Forthcoming events in this series

Thu, 02 Jun 2022
12:00
L5

### Towards multi-dimensional localisation

Krzysztof Ciosmak
(University of Oxford)
Abstract

Localisation is a powerful tool in proving and analysing various geometric inequalities, including isoperimertic inequality in the context of metric measure spaces. Its multi-dimensional generalisation is linked to optimal transport of vector measures and vector-valued Lipschitz maps. I shall present recent developments in this area: a partial affirmative answer to a conjecture of Klartag concerning partitions associated to Lipschitz maps on Euclidean space, and a negative answer to another conjecture of his concerning mass-balance condition for absolutely continuous vector measures. During the course of the talk I shall also discuss an intriguing notion of ghost subspaces related to the above mentioned partitions.

Thu, 26 May 2022

17:00 - 18:00
Online

### The Cauchy problem for the ternary interaction of impulsive gravitational waves

Maxime Van de Moortel
(Princeton University)
Further Information

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

In General Relativity, an impulsive gravitational wave is a localized and singular solution of the

Einstein equations modeling the spacetime distortions created by a strongly gravitating source.
I will present a comprehensive theory allowing for ternary interactions of such impulsive gravitational waves in translation-symmetry, offering the first examples of such an interaction.

The proof combines new techniques from harmonic analysis, Lorentzian geometry, and hyperbolic PDEs that are helpful to treat highly anisotropic low-regularity questions beyond the considered problem.

This is joint work with Jonathan Luk.

Thu, 19 May 2022

12:00 - 13:00
L5

### Non-branching in RCD(K,N) Spaces

Qin Deng
(MIT)
Abstract

On a smooth Riemannian manifold, the uniqueness of a geodesic given initial conditions follows from standard ODE theory. This is known to fail in the setting of RCD(K,N) spaces (metric measure spaces satisfying a synthetic notion of Ricci curvature bounded below) through an example of Cheeger-Colding. Strengthening the assumption a little, one may ask if two geodesics which agree for a definite amount of time must continue on the same trajectory. In this talk, I will show that this is true for RCD(K,N) spaces. In doing so, I will generalize a well-known result of Colding-Naber concerning the Hölder continuity of small balls along geodesics to this setting.

Thu, 12 May 2022

12:00 - 13:00
L5

### Quantitative De Giorgi methods in kinetic theory for non-local operators

Amélie Loher
(University of Cambridge)
Abstract

We derive quantitatively the weak and strong Harnack inequality for kinetic Fokker--Planck type equations with a non-local diffusion operator for the full range of the non-locality exponents in (0,1).  This implies Hölder continuity.  We give novel proofs on the boundedness of the bilinear form associated to the non-local operator and on the construction of a geometric covering accounting for the non-locality to obtain the Harnack inequalities.  Our results apply to the inhomogeneous Boltzmann equation in the non-cutoff case.

Thu, 27 Jan 2022

12:00 - 13:00
L6

### Regularity results for Legendre-Hadamard elliptic systems

Christopher Irving
(Oxford University)
Abstract

I will discuss the regularity of solutions to quasilinear systems satisfying a Legendre-Hadamard ellipticity condition. For such systems it is known that weak solutions may which fail to be C^1 in any neighbourhood, so we cannot expect a general regularity theory. However if we assume an a-priori regularity condition of the solutions we can rule out such counterexamples. Focusing on solutions to Euler-Lagrange systems, I will present an improved regularity results for solutions whose gradient satisfies a suitable BMO / VMO condition. Ideas behind the proof will be presented in the interior case, and global consequences will also be discussed.

Thu, 02 Dec 2021

12:00 - 13:00
Virtual

### Controllability for the (multi-dimensional) Burgers equation with localised one-dimensional control

Ana Djurdjevac
(Zuse Institute Berlin)
Further Information

Abstract

We will consider the viscous Burgers driven by a localised one-dimensional control. The problem is considered in a bounded domain and is supplemented with the Dirichlet boundary condition. We will prove that any solution of the equation in question can be exponentially stabilised. Combining this result with an earlier result on local exact controllability we will show global exact controllability by a localised control. This is a joint work with A. Shirikyan.

