface position has been determined this is needed for the recycle of the clear water for water conservation.

Ashfaq A. Khan, Yan Ding

### Regular and Singular Behaviours and New Morphologies in the Rayleigh Taylor Instability

The Rayleigh Taylor Instability is a fluid instability that develops when fluids of different densities are accelerated against their density gradient. Its applications include inertial confinement fusion, supernovae explosion, fossil fuel extraction and nano fabrication.We study Rayleigh Taylor instability developing at an interface with a spatially periodic perturbation under a time varying acceleration using group theoretic methods. For the first time, to our knowledge, both regular and singular nonlinear solutions are found, which correspond to the structure of bubbles and spikes emerging at the interface. We find that the dynamics of bubbles is regular, and the dynamics of spikes is singular in an asymptotic time-regime. The parameters affecting the behaviour of both bubble and spikes are discussed, including the inter-facial shear, which is shown to have a profound effect. The results set key theoretical benchmarks for future analysis.

Kurt Williams, Desmond L. Hill, Snezhana I. Abarzhi

### The extended Prandtl closure model applied to the two-dimensional turbulent classical far wake

Prandtl’s mixing length closure model has been used extensively in turbulent wake flows. Although the simplicity of this model is advantageous, it contains mathematical and physical limitations. In particular, this model results in a poor estimation of the flow on the center-line and near the wake boundary. Prandtl constructed an improved model, which will be referred to as the extended mixing length model, in an attempt to address many of the limitations of the original model. In this work, the extended Prandtl model is considered. A similarity solution that leaves both the governing equation for the stream-wise mean velocity deficit and the conserved quantity invariant is obtained. The governing partial differential equation is reduced to an ordinary differential equation. The ordinary differential equation,which must be solved subject to appropriate boundary conditions and the conserved quantity, cannot be solved analytically and thus a double-shooting method is developed to obtain the stream-wise mean velocity deficit. A plot of the mean velocity deficit is then produced.

Ashleigh J. Hutchinson

### Mixing, tunnelling and the direction of time in the context of Reichenbach’s principles

This work reviews the understanding of the direction of time introduced by Hans Reichenbach, including the fundamental relation of the perceived flow of time to the second law of thermodynamics (i.e. the Boltzmann time hypothesis), and the principle of parallelism of entropy increase. An example of a mixing process with quantum effects, which is advanced here in conjunction with Reichenbach’s ideas, indicates the existence of a physical mechanism that reflects global conditions prevailing in the universe and enacts the direction of time locally (i.e. the “time primer”). Generally, this mechanism, whose effects are often enacted by presuming antecedent causality, remains unknown at present. The possibility of experimental detection of the time primer is also discussed: if the time primer is CPT-invariant,its detection may be possible in high-energy experiments under the current level of technology.

Alexander Y. Klimenko

### Controlling stability of longwave oscillatory Marangoni patterns

We apply nonlinear feedback control to govern the stability of long-wave oscillatory Marangoni patterns. We focus on the patterns caused by instability in thin liquid film heated from below with a deformable free surface. This instability emerges in the case of substrate of low thermal conductivity, when two monotonic long-wave instabilities, Pearson’s and deformational, are coupled. We provide weakly nonlinear analysis within the amplitude equations, which govern the evolution of the layer thickness and the temperature deviation. The action of the nonlinear feedback control on the nonlinear interaction of two standing waves is investigated. It is shown that quadratic feedback control can produce additional stable structures (standing rolls and standing squares), which are subject to instability leading to traveling wave in the uncontrolled case.

