| Date | Speaker | Title/Abstract |
|---|---|---|
| 6 Feb | Hardy Chan ETH Zürich |
Fractional elliptic gluing We consider nonlocal analogues of elliptic PDEs arising from phase transition and conformal geometry. First, we adapt the infinite dimensional Lyapunov--Schmidt reduction method to the fractional Laplacian with exponent s ∈ (½,1) and construct solutions of the fractional Allen--Cahn equation vanishing near a deformed catenoid. The novelty lies in the computation with the fractional Fermi coordinates and a localized inner-outer gluing scheme. This is a joint work with Yong Liu and Juncheng Wei. Next, we construct singular solutions to the fractional Yamabe problem (with s ∈ (0,1)) that blow up exactly on submanfiolds of dimension k∈(0,(n-2s)/2). A crucial ingredient is the almost explicit inverse of fractional Hardy operators. This is a joint work with Weiwei Ao, Azahara DelaTorre, Marco Fontelos, María del Mar González and Juncheng Wei. Recent progress on related problems are also discussed. |
| 13 Feb | No seminar |
|
| 20 Feb in 6W1.2 |
Roger Moser University of Bath |
The ∞-elastica problem Consider curves in a Euclidean space with fixed length, fixed end points, and fixed tangents at the end points. Suppose that we wish to minimise the (possibly weighted) L∞-norm of the curvature under these constraints. Even though the underlying functional has no Gateaux derivative anywhere, it turns out that there is a natural way to define critical points, which are characterised by a system of differential equations. Analysing the equations, we can further classify the solutions. |
| 27 Feb | Aris Daniilidis University of Chile |
Detecting and controlling the size of critical values: from Classical to Nonsmooth Analysis In the beginning of the talk I will survey classical results concerning the shape and size of critical points/values from the optimization viewpoint. I will progressively move from the smooth case to the case of Lipschitz continuous functions, where tools of nonsmooth analysis, such as the Clarke subdifferential, come into play. This brings us naturally to consider the notion of nonsmooth criticality. I will first explain, at a general level, how a structural assumptions on the function, such as being semi-algebraic, or being a continuous selection over a finite or countable family of smooth functions, ensures powerful results on the size of generalized critical values. I will also discuss theoretical results on how to measure pathological situations, involving Baire-categoric arguments or purely algebraic tools. If time allows, I will discuss on the efficiency of (nonsmooth) first-order methods to detect (generalized) critical points. The talk is built upon a series of works obtained in collaboration with Barbet, Bolte, Dambrine, Drusviatskiy, Ley, Lewis, Mazet, Rifford, Shiota. |
| 5 Mar | Juncheng Wei University of British Columbia |
Gross-Pitaevski, Kadomtsev-Petviashvili and Adler-Moser In this talk, I will discuss the connections between Gross-Pitaevski equation, Kadomtsev-Petviashvili I equation and Adler-Moser polynomials. |
| 12 Mar | Antonin Monteil University of Bristol |
Ginzburg-Landau relaxation for harmonic maps valued into manifolds We will look at the classical problem of minimizing the Dirichlet energy of a map u :Ω⊂ R2 → N valued into a compact Riemannian manifold N and subjected to a Dirichlet boundary condition u=γ on ∂ Ω. It is well known that if γ has a non-trivial homotopy class in N, then there are no maps in the critical Sobolev space H1(Ω,N) such that u=γ on ∂ Ω. To overcome this obstruction, a way is to rather consider a relaxed version of the Dirichlet energy leading to singular harmonic maps with a finite number of topological singularities in Ω. This was done in the 90's in a pioneering work by Bethuel-Brezis-Helein in the case N=S1, related to the Ginzburg-Landau theory. In general, we will see that minimizing the energy leads at main order to a non-trivial combinatorial problem which consists in finding the energetically best topological decomposition of the boundary map γ into minimizing geodesics in N. Moreover, we will introduce a renormalized energy whose minimizers correspond to the optimal positions of the singularities in Ω. |
| 19 Mar | University of Bath |