Thu, 11 Nov 2021

16:00 - 17:00
L5

### Approximation of mean curvature flow with generic singularities by smooth flows with surgery

Joshua Daniels-Holgate
(University of Warwick)
Abstract

We construct smooth flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We demonstrate an application of this surgery flow by improving the entropy bound on the low-entropy Schoenflies conjecture.

Thu, 28 Oct 2021

12:00 - 13:00
C1

### Symmetry breaking and pattern formation for local/nonlocal interaction functionals

Sara Daneri
(Gran Sasso Science Institute GSSI)
Abstract

In this talk I will review some recent results obtained in collaboration with E. Runa and A. Kerschbaum on the one-dimensionality of the minimizers
of a family of continuous local/nonlocal interaction functionals in general dimension. Such functionals have a local term, typically the perimeter or its Modica-Mortola approximation, which penalizes interfaces, and a nonlocal term favouring oscillations which are high in frequency and in amplitude. The competition between the two terms is expected by experiments and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used  to model pattern formation, either in material science or in biology. The difficulty in proving the emergence of such structures is due to the fact that the functionals are symmetric with respect to permutation of coordinates, while in more than one space dimensions minimizers are one-dimesnional, thus losing the symmetry property of the functionals. We will present new techniques and results showing that for two classes of functionals (used to model generalized anti-ferromagnetic systems, respectively  colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers are one-dimensional and periodic, in general dimension and also while imposing a nontrivial volume constraint.

Thu, 17 Jun 2021

12:00 - 13:00
Virtual

### Willmore Surfaces: Min-Max and Morse Index

Alexis Michelat
(University of Oxford)
Further Information

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

The integral of mean curvature squared is a conformal invariant that measures the distance from a given immersion to the standard embedding of a round sphere. Following work of Robert Bryant who showed that all Willmore spheres in the 3-sphere are conformally minimal, Robert Kusner proposed in the early 1980s to use the Willmore energy to obtain an “optimal” sphere eversion, called the min-max sphere eversion.

We will present a method due to Tristan Rivière that permits to tackle a wide variety of min-max problems, including ones about the Willmore energy. An important step to solve Kusner’s conjecture is to determine the Morse index of branched Willmore spheres, and we show that the Morse index of conformally minimal branched Willmore spheres is equal to the index of a canonically associated matrix whose dimension is equal to the number of ends of the dual minimal surface.

Thu, 10 Jun 2021

17:00 - 18:00
Virtual

### Simple motion of stretch-limited elastic strings

Casey Rodriguez
(MIT)
Further Information

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

Elastic strings are among the simplest one-dimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (tensile force produces elongation) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible), and after a certain amount of tensile force is applied the stretch of the string is maximized (becomes inextensible). In this talk, we discuss a model for these stretch-limited elastic strings, in what way they model elastic behavior, the well-posedness and asymptotic stability of certain simple motions, and (many) open questions.

Thu, 13 May 2021

12:00 - 13:00
Virtual

### Deep Neural Networks for High-Dimensional PDEs in Stochastic Control and Games

Yufei Zhang
(Oxford University)
Further Information

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

In this talk, we discuss the feasibility of algorithms based on deep artificial neural networks (DNN) for the solution of high-dimensional PDEs, such as those arising from stochastic control and games. In the first part, we show that in certain cases, DNNs can break the curse of dimensionality in representing high-dimensional value functions of stochastic control problems. We then exploit policy iteration to reduce the associated nonlinear PDEs into a sequence of linear PDEs, which are then further approximated via a multilayer feedforward neural network ansatz. We establish that in suitable settings the numerical solutions and their derivatives converge globally, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration. Numerical experiments on Zermelo's navigation problem and on consensus control of interacting particle systems are presented to demonstrate the effectiveness of the method. This is joint work with Kazufumi Ito, Christoph Reisinger and Wolfgang Stockinger.

Thu, 11 Mar 2021

12:00 - 13:00
Virtual

### Regularity for non-uniformly elliptic equations

Mathias Schäffner
(Technische Universität Dortmund)
Further Information

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

I will discuss regularity properties for solutions of linear second order non-uniformly elliptic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results due to Trudinger from the 1970s.

