The PDEs and Applications seminar

Tuesdays 10.15, room 64119, Ångström

Supported by the Swedish Research Council

Next seminar(s):

17/12: Gabriele Messori: Of Lorenz Butterflies and other animals: atmospheric predictability from dynamical systems and machine learning perspectives

Abstract: Atmospheric flows are characterized by chaotic dynamics and recurrent large-scale patterns. These two characteristics point to the existence of an atmospheric attractor, defined by Lorenz as: “the collection of all states that the system can assume or approach again and again, as opposed to those that it will ultimately avoid”. By leveraging recent developments in dynamical systems theory, we can diagnose local properties of states on the atmospheric attractor, which provide us with powerful insights into atmospheric predictability. Atmospheric predictability can also be studied by using machine learning algorithms. These are often likened to a "black box", where the user knows what goes in and what comes out but is not privy to the box's internal workings. In this sense, machine learning sits at the opposite end of the spectrum to dynamical systems theory. In my talk, I will argue that the two approaches are both valuable for the study of atmospheric predictability, and that they provide largely complementary information. I will mainly focus on practical applications to idealised and real-world atmospheric datasets, rather than on technical or theoretical considerations, but I will nonetheless attempt to introduce the basic concepts underlying the dynamical systems framework.

Upcoming seminars:

14/1: Niklas Lundström, Umeå Universitet: TBA

21/1: David Lundberg, Uppsala: TBA

4/2: Moritz Egert, Universite Paris-Sud: TBA

11/2: Håkan Hedenmalm, KTH: TBA

Past seminars:

10/12, 10.15: Iwona Chlebicka, University of Warsaw: Anisotropic elliptic problems

Abstract: Let us consider a nonlinear elliptic Dirichlet problem with nonstandard growth expressed by the means of fully anisotropic Orlicz function on which no condition of doubling type is imposed. This type of equations are related to elasticity problems of materials of complicated structure or steady flow of non-Newtonian fluids. I shall concentrate on explanation what this anisotropy means, what type of operators are included in the study and, consequently, what is the meaning of the equation we study. When the datum is regular, we prove existence of weak solutions. For measure data we consider a generalized notion of solutions for which we infer existence and anisotropic regularity in Orlicz-Marcinkiewicz spaces extending the known results for anisotropic p-Laplace equation. In the case of datum absolutely continuous with respect to Lebesgue's measure we prove also uniqueness. Based on joint work: with Angela Alberico, Andrea Cianchi and Anna Zatorska-Goldstein, Fully anisotropic elliptic problems with minimally integrable data Calc. Var. PDEs (2019) 58:186.

10/12, 11.15: Vesa Julin, University of Jyväskylä: The surface diffusion flow with elasticity

Abstract: I will discuss about short-time existence of a smooth solution to the surface diffusion equation with an elastic term. The equation models the evolution of material voids inside a stressed elastic solid, but from mathematical point of view it can be regarded as canonical nonlocal perturbation of the surface diffusion by an additive elastic contribution. The elastic part is not, in general, lower order with respect to the area and therefore in the physical literature people often considers a regularized variant of the flow with an additional curvature term. In our recent result we prove the short time existence without any additional curvature regularization. I will also discuss about the asymptotic stability of strictly stable stationary sets. This is a joint work with Nicola Fusco (Napoli) and Massimiliano Morini (Parma).

3/12: Alan Sola, SU: Scaling limits in conformal Laplacian random growth models

Abstract: This talk will give an overview of joint work with A. Turner (Lancaster) and F. Viklund (KTH) on scaling limits in Laplacian random growth processes defined in terms of conformal maps. These are processes defined in terms of conformal maps of the plane where local growth at stage $n$ depends on the derivative of the mapping function at stage $n-1$ . I will survey results obtained in the last few years on instances of these models where feedback is very weak and very strong: in the first instance, the limiting shapes under a suitable scaling are disks, and in the latter, they are one-dimensional.

26/11, 10.15: Irina Mitrea, Temple University: TBA

26/11, 11.15: Andrey Bagrov, Radboud University: TBA

19/11, 10.15: David Maxwell, University of Alaska Fairbanks : Interior estimates for elliptic operators associated with low regularity Riemannian metrics

Abstract: Work in general relativity sometimes requires using elliptic operators associated with metrics having low Sobolev regularity. In this talk we describe the mapping properties of these operators, and an elementary approach to establishing interior elliptic estimates for these operators from scaling arguments, without resort to pseudo differential operators with rough coefficients. This is joint work with Michael Holst and Gantumur Tsogtgerel.

