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Large networks are becoming increasingly prominent in many areas such as physics, applied mathematics, and mathematical biology. The often data-driven analysis of such large networks can yield great insight but also pose new computational challenges. One way to sidestep this is to attempt to understand the behaviour of the continuum limit of such networks, both as a generative model for large random graphs as well as a mathematical object in its own right. Here, we combine recent work on one such limit object - the graphon - with Master Stability Function analysis, a powerful tool for studying the stability of synchronous oscillatory or chaotic states in coupled dynamical systems. In doing so, our network dynamical system is converted into a nonlocal partial differential equation, where the connectivity matrix is replaced by an integral operator with the graphon as its kernel. One important example application of this framework is that it yields a natural method to derive a neural field model from a network of neurons. Furthermore, we find that this method can also be applied more generally than graphon integral operators, which we demonstrate in a reaction-diffusion system. Here the framework allows us to perform Turing pattern-like analysis around a spatially homogeneous oscillatory state (instead of a steady state). Finally, the generality of the method allows the effect of other types of diffusion such as anisotropic or fractional diffusion to be considered in the same way.
In 1986, Gromov introduced the notion of 'partial differential relation' (PDR) as well as a very interesting notion of 'ellipticity' which is loosely related to the traditional meaning of this word in PDE but also applies to fully nonlinear problems. My plan for the tutorial is as follows:
1. I begin with a hiker's-guide introduction to these notions, guided by beautiful examples each of which can be recast as a solution set to a PDR: (i) holomorphic functions (ii) rigid motions (iii) microstructures in the Ball-James model of solid-solid phase transitions (iv) Euler equations of fluid dynamics.
2. I then give a brief introduction to how 'wild' solutions to non-elliptic PDR can be constructed (for (iii) this is elementary and for (iv) this yields a celebrated theorem DeLellis and Szekelyhidi on the existence of nontrivial weak solutions to the Euler equations which are compactly supported in spacetime).
3. I close by discussing the deep question whether elliptic PDRs admit approximate solutions not close to exact solutions. The intuitive answer (No) is correct, e.g., for example (ii) (this is a theorem by Friesecke, James and Mueller) but in general wrong, as I will explain.
In this talk I will discuss the problem of the best approximation of the three-dimensional Lebesgue measure by a discrete measure supported on a Bravais lattice. Here 'best approximation' means best approximation with respect to the Wasserstein metric W_p, p \in [1,\infty). This problem is known as the quantization problem and it arises in numerical integration, electrical engineering, discrete geometry, and statistics.
This talk is about fermionic 1-body reduced density matrices, and how they are restricted 'beyond the Pauli principle'. I will argue that such restrictions are mostly a low-particle number, low-Hilbert space dimension effect. The talk is chiefly mathematical, though some physics motivation is included.
Inference to the best explanation (IBE) may lead to incoherent outcomes when we use it to assess multiple levels of explanation of the same phenomenon. This problem occurs both when IBE is used for full beliefs or for credences, as recently demonstrated (Climenhaga, 2017}. I will discuss a solution of this problem: inference to the best of the best explanations. We first identify a superset of all relevant explanations of the same phenomenon by considering their causal proximity to the explanandum. We then compare the explanatory power of all of them, in order to find the best among the best. I will also show that this is the intuitive way IBE works.
It is known that the percolation threshold for percolation on Z^d can be written as an asymptotic series in 1/d, for both bond and site percolation. The first four terms of both series were computed in the 1970s, but the series have not been extended since then. I will show how brute-force enumerations, combinatorial identities and a new approach based on Padé approximants can be deployed to compute more terms of the series.In the second part of the talk I will present a startling symmetry in the number of percolating configurations. This symmetry implies, for example, that the number of percolating configurations on finite subsets of Z^d is always odd.
A learning approach for optimal feedback gains to nonlinear continuous time control systems is proposed and analysed. The goal is to establish a rigorous framework for computing approximating optimal feedback gains using neural networks. The approach rests on two main ingredients. First, an optimal control formulation involving an ensemble of state trajectories with control variable given by the feedback gain functions. Second, an approximation to the set of admissible feedback laws by neural networks. Based on universal approximation properties we prove the existence of optimal stabilizing neural network feedback controllers and characterize them through first order necessary optimality conditions. Qualitative convergence results towards optimal feedback gains are discussed. The talk is completed by numerical examples highlighting the practical applicability of the presented approach. This is joint work with Karl Kunisch.
The Dean-Kawasaki equation was proposed in physics model for colloids and glassy materials. Mathematically, it is a singular super critical stochastic PDE with multiplicative noise. We show that the equation has a remarkable rigidity in terms of well-posedness allowing for solutions in only discrete parameter regimes corresponding to the case of finite particle systems. We also present corrections to the SPDE allowing for a rich class of non trivial mathematically rigorous models with complex patterns to emerge. Our methods combine classical arguments from measure valued Markov processes as well as Dirichlet form theory in infinite dimensions.