As an application of the local boundedness result, we deduce a quenched invariance principle for random walks among random degenerate conductances. If time permits I will discuss further regularity results for nonlinear non-uniformly elliptic variational problems.

Thu, 25 Feb 2021

12:00 - 13:00
Virtual

### Homogenization in randomly perforated domains

Arianna Giunti
(Imperial College London)
Further Information

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

We consider the homogenization of a Stokes system in a domain having many small random holes. This model mainly arises from problems of solid-fluid interaction (e.g. the flow of a viscous and incompressible fluid through a porous medium). We aim at the rigorous derivation of the homogenization limit both in the Brinkmann regime and in the one of Darcy’s law. In particular, we focus on holes that are distributed according to probability measures that allow for overlapping and clustering phenomena.

Thu, 18 Feb 2021

17:00 - 18:00
Virtual

### Quantitative inviscid limits and universal shock formation in scalar conservation laws

Cole Graham
(Stanford University)
Further Information

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact Benjamin Fehrman.

Abstract

We explore one facet of an old problem: the approximation of hyperbolic conservation laws by viscous counterparts. While qualitative convergence results are well-known, quantitative rates for the inviscid limit are less common. In this talk, we consider the simplest case: a one-dimensional scalar strictly-convex conservation law started from "generic" smooth initial data. Using a matched asymptotic expansion, we quantitatively control the inviscid limit up to the time of first shock. We conclude that the inviscid limit has a universal character near the first shock. This is joint work with Sanchit Chaturvedi.

Thu, 04 Feb 2021

12:00 - 13:00
Virtual

### Interacting particle systems and phase transitions

(Univesity of Oxford)
Abstract

Phase transitions are present in a wide array of systems ranging from traffic to machine learning algorithms. In this talk, we will relate the concept of phase transitions to the convexity properties of the associated thermodynamic energy. Motivated by noisy stochastic gradient descent in supervised learning, we will consider the problem of understanding the thermodynamic limit of exchangeable weakly interacting diffusions (AKA propagation of chaos) from an energetic perspective. The strategy will be to exploit the 2-Wasserstein gradient flow structure associated with the thermodynamic energy in the infinite particle setting. Using this perspective, we will show how the convexity properties of the thermodynamic energy affects the homogenization limit or the stability of the log-Sobolev inequality.

Thu, 21 Jan 2021

12:00 - 13:00
Virtual

### Numerical analysis of a topology optimization problem for Stokes flow / Estimates and regularity for a class of augmented Hessian equations, and a fully nonlinear generalisation of the Yamabe problem

(University of Oxford)
Abstract

A topology optimization problem for Stokes flow finds the optimal material distribution of a fluid in Stokes flow that minimizes the fluid’s power dissipation under a volume constraint. In 2003, T. Borrvall and J. Petersson [1] formulated a nonconvex optimization problem for this objective. They proved the existence of minimizers in the infinite-dimensional setting and showed that a suitably chosen finite element method will converge in a weak(-*) sense to an unspecified solution. In this talk, we will extend and refine their numerical analysis. In particular, we will show that there exist finite element functions, satisfying the necessary first-order conditions of optimality, that converge strongly to each isolated local minimizer of the problem.

/

Fully nonlinear PDEs involving the eigenvalues of matrix-valued differential operators (such as the Hessian) have been the subject of intensive study over the last few decades, since the seminal work of Caffarelli, Kohn, Nirenberg and Spruck. In this talk I will discuss some recent joint work with Luc Nguyen on the regularity theory for a large class of these equations, with a particular emphasis on a special case known as the sigma_k-Yamabe equation, which arises in conformal geometry.

[1] T. Borrvall, J. Petersson, Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77–107. doi:10.1002/fld.426.