19/11, 11.15: Andrew Morris, University of Birmingham: The minimal regularity Dirichlet problem for degenerate elliptic PDEs beyond symmetric coefficients

Abstract: We prove that the Dirichlet problem for degenerate elliptic equations with nonsymmetric coefficients on Lipschitz domains is solvable when the boundary data is in weighted $L^p$ for some $p<\infty$. The result is achieved without requiring any structure on the coefficient matrix, thus allowing for nonsymmetric coefficients, in which case $p>2$ becomes necessary. We build on the groundbreaking result obtained by Hofmann, Kenig, Mayboroda and Pipher for uniformly elliptic equations, by allowing for the bound and ellipticity on the coefficients to degenerate under the control of a Muckenhoupt weight. We also adopt an alternative strategy, which is outlined in their work, although the crucial technical estimate is not at all an obvious extension of the uniformly elliptic theory. In this approach, a Carleson measure estimate for bounded solutions is established directly. This allows us to avoid good-$\lambda$ inequalities entirely, and thus apply a Dhalberg--Kenig--Stein pull-back based on an $L^2$-Hodge decomposition instead of an $L^{2+\epsilon}$-version. The result is then combined with an oscillation estimate for solutions, which allows us to avoid the method of $\epsilon$-approximability, to deduce that the degenerate harmonic measure is in the $A_\infty$-class with respect to weighted Lebesgue measure on the domain boundary. This is joint work with Steve Hofmann and Phi Le.

12/11: Attila Szilva, Department of Physics and Astronomy, Materials Theory, UU: Smart and Sustainable: the Universal Scaling Laws of Organisms and Cities.

Abstract: Animals from rats to the blues whales are built up from cells arranged in networks. The topology of the underlying networks explains the so-called Kleiber's scaling law, which states that an animal's metabolic rate scales to the 3/4 power of the animal's mass (a cat having a mass 100 times that of a mouse will consume only about 32 times the energy the mouse uses). This scaling is sublinear because the power is less than 1. In the presentation, it will be shown that the infrastructure of cities (the length of electric cables or the number of gas stations) also follows universal sublinear scaling law while in socio-economic dimensions (GDP per capita, innovation, crime) cities are superlinear. They are as a result of the individual interactions proven by a large set of mobile phone data. The concept of scaling and universality is originated in statistical physics where a large complex system emerges from simple interactions, and its behavior is almost totally independent of its microscopic structure.

5/11: Nikos Katzourakis, University of Reading: On the existence and uniqueness of vectorial absolute minimisers in Calculus of Variations in L-infinity

Abstract: Calculus of Variations in the space L-infinity has a relatively short history in Analysis. The scalar-valued theory was pioneered by the Swedish mathematician Gunnar Aronsson in the 1960s and since then has developed enormously. The general vector-valued case delayed a lot to be developed and its systematic development began in the 2010s. One of the most fundamental problems in the area which was completely open until today (and has been attempted by many researchers) concerned that of the title. In this talk I will discuss the first result in this direction.

22/10: Douglas Lundholm, Uppsala: Mathematical methods for uncertainty and exclusion in quantum mechanics

Abstract: The success of quantum mechanics over classical mechanics in explaining properties of matter such as the periodic table of the elements, stability against collapse, etc, rests on two fundamental principles: the uncertainty principle and the exclusion principle. In 1975 Elliott Lieb and Walter Thirring found a powerful functional inequality, the Lieb-Thirring inequality, which incorporates these two principles and was used to prove stability for "ordinary matter" which is subject to fermionic statistics and Pauli's exclusion principle. Recently this inequality has been extended to situations where only weaker forms of exclusion apply, such as for certain contact interactions. If there is time I will also discuss exotic forms of particle statistics possible in two dimensions, namely "anyons" and their exclusion properties.

15/10: Per Sjölin, KTH: Convergence and localization of Schrödinger means.

Abstract: We shall study localization of Schrödinger means and convergence of sequences of Schrödinger means. We shall also replace sequences by uncountable sets, for instance Cantor sets. The convergence results have been obtained in joint work with Jan-Olov Strömberg.