We investigate synonymy in the strong sense of content identity (and not just meaning similarity). This notion is central not only in the philosophy of language but also in applications of logic (say, which logical laws govern a neural network). Finding a good such notion is non-trivial due to a no-go result: if synonymy is to preserve subject matter, then either conjunction and disjunction lose an essential property or a very weak logical law is violated. We motivate, uniformly axiomatize, and characterize five “benchmark” notions of synonymy. This helps to understand the derlying logic in a given application, and it provides novel arguments - independent of a particular semantic framework - for each notion of synonymy discussed.
A mathematical analysis of irrevisible behavior in various physical systems is presented. Often irrevisibility manifests itself in the form of "entropy production". This motivates to begin the lecture with a brief review of relative entropy and with a sketch of some of the sources of irrevisible behavior. Subsequently, the Second Law of thermodynamics in the forms given to it by Clausius and Carnot is derived from quantum statistical mechanics. In a third part, a derivation of Brownian motion of a quantum particle interacting with a quantum-mechanical thermal reservoir is sketched. This is followed by an outline of a theory of Hamiltonian Friction. To conclude, the fundamental arrow of time inherent in quantum mechanics is highlighted.
We introduce and study a variant of the Wasserstein distance on the space of probability measures, specially designed to deal with measures whose support has a dendritic, or treelike structure with a particular direction of orientation. Our motivation is the comparison of and interpolation between plants' root systems. We characterize barycenters with respect to this metric, and establish that the interpolations of root-like measures, using this new metric, are also root like, in a certain sense; this property fails for conventional Wasserstein barycenters. We also establish geodesic convexity with respect to this metric for a variety of functionals, some of which we expect to have biological importance. This represents joint work with Young-Heon Kim and Dave Schneider.
The talk we discuss the behavior of the concentration of some bacteria swimming in water (for example of the species Bacillus subtilis), whose otherwise random motion is partially directed towards higher concentrations of a signaling substance (oxygen) they consume. After a transition phase, the system can be described using a chemotaxis-consumption model on a bounded domain. Previous studies of chemotaxis models with consumption of the chemoattractant (with or without fluid) have not been successful in explaining pattern formation even in the simplest form of concentration near the boundary, which had been experimentally observed.
Following the suggestions that the main reason for that is usage of inappropriate boundary conditions, this talks considers no-flux boundary conditions for the bacteria density and the physically meaningful Robin boundary conditions for the signaling substance and Dirichlet boundary conditions for the flow.
In the talk, we study the existence of a global (weak) solution. Moreover, we discuss how to show that there exists a unique stationary solution for any given mass assuming that the flow vanishes. This solution is non-constant. In the radial symmetric case, the densities are strictly convex.
Heteroclinic connections are trajectories that link invariant sets of a dynamical system. For example, they can form robust networks of connections between equilibria in autonomous systems with flow-invariant subspaces. This talk is about joint work with Sofia Castro (University of Porto) and Peter Ashwin (University of Exeter) on the relation between the structure of heteroclinic networks and directed graphs between nodes. We consider realizations of a large class of transitive digraphs as robust heteroclinic networks and show that although these realizations are typically not complete (i.e. not all unstable manifolds of its nodes are part of the network), they can be almost complete (i.e. complete up to a set of measure zero) and equable (i.e. all sets of connections from a node have the same dimension).
The Vlasov-Poisson system with massless electrons (VPME) is widely used in plasma physics to model the evolution of ions in a plasma. It differs from the classical Vlasov-Poisson system (VP) in that the Poisson coupling has an exponential nonlinearity that creates several mathematical difficulties. We will discuss a recent result proving uniqueness for VPME in the class of solutions with bounded density, and global existence of solutions with bounded density for a general class of initial data, generalising to this setting all the previous results known for VP. Moreover we will talk about a mean field derivation of the VPME and a rigorous quasi neutral limit for initial data that are close to analytic data deriving the Kinetic Isothermal Euler (KIE) system from the VPME in dimensions d=1,2,3. Lastly, we combine these two singular limits in order to show how to obtain the KIE system from an underlying particle system.
Dies ist ein Habilitationsvortrag. Er wird von homologischer Algebra, Geometrie und einfachen Bausteinen handeln. Nebenbei wird enthüllt, und mit gewissen elementaren Komplexen erklärt, was manche erwachsene MathematikerInnen im gegenseitigen Einverständnis hinter verschlossenen Türen tun.