Thu, 19 Nov 2020
12:00
Virtual

### Explicit bounds for the generation of a lift force exerted by steady-state Navier-Stokes flows over a fixed obstacle

Ph.D. Gianmarco Sperone
(Charles University in Prague)
Abstract

We analyze the steady motion of a viscous incompressible fluid in a two- and three-dimensional channel containing an obstacle through the Navier-Stokes equations under different types of boundary conditions. In the 2D case we take constant non-homogeneous Dirichlet boundary data in a (virtual) square containing the obstacle, and emphasize the connection between the appearance of lift and the unique solvability of Navier-Stokes equations. In the 3D case we consider mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet. In the absence of external forcing, explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity. In the 3D case, this solenoidal extension is built through the Bogovskii operator and explicit bounds on its Dirichlet norm (in terms of the geometric parameters of the obstacle) are found by solving a variational problem involving the infinity-Laplacian.

The talk accounts for results obtained in collaboration with Filippo Gazzola and Ilaria Fragalà (both at Politecnico di Milano).

Thu, 05 Nov 2020
12:00
Virtual

### A bi-fidelity method for multi-scale kinetic models with uncertain parameters

Prof. Liu Liu
(The Chinese University of Hong Kong)
Abstract

Solving kinetic or related models with high-dimensional random parameters has been a challenging problem. In this talk, we will discuss how to employ the bi-fidelity stochastic collocation and choose efficient low-fidelity models in order to solve a class of multi-scale kinetic equations with uncertainties, including the Boltzmann equation, linear transport and the Vlasov-Poisson equation. In addition, some error analysis for the bi-fidelity method based on these PDEs will be presented. Finally, several numerical examples are shown to validate the efficiency and accuracy of the proposed method.

Thu, 22 Oct 2020
12:00
Virtual

### A nonlinear open mapping principle, with applications to the Jacobian determinant / A general nonlinear mapping theorem and applications to the incompressible Euler equations

André Guerra / Lukas Koch
(University of Oxford)
Abstract

I will present a nonlinear version of the open mapping principle which applies to constant-coefficient PDEs which are both homogeneous and weak* stable. An example of such a PDE is the Jacobian equation. I will discuss the consequences of such a result for the Jacobian and its relevance towards an answer to a long-standing problem due to Coifman, Lions, Meyer and Semmes. This is based on joint work with Lukas Koch and Sauli Lindberg.

/

I present a general nonlinear open mapping principle suited to applications to scale-invariant PDEs in regularity regimes where the equations are stable under weak* convergence. As an application I show that, for any $p < \infty$, the set of initial data for which there are dissipative weak solutions in $L^p_t L^2_x$ is meagre in the space of solenoidal L^2 fields. This is based on joint work with A. Guerra (Oxford) and S. Lindberg (Aalto).

Thu, 15 Oct 2020
12:00
Virtual

### (Non-)unique limits of geometric flows / The Landau equation as a gradient flow

James Kohout / Jeremy Wu
(University of Oxford)
Abstract

In the study of geometric flows it is often important to understand when a flow which converges along a sequence of times going to infinity will, in fact, converge along every such sequence of times to the same limit. While examples of finite dimensional gradient flows that asymptote to a circle of critical points show that this cannot hold in general, a positive result can be obtained in the presence of a so-called Lojasiewicz-Simon inequality. In this talk we will introduce this problem of uniqueness of asymptotic limits and discuss joint work with Melanie Rupflin and Peter M. Topping in which we examined the situation for a geometric flow that is designed to evolve a map describing a closed surface in a given target manifold into a parametrization of a minimal surface.

/

The Landau equation is an important PDE in kinetic theory modelling plasma particles in a gas. It can be derived as a limiting process from the famous Boltzmann equation. From the mathematical point of view, the Landau equation can be very challenging to study; many partial results require, for example, stochastic analysis as well as a delicate combination of kinetic and parabolic theory. The major open question is uniqueness in the physically relevant Coulomb case. I will present joint work with Jose Carrillo, Matias Delgadino, and Laurent Desvillettes where we cast the Landau equation as a generalized gradient flow from the optimal transportation perspective motivated by analogous results on the Boltzmann equation. A direct outcome of this is a numerical scheme for the Landau equation in the spirit of de Giorgi and Jordan, Kinderlehrer, and Otto. An extended area of investigation is to use the powerful gradient flow techniques to resolve some of the open problems and recover known results.