8/10: Oliver Lindblad Petersen, University of Hamburg: Wave equations close to Cauchy and event horizons and applications in general relativity

Abstract: The fundamental equations in Einstein's theory of general relativity are wave equations on curved spaces. The behavior of waves becomes particularly interesting close to event horizons of black holes or close to so called Cauchy horizons. I will explain new results on this topic and relate them to the following classical open questions:
Is there a general mathematical obstruction to time travel?
Is the known model for a stationary black hole the only one mathematically possible?

1/10: Anders Israelsson, Uppsala: Boundedness of Schrödinger Integral Operators in Sobolev Spaces

Abstract: Schrödinger integral operators (SIO) are a type of oscillatory integral operators that arise naturally in the analysis of linear partial differential equations of Schrödinger type. Hence, if one wants to say something about the regularity of the solution to a PDE of Schrödinger type, then it is enough to investigate the corresponding SIO. Until a couple of years ago not much was known about the regularity of SIO's, but as a new Littlewood-Paley type decomposition was developed the Sobolev space regularity problem was fully solved. In this talk I will sketch the proof of this and mention some of the immediate consequences.

24/9: Anton Savostianov, Uppsala: Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations

Abstract: It is well known that long time behaviour of a dissipative dynamical system generated by an evolutionary PDE can be described in terms of attractor, an attracting set which is essentially thinner than a ball of the corresponding phase space of the system. In this talk we compare long time behaviour of damped anisotropic wave equations with the corresponding homogenised limit in terms of their attractors. First we will formulate order sharp estimates between the trajectories of the corresponding systems and will see that the hyperbolic nature of the problem results in extra correction comparing with parabolic equations. Then, after brief review on previous results on homogenisation of attractors, we will discuss new results. It appears that the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts, in suitable norms, can be estimated via operator norm of the difference of the resolvents of the corresponding elliptic operators. Furthermore, we show that the homogenised attractor admits first-order correction suggested by the natural asymptotic expansion. The corrected homogenised attractors, as expected, are close to the anisotropic attractors already in the strong energy norm. The corresponding quantitative estimates on the Hausdorff distance between the corrected homogenised attractors and anisotropic ones, with respect to the strong energy norm, are also obtained. Our results are applied to Dirchlet, Neumann and periodic boundary conditions. This is joint work with Shane Cooper.

17/9: Rostyslav Kozhan, Uppsala: Central Limit Theorem for linear statistics for orthogonal polynomial ensembles on the unit circle

Abstract: We prove a Central Limit Theorem for linear statistics for orthogonal polynomial ensembles on the unit circle. This can also be viewed as a relative version of the famous Strong Szego Theorem for Toeplitz determinants. The talk will include an introduction to Toeplitz determinants and to orthogonal polynomials on the unit circle. Joint work with M.Duits (KTH).

10/9: Erik Ekström, Uppsala: Degenerate Equations and Boundary Conditions in Finance

Abstract: Pricing rules in Finance are often given as expected values of diffusion processes. Using the Feynman-Kac theorem, this translates into a formulation in terms of PDEs. However, the equations often have degenerate coefficients at the boundary of the state space, and additional care is needed to establish the Feynman-Kac connection. We study backward and forward equations for a couple of problems appearing in Finance, with a special focus on boundary conditions.

3/9: Andreas Strömbergsson, Uppsala: Kinetic theory for the low-density Lorentz gas

Abstract: The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers, whose radii are small compared to their mean separation. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that its macroscopic transport properties should be governed by a linear Boltzmann equation. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. In the talk I will describe an approach that applies for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas. The limiting transport process is expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action. (Joint work with Jens Marklof)

11/6: Felix del Teso, BCAM, Bilbao: The Liouville Theorem for Nonlocal Operator (and its relation to irrational numbers and subgroups of R^N.

The classical Liouville Theorem states that bounded harmonic functions are constant. In this talk we will revisit this result for a class of nonlocal operators. This class of operators naturally includes the fractional Laplacian, Relativistic Schrodinger operators, convolution operators, as well as discretizations of both local and nonlocal symmetric diffusion operators.
First, we will treat the one dimensional case. Here we give a precise classification for which the Liouville Theorem holds. The condition will be related to irrational numbers ([1]).
In the multi dimensional case such a characterization is also proved. This time it will be given in terms of additives subgroups of R^N ([1]).
This nonlocal result will allow us ([2]) to give a full characterization of the Liouville property for any linear operator (local + nonlocal, and not necessarily symmetric) with constant coefficients satisfying the maximum principle (see [3]).

