Lecture notes and the manuscript for Lecture 2.

Abstract, lecture notes and the manuscript for Lecture 1

References
[1] F. Bethuel, H. Brezis, F. Helein, Ginzburg-Landau vortices, Birkhauser, Boston, 1994.
[2] H. Brezis, L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries,
Selecta Math. (N.S.) 1 (1995), 197{263.
[3] R. Ignat, R.L. Jerrard, Renormalized energy between vortices in some Ginzburg-Landau models
on 2-dimensional Riemannian manifolds, Arch. Ration. Mech. Anal. 239 (2021), 1577{1666.
[4] R. Ignat, L. Nguyen, V. Slastikov, A. Zarnescu, On the uniqueness of minimisers of Ginzburg-
Landau functionals, Ann. Sci. Ec. Norm. Super. 53 (2020), 589{613.
[5] R.L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math. Anal.
30 (1999), 721-746.
[6] R.L. Jerrard, H.M. Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. PDE 14
(2002), 151-191.
[7] E. Sandier, Lower bounds for the energy of unit vector elds and applications J. Funct. Anal.
152 (1998), 379-403.
[8] E. Sandier, S. Serfaty, Vortices in the magnetic Ginzburg-Landau model, Birkhauser, 2007.

The Atiyah-Floer conjecture asserts the instanton Floer homology of a closed three-manifold (constructed via gauge theory) is isomorphic to the Lagrangian Floer homology of a pair of Lagrangian submanifolds associated to a splitting of the three manifold (constructed via symplectic geometry). This conjecture has remained open for more than three decades. In this talk I will explain two compactness results for the SO(3) case of the conjecture in the neck-stretching process. One result is related to the construction of a natural bounding chain in the Lagrangian Floer theory and a conjecture of Fukaya.

I'll be presenting my PhD work, in which I define two new algebraic structures on the equivariant symplectic cohomology of a convex symplectic manifold. The first is a collection of shift operators which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus T, we assign to a cocharacter of T an endomorphism of (S1 × T)-equivariant Floer cohomology based on the equivariant Floer Seidel map. The second is a connection which is a multivariate version of Seidel’s q-connection on S1 -equivariant Floer cohomology and generalises the Dubrovin connection on equivariant quantum cohomology.

We will discuss recent developments of the theory of a-contraction with shifts to study the stability of discontinuous solutions of systems of equations modeling inviscid compressible flows, like the compressible Euler equation.

In the one dimensional configuration, the Bressan theory shows that small BV solutions are stable under small BV perturbations (together with a technical condition known as bounded variations on space-like curve).

The theory of a-contraction allows to extend the Bressan theory to a weak/BV stability result allowing wild perturbations fulfilling only the so-called strong trace property. Especially, it shows that the technical condition of BV on space-like curve is not needed. (joint work with Sam Krupa and Geng Chen). 

We will show several applications of the theory of a-contraction with shifts on the barotropic Navier-Stokes equation. Together with Moon-Jin Kang and Yi wang, we proved the conjecture of Matsumura (first mentioned in 1986). It consists in proving the time asymptotic stability of composite waves of viscous shocks and rarefactions. 

Together with Moon-Jin Kang, we proved also that inviscid shocks of the Euler equation, are stable among the family of inviscid limits of Navier-Stokes equation (Inventiones 2021). This stability result holds in the class of wild perturbations of inviscid limits, without any regularity restriction (not even strong trace property). This shows that the class of inviscid limits of Navier-Stokes equations is better behaved that the class of weak solutions to the inviscid limit problem.

This is obtained thanks to a stability result at the level of Navier-Stokes, which is uniform with respect to the viscosity, allowing asymptotically infinitely large perturbations (JEMS 2021).

A first multi D result of stability of contact discontinuities without shear, in the class of inviscid limit of Fourier-Navier-Stokes, shows that the same property is true for some situations even in multi D (joint work with Moon-jin Kang and Yi Wang). 

A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. In previous works, we have established stable generic shock formation for the compressible Euler system, showing that at the first singularity the solution has precisely C^{1/3} Holder regularity, a so-called preshock. The focus of this talk is a complete space-time description of the solution after this initial singularity. We show that three surfaces of discontinuity emerge simultaneously and instantaneously from the preshock: the classical shock discontinuity that propagates by the Rankine–Hugoniot conditions, together with two distinct surfaces in space-time, along which C^{3/2} cusp singularities form.

We study the minimization of functionals of the form $$ u  \mapsto \int_\Omega  f(\nabla u) \, dx $$

with a convex integrand $f$ of linear growth (such as the area integrand), among all functions in the Sobolev space W$^{1,1}$ with prescribed boundary values. Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and might in fact fail, as it is well-known already for the non-parametric minimal surface problem. In such cases, the functional is extended suitably to the space BV of functions of bounded variation via relaxation, and for the relaxed functional one can in turn guarantee the existence of minimizers. However, in contrast to the original minimization problem, these BV minimizers might in principle have interior jump discontinuities or not attain the prescribed boundary values.

After a short introduction to the problem I want to focus on the question of regularity of BV minimizers. In past years, Sobolev regularity was established provided that the lack of ellipticity -- which is always inherent for such linear growth integrands -- is mild, while, in general, only some structure results seems to be within reach. In this regard, I will review several results which were obtained in cooperation with Miroslav Bulíček (Prague), Franz Gmeineder (Bonn), Erika Maringová (Vienna), and Thomas Schmidt (Hamburg).

Numerical relativity allows us to simulate the behaviour of regions of space and time where gravity is strong and dynamical. For example, it allows us to calculate precisely the gravitational waveform that should be generated by the merger of two inspiralling black holes. Since the first detection of gravitational waves from such an event in 2015, banks of numerical relativity “templates” have been used to extract further information from noisy data streams. In this talk I will give an overview of the field - what are we simulating, why, and what are the main challenges, past and future.

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