Thu, 18 Jun 2020
12:00
Virtual

### A variational approach to fluid-structure interactions

Sebastian Schwarzacher
(Charles University in Prague)
Abstract

I introduce a recently developed variational approach for hyperbolic PDE's. The method allows to show the existence of weak solutions to fluid-structure interactions where a visco-elastic bulk solid is interacting with an incompressible fluid governed by the unsteady Navier Stokes equations. This is a joint work with M. Kampschulte and B. Benesova.

Thu, 11 Jun 2020
12:00
Virtual

### On dynamic slip boundary condition

Erika Maringova
(Vienna University of Technology)
Abstract

In the talk, we study the Navier–Stokes-like problems for the flows of homogeneous incompressible fluids. We introduce a new type of boundary condition for the shear stress tensor, which includes an auxiliary stress function and the time derivative of the velocity. The auxiliary stress function serves to relate the normal stress to the slip velocity via rather general maximal monotone graph. In such way, we are able to capture the dynamic response of the fluid on the boundary. Also, the constitutive relation inside the domain is formulated implicitly. The main result is the existence analysis for these problems.

Thu, 28 May 2020
15:00
Virtual

### Boundary regularity of area-minimizing currents: a linear model with analytic interface

Zihui Zhao
(University of Chicago)
Abstract

Given a curve , what is the surface  that has smallest area among all surfaces spanning ? This classical problem and its generalizations are called Plateau's problem. In this talk we consider area minimizers among the class of integral currents, or roughly speaking, orientable manifolds. Since the 1960s a lot of work has been done by De Giorgi, Almgren, et al to study the interior regularity of these minimizers. Much less is known about the boundary regularity, in the case of codimension greater than 1. I will speak about some recent progress in this direction.

Thu, 14 May 2020

12:00 - 13:00
Virtual

### Augmented systems and surface tension

Prof. Didier Bresch
(Savoie University)
Abstract

In this talk, I will present different PDE models involving surface tension where it may be efficient to consider augmented versions.

Thu, 07 May 2020

12:00 - 13:00
Virtual

### Vectorial problems: sharp Lipschitz bounds and borderline regularity

Cristiana De FIlippis
(University of Oxford)
Abstract

Non-uniformly elliptic functionals are variational integrals like
$(1) \qquad \qquad W^{1,1}_{loc}(\Omega,\mathbb{R}^{N})\ni w\mapsto \int_{\Omega} \left[F(x,Dw)-f\cdot w\right] \, \textrm{d}x,$
characterized by quite a wild behavior of the ellipticity ratio associated to their integrand $F(x,z)$, in the sense that the quantity
$$\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}}\mathcal R(z, B):=\sup_{\substack{x\in B \\ B\Subset \Omega \ \small{\mbox{open ball}}}} \frac{\mbox{highest eigenvalue of}\ \partial_{z}^{2} F(x,z)}{\mbox{lowest eigenvalue of}\ \partial_{z}^{2} F(x,z)}$$
may blow up as $|z|\to \infty$.
We analyze the interaction between the space-depending coefficient of the integrand and the forcing term $f$ and derive optimal Lipschitz criteria for minimizers of (1). We catch the main model cases appearing in the literature, such as functionals with unbalanced power growth or with fast exponential growth such as
$$w \mapsto \int_{\Omega} \gamma_1(x)\left[\exp(\exp(\dots \exp(\gamma_2(x)|Dw|^{p(x)})\ldots))-f\cdot w \right]\, \textrm{d}x$$
or
$$w\mapsto \int_{\Omega}\left[|Dw|^{p(x)}+a(x)|Dw|^{q(x)}-f\cdot w\right] \, \textrm{d}x.$$
Finally, we find new borderline regularity results also in the uniformly elliptic case, i.e. when
$$\mathcal{R}(z,B)\sim \mbox{const}\quad \mbox{for all balls} \ \ B\Subset \Omega.$$

The talk is based on:
C. De Filippis, G. Mingione, Lipschitz bounds and non-autonomous functionals. $\textit{Preprint}$ (2020).