References:
[1] N. Alibaud, F. del Teso, J. Endal, and E. R. Jakobsen. Characterization of nonlocal diffusion operators satisfying the Liouville theorem. Irrational numbers and subgroups of R^d. Preprint: arXiv:1807.01843.
[2] N. Alibaud, F. del Teso, J. Endal, and E. R. Jakobsen. The Liouville theorem and linear operators satisfying the maximum principle. A complete characterization in the constant coefficient case. Work in progress.
[3] P. Courrege. Generateur infinitesimal d'un semi-groupe de convolution sur R^n, et formule de Levy-Khinchine. Bull. Sci. Math. (2), 88:3-30, 1964. Joint work with N. Alibaud (Laboratorie de Mathematiques de Besancon), J. Endal and E. R. Jakobsen (Norwegian University of Science and Technology).

21/5: Kaj Nyström, Uppsala: On a strongly degenerate parabolic equation of Kolmogorov.

Abstract: Strongly degenerate parabolic equations of Kolmogorov type appear naturally in models for particle dispersion in random media as well in finance. In this talk I will discuss results in the direction of our goal to develop a harmonic analysis approach to boundary value problems for these equations in appropriate geometries. Topics to be discussed include boundary comparison principles and the fine properties of associated parabolic measures.

21/5: Julie Rowlett, Göteborg: Analysis and geometry on asymptotically hyperbolic manifolds and convex-cocompact manifolds with variable negative curvature

Abstract: A broad class of Riemannian manifolds are those which are complete, geometrically finite without cusps, and have bounded, negative sectional curvatures. These are known as convex-cocompact. Conformally compact, asymptotically hyperbolic, and almost hyperbolic manifolds with negative sectional curvatures are all examples of convex co-compact manifolds. We will consider the geodesic flow on these manifolds and connections to the spectral theory of their Laplacians.

7/5: Eric Schippers: Schiffer operators on Riemann surfaces and the jump formula for quasicircles

Abstract: Consider a Riemann surface $R$ separated into two connected components by a smooth closed curve. The jump formula states that, given a sufficiently regular function $u$ on this curve satisfying certain algebraic conditions, there are holomorphic functions on the components whose difference on the curve is $u$. We extend this classical result to quasicircles, which are not rectifiable, and functions $u$ in a certain conformally invariant function space related to the Dirichlet space. This is accomplished by relating the problem to the Schiffer operators, which are singular integral operators on $L^2$ holomorphic or anti-holomorphic one-forms on the components. Joint work with Wolfgang Staubach.

23/4, 10.15: Erik Lindgren, Uppsala: Infinity-harmonic potentials in convex rings

Abstract: In this talk, I will discuss certain solutions of the Infinity-Laplace equation in planar convex rings. Focus will be on their streamlines. It turns out that their ascending streamlines are unique while the descending ones may bifurcate. I will explain why bifurcation occurs quite often and the connection to regularity issues. Finally, I will discuss the solution in a punctured square more in detail. The talk is based on joint work with Peter Lindqvist (NTNU, Trondheim).

9/4, 10.15: Stephen McCormick: Quasi-local mass in general relativity and some recent developments on Bartnik's definition

Abstract: The problem of quasi-local mass in general relativity is the problem of assigning some meaningful notion of the total mass (or energy) contained in a bounded domain (compact Riemannian manifold with boundary). We first will give an introduction to the problem and discuss some proposed definitions of quasi-local mass, before turning to discuss some recent progress on understanding the definition due to Bartnik. We will discuss a method to obtain estimates of the Bartnik mass in the CMC case, and briefly mention how these estimates can be used to obtain estimates outside of the CMC case. The work presented here include results obtained in collaboration with Armando Cabrera Pacheco, Carla Cederbaum, and Pengzi Miao.

2/4, 10.15: Benny Avelin, Uppsala: Introduction to deep learning, continued

26/3, 10.15: Alejandro Luque, Uppsala: Oscillatory integrals and the problem of splitting of separatrices

Abstract: A fundamental problem in Dynamical Systems is to ascertain whether a given system is chaotic. One of the main tools to address this question, which dates back to Poincare, is to consider perturbations of a homoclinic connection of the system to produce transverse intersections of a stable manifold and an unstable manifold. The displacement (and hence the splitting) of such manifolds is typically approximated using an oscillatory integral called the Melnikov function. In this talk, we study the Melnikov function of a mechanical system that presents fast oscillations in time and space. Using stationary phase methods and the geometric constrains of the problem, we provide an explicit formula for the splitting which is valid in a very general setting.