The behaviour of sparse regularization using the Lasso method is well understood when dealing with discretized linear models. However, the behaviour of Lasso is poor when dealing with models with very large parameter spaces and in recent years, there has been much interest in the use of “off-the-grid” approaches, using a continuous parameter space in conjunction with a convex optimization problem over Radon measures. I will present some recent results which explain the behaviour of this method in arbitrary dimensions. Some highlights include the use of the Fisher metric to study the performance of Blasso over general domains [1] and the geometry of super-resolution in the multivariate setting [2]. This is joint work with Nicolas Keriven and Gabriel Peyré. Bibliography: [1] C. Poon, N. Keriven, G. Peyré. Support Localization and the Fisher Metric for off-the-grid Sparse Regularization. In Proc. AISTATS'19, 2019. [2] C. Poon, G. Peyré. Multi-dimensional Sparse Super-resolution. SIAM Journal on Mathematical Analysis, 51(1), pp. 1–44, 2019. [3] C. Poon, G. Peyré. Degrees of freedom for off-the-grid sparse estimation. Arxiv, 2019. |
| 26 Mar | London School of Economics |
Given a linear, constant coefficient partial differential equation in Rd+1, where one independent variable plays the role of ‘time’, a distributional solution is called a null solution if its past is zero. Motivated by physical considerations, we consider distributional solutions that are tempered in the spatial directions alone (and do not impose any restriction in the time direction). Considering such spatially tempered distributional solutions, we give an algebraic-geometric characterization, in terms of the polynomial describing the PDE at hand, for the null solution space to be trivial (that is, consisting only of the zero distribution). |
| 2 Apr | Heriot-Watt University Edinburgh |
|
| 23 Apr | Pierre Raphael University of Cambridge |
On blow up for the defocusing nls and three dimensional
viscous compressible fluids [Via zoom] Global existence and scattering for the defocusing nonlinear Schrodinger equation is a celebrated result by Ginibre-Velo in the early 80’s in the strictly energy sub critical case, and Bourgain in 94 in the energy critical case. In the energy super critical setting, the defocusing energy is conserved and controls the energy norm, but this is too weak to conclude to global existence which yet had been conjectured by many and confirmed by numerical computations. This is a canonical super critical problem which typically arises similarly in fluid mechanics, and there global existence is either completely open or a direct consequence of the existence of additional conservation laws. In this talk based on recent joint works with Merle (IHES), Rodnianski (Princeton) and Szeftel (Paris Sorbonne), I will describe the construction of new smooth and finite energy highly oscillatory blow up solutions for the defocusing NLS in suitable energy super critical regimes, and explain how these new bubbles are connected to the also new description of implosion mechanisms for viscous three dimensional compressible fluids. |
| 30 Apr | Michele Coti Zelati Imperial College London |
Inviscid damping and enhanced dissipation in 2d fluids [Via zoom] We review some recent results on the asymptotic stability of stationary solutions to the two-dimensional Euler and Navier-Stokes equations of incompressible flows. In many cases, sharp decay rates for the linearized problem imply some sort of nonlinear asymptotic stability, both in the Euler equations (through the so-called inviscid damping) and the Navier-Stokes equations (undergoing enhanced dissipation). However, we will see that in the case of the 2D square periodic domain, the so-called Kolmogorov flow exhibits much more complex behavior: in particular, linear asymptotic stability holds, while nonlinear asymptotic stability is not true even for analytic perturbations. |
| Date | Speaker | Title/Abstract |
|---|---|---|
| 3 Oct | Yuxia Guo Tsinghua |
Non-degeneracy of Multi-bubbling Solutions We consider the following prescribed scalar curvature equations in RN:-Δu = K(|y|))u 2*-2, u > 0 in RN, u ∈ D1,2(RN); where K(r) is a positive function, 2*=2N/(N-2). We first prove a non-degeneracy result for the positive multi-bubbling solutions for the above equation by using the local Pohozaev identities. Then we use this non-degeneracy result to glue together bubbles with different concentration rate to obatin new solutions. Joint works with M.Musso, S.Peng and S.Yan. |