Anna Samoilova, Alexander Nepomnyashchy

### Rigorous modelling of nonlocal interactions determines a macroscale advection-diffusion PDE

A slowly-varying or thin-layer multiscale assumption empowers macroscale understanding of many physical scenarios from dispersion in pipes and rivers, including beams, shells, and the modulation of nonlinear waves, to homogenisation of micro-structures. Here we begin a new exploration of the scenario where the given physics has non-local microscale interactions. We rigorously analyse the dynamics of a basic example of shear dispersion. Near each cross-section, the dynamics is expressed in the local moments of the microscale non-local effects. Centre manifold theory then supports the local modelling of the system’s dynamics with coupling to neighbouring cross-sections as a non-autonomous forcing. The union over all cross-sections then provides powerful new support for the existence and emergence of a macroscale model advection-diffusion PDE global in the large, finite-sized, domain. The approach quantifies the accuracy of macroscale advection-diffusion approximations, and has the potential to open previously intractable multiscale issues to new insights.

A. J. Roberts

### Influence of an oblique magnetic field on planar flame front instability

We investigate the effect of external magnetic field on the Darrieus-Landau instability (DLI), the linear instability of a planar premixed flame front,in an electrically conducting fluid. This setting has applicability to combustion phenomena of the astrophysical scale. Without magnetic field, the planar flame front is necessarily unstable. Previous investigation treated independently the normal and tangential magnetic fields. Here we focus on the case of their simultaneous application,namely, oblique magnetic field. Rederiving the jump conditions, across the flame front, of the physical variables based on the ideal magnetohydrodyamics equations,we correct the previous treatment of the Markstein effect and extend it to incorporate the disparity of the magnetic permeability. A genuinely oblique magnetic field has an unusual characteristics that discontinuity in tangential velocity across the flame is induced. It is found that the Kelvin-Helmholtz instability takes over the stabilizing effect on the DLI in a limited parameter regime when the normal Alfv´en speed exceeds the normal fluid velocity in both the unburned and burned regions.

Mako Sato, Yasuhide Fukumoto

### Numerical Study of Crystal Growth in Reaction-Diffusion Systems Using Front Tracking

We study the crystal growth in a Reaction-Diffusion System for the generic reaction A + B → C. Reactants A and B react to form the product C which then undergoes phase transition. We have used the Lagrangian Front Tracking to explicitly track the crystal surface. The evolution of the concentrations of A, B and C is described by a system of three partial differential equations. This system is solved using finite difference method. Main focus of the study is on observing the effects of different parameters on the crystal growth, namely the diffusion coefficients, homogeneous reaction constant, heterogeneous reaction constant and the equilibrium concentration.

Saurabh Joglekar, Xiaolin Li

### Numerical Study of Center of Reaction Front for Reaction-Diffusion System nA + mB → C with Arbitrary Diffusivities

We study the movement of the center of reaction front in the reaction diffusion system nA + mB → C for arbitrary diffusivities (Da ≠ Db). We present numerical evidence that xf (t) ∝ √t for all t ∈ (0, ∞). Numerical experiments are carried out for (n, m) = (1,1), (1,2), (2,1) and (2,2) and for various $$\frac{{D_{a} }}{{D_{b} }}$$ D a D b . Finite difference method is used. Empahsis is not on asymptotic behaviour or scaling, rather on verifying the stated claim for all t.

Saurabh Joglekar, Xiaolin Li

### Subdivided Claws and the Clique-Stable Set Separation Property

Let $${\mathscr{C}}$$ C be a class of graphs closed under taking induced subgraphs. We say that $${\mathscr{C}}$$ C has the clique-stable set separation property if there exists $$c \in {\mathbb{N}}$$ c ∈ N such that for every graph $$G \in {\mathscr{C}}$$ G ∈ C there is a collection $${\mathscr{P}}$$ P of partitions (X, Y) of the vertex set of G with | $${\mathscr{P}}$$ P | ≤ |V(G)|c and with the following property: if K is a clique of G, and S is a stable set of G, and K ∩ S = $$\emptyset$$ ∅ , then there is (X, Y) ∊ $${\mathscr{P}}$$ P with K ⊆ X and S ⊆ Y. In 1991 M. Yannakakis conjectured that the class of all graphs has the clique-stable set separation property, but this conjecture was disproved by M. Göös in 2014. Therefore it is now of interest to understand for which classes of graphs such a constant c exists. In this paper we define two infinite families $${\mathscr{S}}, {\mathscr{K}}$$ S , K of graphs and show that for every S ∊ $${\mathscr{S}}$$ S and K ∊ $${\mathscr{K}}$$ K , the class of graphs with no induced subgraph isomorphic to S or K has the clique-stable set separation property.