26/3, 11.15: Benny Avelin, Uppsala: Introduction to deep learning, continued

19/3: 10.15: Anna Sakovich, Uppsala: On zero mass limit in mathematical general relativity

Abstract: For Riemannian manifolds and spacetimes with certain asymptotics at infinity there is a rather unique invariant called mass, first discovered in General Relativity. The so called rigidity theorems describe what happens when the mass vanishes. In this talk we will address the related question: what happens when the mass is small? How do Riemannian manifolds and spacetimes converge when the mass goes to zero? We study this question with respect to different notions of convergence.

19/3, 11.15: Benny Avelin, Uppsala: Introduction to deep learning, continued

12/3, 10.15: Linnea Gyllingberg, Uppsala: Deriving macroscopic equations for complex systems

Abstract: Complex systems are commonly investigated through microscopic, Individual-Based Models, which predict the evolution of each individual in time. However, macroscopic models, describing the overall behaviour of the system are often easier to analyse. Establishing the link between microscopic and macroscopic models is the central task of kinetic theory. However, some of the classical techniques from statistical mechanics fail for complex systems due to the specific nature of the individual's interactions. In this seminar I will discuss the challenges and advantages of going from a microscopic to a macroscopic description of complex systems and present how to do this for the application of sperm dynamics.

12/3, 11.15: Benny Avelin, Uppsala: Introduction to deep learning, continued

Abstract: This is a series of lectures essentially making up a mini course on the subject. The idea is to cover the practical as well as the theoretical aspects. In fact, my hope with this series of lectures is to shine light on interesting pure math problems arising in the context of deep learning, that so far has gathered only a minute amount of attention. The second lecture will be slightly more theoretical. We will go through what the optimization problem and the VC dimension problem.

26/2: Vesa Julin, Jyväskylä: The Gaussian Isoperimetric Problem for Symmetric Sets

Abstract: The Gaussian isoperimetric inequality states that among all sets with given Gaussian measure the half-space has the smallest Gaussian surface area. Since the half-space is not symmetric with respect to the origin, a natural question is to restrict the problem among symmetric sets. This problem turns out to be surprisingly difficult. In my talk I will discuss how it is related to probability and to the study of singularities of mean curvature flow, and present our result which partially solves the problem. This is a joint work with Marco Barchiesi.

19/2, 10.15: Benny Avelin, Uppsala: Introduction to deep learning.

Abstract: This is the first instalment of a series of lectures essentially making up a mini course on the subject. The idea is to cover the practical as well as the theoretical aspects. In fact, my hope with this series of lectures is to shine light on interesting pure math problems arising in the context of deep learning, that so far has gathered only a minute amount of attention. The first lecture will cover some of the history of deep learning starting with the perceptron algorithm.

19/2, 11.15: Avner Kiro, Tel Aviv: Power substitution in quasianalytic Carleman classes.

Abstract: In this talk, I will consider power substitutions in quasianalytic Carleman classes, i.e. equations of the form f(x)=g(x^k), where k>1 is an integer and f is a given function in a quasianalytic Carleman class. I will show that if g happens to be a smooth function, then g belongs to a quasianalytic class completely characterized in terms of bounds on the derivatives of g. The talk is based on joint work with L Buhovski and S. Sodin.

12/2: Stefano Borghini, Uppsala: Riemannian methods in potential theory.

Abstract: We discuss two recent methods for establishing the rotational symmetry of solutions to overdetermined elliptic boundary value problems. We illustrate these approaches through a couple of very classical examples arising in potential theory

29/1: Wolfgang Staubach, Uppsala: Oscillatory integrals and partial differential equations

We give an overview of oscillatory integrals and their role in the theory of partial differential equations, highlighting main developments of the theory from past to present.

5/2: Bruno Vergara, ICMAT: Convexity of Whitam's wave of extreme form.

Abstract: Whitham's model of shallow water waves is a non-local dispersive equation featuring travelling wave solutions as well as singularities. In this talk we will be interested in a conjecture of Ehrnström and Wahlen on the profile of solutions of extreme form to this equation. We will see that there exists a highest, cusped and periodic solution that is convex between consecutive peaks, where $C^{1/2}$-regularity has been shown to be optimal. The talk is based on joint work with A. Enciso and J. Gomez-Serrano.

Organizers: Benny Avelin, Erik Lindgren and Kaj Nyström