| 10 Oct | Javier F Rosenblueth UNAM |
Necessity and Uniqueness of Multipliers in Constrained Optimization In mathematical programming, first and second order necessary optimality conditions are strongly related to the uniqueness of Lagrange multipliers. In particular, it is well-known that a strict form of the Mangasarian-Fromovitz constraint qualification is equivalent to the uniqueness of Kuhn-Tucker multipliers and, moreover, it implies the satisfaction of second order necessary conditions at a local minimum. In this talk we shall explain how a surprising and entirely different situation occurs in infinite dimensional problems including problems in calculus of variations and optimal control. |
| 17 Oct | Miles Wheeler University of Bath |
New exact solutions to the steady 2D Euler equations We present a large class of explicit "hybrid" equilibria for the 2D Euler equations, consisting of point vortices embedded in a smooth sea of "Stuart-type" vorticity. Mathematically, these are solutions of a singular Liouville equation with Dirac deltas on the right-hand side, together with an additional constraint at each singularity guaranteeing that corresponding point vortex is stationary. This is joint work with Vikas Krishnamurthy, Darren Crowdy, and Adrian Constantin. |
| 24 Oct | Juan Diego Davila University of Bath |
Helicoidal vortex filaments in the 3-dimensional Ginzburg-Landau equation We construct a family of entire solutions of the 3D Ginzburg-Landau equation with vortex lines given by interacting helices, with degree one around each filament and total degree an arbitrary positive integer. The existence of these solutions was conjectured by del Pino and Kowalczyk (2008), and answers negatively a question of Brezis analogous to the the Gibbons conjecture for the Allen-Cahn equation. This is joint work with Manuel del Pino, María Medina and Remy Rodiac. |
| 31 Oct | No seminar |
No seminar |
| 7 Nov | Ben Pooley Warwick/Bath |
Asymptotics for a regularised local induction model and non-conservation of filaments in passive/active scalar systems We begin by discussing asymptotics for a the evolution of vortex filaments under a regularised Biot-Savart law, in the context of the binormal curvature flow (a.k.a. the local induction approximation) for the 3D Euler equations. In the second part of the talk, we will construct explicit solutions to certain passive and active scalar systems that dramatically fail to conserve the dimension of filaments but where the velocity is nonetheless (weakly) divergence-free, continuous, and within the regime of Di Perna and Lions theory. This is based on recent works with Charles Fefferman and Jose Rodrigo. |
| 14 Nov | Ben Sharp University of Leeds |
Quantitative estimates on the index of a minimal hypersurface A minimal (hyper)surface is a critical point of the area functional. The second variation of area (or the Hessian of this functional) at a minimal surface corresponds to a self-adjoint elliptic operator whose spectrum is bounded from below. The index of this operator (how many negative eigenvalues it has), thus indicates how many ways we can push a minimal surface to reduce its area. For example, an equator on a 2-sphere is a closed minimal curve (a geodesic), but its index is one since we can always roll the equator up the sphere to reduce its length. We will discuss various relationships between the topology, geometry and index of minimal hypersurfaces, with a focus on a linear estimate on the first Betti number in terms of index. The talk will include some joint works with L. Ambrozio, R. Buzano and A. Carlotto. |
| 21 Nov | Aram Karakhanyan University of Edinburgh |
Finite morse index solutions of the Alt-Caffarelli problem In this talk I will introduce the stability operator for the solutions of Alt-Caffarelli problem with quasilinear elliptic operators. Using some ideas from the theory of varifolds and classical minimal surfaces we will classify the global solutions of finite Morse index in two and three dimensions. |
| 28 Nov | cancelled |
cancelled |
| 5 Dec | Antonio Fernandez University of Bath |
Some remarks on a biharmonic NLS with mixed dispersion Abstract in pdf |
| 12 Dec | No seminar |
No seminar |