Maria Chudnovsky, Paul Seymour

### Notes on Tree- and Path-Chromatic Number

Tree-chromatic number is a chromatic version of treewidth, where the cost of a bag in a tree-decomposition is measured by its chromatic number rather than its size. Path-chromatic number is defined analogously. These parameters were introduced by Seymour [JCTB 2016]. In this paper, we survey all the known results on tree- and path-chromatic number and then present some new results and conjectures. In particular, we propose a version of Hadwiger’s Conjecture for tree chromatic number. As evidence that our conjecture may be more tractable than Hadwiger’s Conjecture, we give a short proof that every K5-minor-free graph has tree-chromatic number at most 4, which avoids the Four Colour Theorem. We also present some hardness results and conjectures for computing tree- and path chromatic number.

Tony Huynh, Bruce Reed, David R. Wood, Liana Yepremyan

### Note on Hedetniemi’s Conjecture and the Poljak-Rödl Function

Hedetniemi conjectured in 1966 that Hedetniemi conjectured in 1966 that $$\chi(G \times H) = \min\{\chi(G), \chi(H)\}$$ χ ( G × H ) = min { χ ( G ) , χ ( H ) } for any graphs G and H. Here $$G\times H$$ G × H is the graph with vertex set $$V(G)\times V(H)$$ V ( G ) × V ( H ) defined by putting $$(x,y)$$ ( x , y ) and $$(x^{\prime}, y^{\prime})$$ ( x ′ , y ′ ) adjacent if and only if $$xx^{\prime}\in E(G)$$ x x ′ ∈ E ( G ) and $$yy^{\prime}\in V(H)$$ y y ′ ∈ V ( H ) . This conjecture received a lot of attention in the past half century. It was disproved recently by Shitov. The Poljak-Rodl function is defined as $$f(n) = \min\{\chi(G \times H): \chi(G)=\chi(H)=n\}$$ f ( n ) = min { χ ( G × H ) : χ ( G ) = χ ( H ) = n } . Hedetniemi's conjecture is equivalent to saying $$f(n)=n$$ f ( n ) = n for every integer $$n$$ n . Shitov’s result shows that $$f(n)<n$$ f ( n ) < n when $$n$$ n is sufficiently large. Using Shitov’s result, Tardif and Zhu showed that $$f(n) \le n - (\log n)^{1/4-o(1)}$$ f ( n ) ≤ n - ( log n ) 1 / 4 - o ( 1 ) for sufficiently large $$n$$ n . Using Shitov’s method, He and Wigderson showed that for $$\epsilon \approx 10^{-9}$$ ϵ ≈ 10 - 9 and $$n$$ n sufficiently large, $$f(n) \le (1-\epsilon)n$$ f ( n ) ≤ ( 1 - ϵ ) n . In this note we observe that a slight modification of the proof in the paper of Zhu and Tardif shows that $$f(n) \le (\frac 12 + o(1))n$$ f ( n ) ≤ ( 1 2 + o ( 1 ) ) n for sufficiently large $$n$$ n . On the other hand, it is unknown whether $$f(n)$$ f ( n ) is bounded by a constant. However, we do know that if $$f(n)$$ f ( n ) is bounded by a constant, then the smallest such constant is at most 9. This note gives self-contained proofs of the above mentioned results.