A classical problem in stochastic analysis is the Skorokhod embedding problem: Given a Brownian motion and a probability measure, the task is to stop the trajectories of the process such that the terminal points are distributed according to the given measure. One approach in order to determine a solution for the problem is to construct it as a first hitting time of the Root barrier. Rost proved in 1976 that Root’s solution has the minimal variance among the solutions to Skorokhod embedding problem using methods from probabilistic potential theory. We are going to investigate sufficient conditions such that such an embedding can be made and provide a free boundary characterisation of Root’s solution for a general class of Markov processes.
The multiple scattering theory (MST), also called Korringa-Kohn-Rostoker (KKR) method, is widely used in electronic structure calculations of solid materials. In the MST method, a perfect separation between the atomic potentials and configuration geometries can be achieved. This can be exploited in the simulations to reduce the computational costs for many large-scale systems, including defected and disordered systems. This work studies the MST method by a rigorous numerical analysis and derives a spectral convergence rate for the numerical approximations, which justifies the reliability and efficiency of the MST method.
Running simulations on high-performance computers faces new challenges due to e.g. the stagnating or even decreasing per-core speed. This poses new restrictions and therefore challenges on solving PDEs within a particular time frame. Here, disruptive mathematical reformulations which e.g. exploit additional degrees of parallelism also in the time dimension gained increasing interest over the last two decades.
This talk will give concrete examples of the current cutting edge research on parallel-in-time and other next-generation time integration methods in the context of weather and climate simulations:
* Parallel-in-time rational approximation of exponential integrators (REXI) based on Terry's (T-REXI), Cauchy Contour (CI-REXI) and Butcher Tableau (B-REXI). * Semi-Lagrangian Parareal (SL-Parareal) * Semi-Lagrangian exponential integration (SL-EXP) * Multi-level time integration of spectral deferred correction (ML-SDC) * Parallel Full Approximation Scheme in Space and Time (PFASST)
These methods are mostly realized with numerics similar to the ones used by the European Centre for Medium-Range Weather Forecasts (ECMWF). Our results motivate further investigation for operational weather/climate systems in order to cope with the hardware imposed restrictions of future super computer architectures.
(I gratefully acknowledge contributions and more from Jed Brown, Francois Hamon, Terry S. Haut, Richard Loft, Michael L. Minion, Matthew Normile, Pedro S. Peixoto, Nathanaël Schaeffer, Andreas Schmitt)
In this talk, I am going to present my Master’s thesis in which I have analyzed a nonlinear advection diffusion system with three coupled scalar fields representing the surface water height, the soil moisture, and the biomass distribution of the vegetation of dryland ecosystems of interest. The various parameters of the PDE system allow for modeling different arid and semi-arid ecosystems on our planet that exhibit vegetation pattern formation. The system also exhibits two different time scales, a fast time scale in which water flows downhill and infiltrates into the soil and a slow time scale in which transpiration happens and plants grow. Naturally, the system can then be splitted into a fast and a slow system. The fast system can be solved with simplifications analytically and unsimplified numerically using an implicit-explicit method. The slow system is solved using a similar implicit-explicit method. By coupling the fast system representing an a-few-hour-long rain event with the slow system representing a 182-day-long dry season (this is a modeling step), a Poincaré map can be created and a steady or uphill traveling vegetation stripe pattern is obtained depending on the boundary conditions used when the map is run for many years.
Thermal electrochemical models for porous electrode batteries (such as lithium ion batteries) are widely used. Due to the multiple scales involved, solving the model accounting for the porous microstructure is computationally expensive, therefore effective models at the macroscale are preferable. However, these effective models are usually postulated ad hoc rather than systematically upscaled from the microscale equations. Using the method of periodic homogenisation I will demonstrate that in a suitable limit the equations converge to an effective thermal electrochemical model as the fineness of the microstructure converges to 0.
What does a mathematical proposition mean? Under one standard account, all true mathematical statements mean the same thing, namely True. A more meaningful account is provided by the Propositions-As-Types conception of type theory, according to which the meaning of a proposition is its collection of proofs. The new system of Homotopy Type Theory provides a further refinement: The meaning of a proposition is the homotopy type of its proofs. A homotopy type may be seen as an infinite-dimensional structure, consisting of objects, isomorphisms, isomorphisms of isomorphisms, etc. Such structures represent systems of objects together with all of their higher symmetries. The language of Martin-Löf type theory is an invariant of all such higher symmetries, a fact which is enshrined in the celebrated Principle of Univalence.
Ich werde etwas über die englische Kunst des Wechselläutens erzählen, bei der es darum geht, mit Kirchenglocken nicht etwa Melodien, sondern Permutationen der symmetrischen Gruppe zu läuten. Ich werde eine Einführung und Tonbeispiele geben, dann etwas über die Mathematik dahinter geben - Gruppentheorie und Graphentheorie - und möchte so die Weihnachtsferien einläuten.