Noise is generated in an aerodynamic setting when flow turbulence encounters a structural edge, such as at the sharp trailing edge of an aerofoil. The generation of this noise is unavoidable, however this talk addresses various ways in which it may be mitigated through altering the design of the edge. The alterations are inspired by natural silent fliers: owls. A short review of how trailing-edge noise is modelled will be given, followed by a discussion of two independent adaptations; serrations, and porosity. The mathematical impacts of the adaptations to the basic trailing-edge model will be presented, along with the physical implications they have on noise generation and control.

Lecture notes and the manuscript for Lecture 2.

Abstract, lecture notes and the manuscript for Lecture 1

References
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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.
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[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.