Xuding Zhu

### Notes on Graph Product Structure Theory

It was recently proved that every planar graph is a subgraph of the strongproduct of a path and a graph with bounded treewidth. This paper surveys generalisationsof this result for graphs on surfaces, minor-closed classes, various nonminor-closed classes, and graph classes with polynomial growth. We then explorehow graph product structure might be applicable to more broadly defined graphclasses. In particular, we characterise when a graph class defined by a cartesian orstrong product has bounded or polynomial expansion. We then explore graph productstructure theorems for various geometrically defined graph classes, and presentseveral open problems.Zden

Zdeněk Dvořák, Tony Huynh, Gwenael Joret, Chun-Hung Liu, David R. Wood

### A Note on Bilinear Wave-Schrödinger Interactions

We consider bilinear restriction estimates for wave-Schrödinger interactions and provide a sharp condition to ensure that the product belongs to $$L_{q}^{t}$$ L q t $$L_{x}^{r}$$ L x r in the full bilinear range $$\frac{2}{q} + \frac{d + 1}{r} < d + 1,\,1 \leqslant q,r \leqslant 2$$ 2 q + d + 1 r < d + 1 , 1 ⩽ q , r ⩽ 2 . Moreover, we give a counterexample which shows that the bilinear restriction estimate can fail, even in the transverse setting. This failure is closely related to the lack of curvature of the cone. Finally we mention extensions of these estimates to adapted function spaces. In particular we give a general transference type principle for U2 type spaces that roughly implies that if an estimate holds for homogeneous solutions, then it also holds in U2. This transference argument can be used to obtain bilinear and multilinear estimates in U2 from the corresponding bounds for homogeneous solutions.

Timothy Candy

### A Note on the Scattering for 3D Quantum Zakharov System with Non-radial Data in L2

In this note, we give a remark on the scattering for quantum Zakharov system with non-radial small initial data in L2 with one order additional angular regularity using the generalized Strichartz estimate with wider range and the normal form transformation.

Chunyan Huang

### Algebraic and Symplectic Viewpoint on Compactifications of Two-Dimensional Cluster Varieties of Finite Type

In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be given by a grading on that ring, which can be described by positive polytopes [17]. In the examples we exploit, the cluster variety can be interpreted as the complement of certain divisors in del Pezzo surfaces. In the symplectic viewpoint, they can be described via almost toric fibrations over $$\mathcal{R}^{2}$$ R 2 (after completion). Once identifying them as almost toric manifolds, one can symplectically view them inside other del Pezzo surfaces. So we can identify other symplectic compactifications of the same cluster variety, which we expect should also correspond to different algebraic compactifications. Both viewpoints are presented here and several compactifications have their corresponding polytopes compared. The finiteness of the cluster mutations are explored to provide cycles in the graph describing monotone Lagrangian tori in del Pezzo surfaces connected via almost toric mutation [34].

Man-Wai Mandy Cheung, Renato Vianna

### Observations on Disks with Tropical Lagrangian Boundary

In this survey, we look at some expectations for Lagrangian submanifolds which are built as the lifts of tropical curves from the base of an Lagrangian torus fibration. In particular, we perform a first computation showing that holomorphic triangles can appear with boundary on the Lagrangian submanifold. We speculate how these holomorphic triangles can contribute to the count of holomorphic strips in the Lagrangian intersection Floer cohomology between a tropical Lagrangian submanifold and a fiber of the SYZ fibration.

Jeff Hicks

### Compactifying Torus Fibrations Over Integral Affine Manifolds with Singularities

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.

Helge Ruddat, Ilia Zharkov

### Graphical Neighborhoods of Spatial Graphs

We give a definition of a graphical neighborhood of a spatial graph which generalizes the tubular neighborhood of a link in S3. Furthermore we prove existence and uniqueness of graphical tubular neighborhoods.

Stefan Friedl, Gerrit Herrmann

### Open Problems in the Topology of Manifolds

∎∎∎

Jonathan Bowden, Diarmuid Crowley, Jim Davis, Stefan Friedl, Carmen Rovi, Stephan Tillmann

### Aperiodic Order Meets Number Theory: Origin and Structure of the Field

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.

M. Baake, M. Coons, U. Grimm, J. A. G. Roberts, R. Yassawi

### Delone Sets on Spirals

Motivated by phyllotaxis in botany, the angular development of plants widely found in nature, we give a simple mathematical characterization of Delone sets on spirals.

Shigeki Akiyama

### Topological methods for symbolic discrepancy

In this lecture, we discuss the notion of bounded symbolic discrepancy for infinite words and subshifts, both for letters and factors, from a topological dynamics viewpoint. We focus on three families of words, namely hypercubic words, words generated by substitutions, and dendric words.

Valérie Berthé

### Extended symmetry groups of multidimensional subshifts with hierarchical structure

In this contribution, we discuss the automorphism group, i.e., the centralizer of the shift action inside the group of self-homeomorphisms of a subshift, together with the extended symmetry group (the corresponding normalizer) of certain Zd subshifts with a hierarchical structure, like bijective substitutive subshifts and the Robinson tiling.

Álvaro Bustos

### Algebraic Invariants for Group Actions on the Cantor Set

The algebraic invariants associated to the group actions on the Cantor set provide an interesting connection between the fields of dynamical systems and group theory. For instance, Giordano, Putnam and Skau have shown in [29] that the dimension group (see [24] for an introduction about dimension groups) of a minimal Z-action on the Cantor set completely determines its strong orbit equivalence class. Furthermore, the topological full group of such a system, which is known from Juschenko and Monod [38] to be amenable, determines its flip-conjugacy class (see [6] and [30] for more details). On the other hand, the amenability of the topological full groups of minimal Z-actions together with their properties shown in [41] by Matui make them the first known examples of infinite groups which are at the same time amenable, simple and finitely generated. Recently, another algebraic invariant, the group of automorphisms of actions on the Cantor set, has caught the eye of several researchers working in the field [13, 15, 16, 17, 14, 19, 20]. In [5], Boyle, Lind and Rudolph focused their attention on the group of automorphisms of subshifts of finite type, showing that these groups are always countable and residually finite. At the same time, they gave an example of a minimal Z-action on the Cantor set whose group of automorphisms contains Q, which implies that the automorphism group of a minimal action may be a non-residually finite group (recall that the Z-subshifts of finite type are not minimal). This leads to the natural question about the relation between the algebraic properties of the group of automorphisms and the dynamics of the system. Indeed, the residually finite property of the group of automorphisms of the subshifts of finite type is a consequence of the existence of periodic points.

Marìa Isabel Cortez

### Lyapunov Exponents: Recent Applications of Ḟurstenberg’s Theorem in Spectral Theory

We discuss the phenomenon of Anderson localization and a new proof of it in one space dimension. This proof is due to V. Bucaj, D. Damanik, J. Fillman, V. Gerbuz, T. VandenBoom, F.Wang, Z. Zhang, and it is centered around the positivity of and large deviation estimates for the Lyapunov exponent — a strategy originally developed in non-random settings by J. Bourgain, M. Goldstein, W. Schlag.

David Damanik

### Extended Symmetries of Markov Subgroups

A symmetry of a tessellation is an isometry of the plane, or space, preserving the tessellation. What symmetry groups can one get? This is a classical problem in geometry, leading to the wallpaper groups of the plane or crystallographic groups in higher dimensions.

Robbert Fokkink

### Renormalisation for Inflation Tilings I:General Theory

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.

F. G¨ahler

### Problems in number theory related to aperiodic order

This talk concerns properties of dilated floor functions fα(x) = [αx], where α takes a fixed real value.

Jeffrey C. Lagarias

### Pure point spectrum and regular model sets in substitution tilings on ℝd

It has long been known that every regular model set has pure point spectrum, but the converse is not true in general.

Jeong-Yup Lee

### Automatic sequences are orthogonal to aperiodic multiplicative functions

In 2010, P. Sarnak [7] formulated the following conjecture: For each zero entropy topological dynamical system (X, T).

Mariusz Lemańczyk

### Similarity isometries of shifted lattices and point packings

A lattice $$\Gamma$$ Γ (of rank and dimension d) is a discrete subset of ℝd that is the $$\mathbb{Z}$$ Z -span of d linearly independent vectors v1, . . . ,vd $$\in$$ ∈ ℝd over ℝ. The set {v1, . . . ,vd} is called a basis for $$\Gamma$$ Γ , and $$\Gamma$$ Γ = $$\mathbb{Z}$$ Z v1 ⊕···⊕ $$\mathbb{Z}$$ Z vd.

Manuel Joseph C. Loquias

### Renormalisation for inflation tilings II: Connections to number theory

In the study of spectral properties of a d-dimensional aperiodic tiling which arises from an inflation rule ρ on a finite set of prototiles.

Neil Mañibo

### Scaling properties of the Thue–Morse measure: A summary

This is an extended abstract of the paper ‘Scaling properties of the Thue–Morse measure’ by Baake, Gohlke, Kesseb¨ohmer and Schindler [1].

Tanja Schindler

### Scaling Properties of the Thue–Morse Measure: A Summary

This is an extended abstract of the paper ‘Scaling properties of the Thue–Morse measure’ by Baake, Gohlke, Kesseb¨ohmer and Schindler [1].

Tanja Schindler

### Weak Model Sets

In this talk, which is based on joint work with M. Baake and C. Huck, we will review the properties of weak model sets of extremal density. These results have been proved independently in [3, 6], and we recommend these for more details.

Nicolae Strungaru

### Doubly Sparse Measures on Locally Compact Abelian Groups

In this work, joint with Michael Baake and Nicolae Strungaru [3], we are interested in doubly sparse measures on a locally compact Abelian group (LCAG) G.

Venta Terauds

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Franco Vivaldi

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Peter Zeiner

### A Diffraction Abstraction

For some time now, I have been trying to understand the complexity of integer sequences from a variety of different viewpoints and, at least at some level, trying to reconcile these viewpoints. However vague that sounds—and it certainly is vague to me—in this short note, I hope to explain this sentiment.

Michael Coons

### Extrinsic curvature flows and applications

These notes arose from a mini lecture series the author gave at the Early Career Researchers Workshop on Geometric Analysis and PDEs, held in January 2020 at The Mathematical Research Institute MATRIX. We discussed some classical aspects of expanding curvature flows and obtained first applications. In these notes we will give a detailed account on what was covered during the lectures.

Julian Scheuer

### Short time existence for higher order curvature flows with and without boundary conditions

We prove short time existence for higher order curvature flows of plane curves with and without generalised Neumann boundary condition.

Yuhan Wu

### Hankel transforms and weak dispersion

This survey is concerned with a general strategy, based on Hankel transforms and special functions decompositions, to prove weak dispersive estimates for a class of PDE's. Inspired by [2], we show how to adapt the method to some scaling critical dispersive models, as the Dirac-Coulomb equation and the fractional Schr¨odinger and Dirac equation in Aharonov-Bohm field.

Federico Cacciafesta, Luca Fanelli

### A priori bounds for the kinetic DNLS

In this note, we consider the kinetic derivative nonlinear Schrödinger equation (KDNLS), which arises as a model of propagation of a plasma taking the effect of the resonant interaction between the wave modulation and the ions into account. In contrast to the standard derivative NLS equation, KDNLS does not conserve the mass and the energy. Nevertheless, the dissipative structure of KDNLS enables us to show an a priori bound in the energy space and a lower bound of the L2 norm for its solution, as we see in this note. Combined with the local wellposedness result, which we plan to show in a forthcoming paper, these bounds will give a global existence result in the energy space for small initial data.

Federico Cacciafesta, Yoshio Tsutsumi
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