Two Calabi-Yau threefolds 𝑋 and 𝑌 are called mirror manifolds if type IIA string theory compactified on 𝑋 and type IIB string theory compactified on 𝑌 yield the same physical theory. We will describe a rough mathematical principle that follows from mirror duality of 𝑋 and 𝑌, and derive from this a relationship between the Hodge numbers of 𝑋 and 𝑌. This is an introductory talk, with (almost) entirely mathematical content. Knowledge of physics is emphatically not required; all buzzwords above will be either defined or ignored.

We study the regularity of the free boundary in the fully nonlinear thin obstacle problem. Prior to our work, the only known results concerned with the regularity of solutions, but nothing was known about the regularity of free boundaries for this problem. The main difficulty was the lack of monotonicity formulas for such problem. We present a new approach towards the regularity of thin free boundaries, based on a joint work with J. Serra. Our main result establishes that the free boundary is near regular points.

In the classical result of Morse theory, a smooth real-valued function with nondegenerate critical points can be used to give a handle decomposition of a compact manifold. This can be generalized to certain types of spaces with non-isolated singularites, such as complex varieties, which admit a Whitney stratification. I'll explain stratifications and present Goresky and MacPherson's fundamental theorem of stratified Morse theory, which describes the computation of local Morse data for a Whitney-stratified space.

Let S be a closed orientable surface of genus at least 2, and let G be a semisimple Lie group of non-compact type. Positively ratioed representations are Anosov representations from the fundamental group of S to G, that satisfy an additional cross ratio condition. In the this talk, I will explain what are positively ratioed representations, give examples of such representations, and explain a systolic inequality that we proved using geodesic currents. This is joint work with Giuseppe Martone.

Subwavelength apertures and holes on surfaces of metals induce strong electromagnetic field enhancement and extraordinary optical transmission. This remarkable phenomenon can lead to potentially significant applications in biological and chemical sensing, spectroscopy, and other novel optical devices. In this talk, I will present mathematical studies of the enhancement mechanism for the scattering of narrow slits perforated in a slab of perfect conductor. Both the single slit and an array of slits will be discussed. It is demonstrated that the enhancement of electromagnetic fields for a single slit can be induced either by scattering resonances or certain non-resonant effect in the quasi-static regime. We derive the asymptotic expansions of resonances and quantitatively analyze the field enhancement at resonant frequencies. The field enhancement at non-resonant frequencies in the quasi-static regime is also investigated, and it is shown that the fast transition of the magnetic field in the slit induces strong electric field enhancement. For a periodic array of slits, we show that additional enhancement mechanism arises at the Wood’s anomaly.

Schramm-Loewner Evolution (SLE) is the scaling limit for many interfaces in classical models of statistical physics (such as Ising and FK percolation), as well as the scaling limit for many objects of purely mathematical interest (for instance the loop-erased random walk and the uniform spanning tree). I will try to motivate everything first, then introduce the necessary complex analysis before talking about SLE and its properties. If time permits, I will also talk about the Gaussian Free Field, which is a Gaussian process that is a natural generalization of Brownian motion indexed by more than just one time parameter. I will also explain some natural couplings of the SLE and the GFF. A few proofs will be given in simple cases, but most of the talk will just try to convey qualitatively why these objects are important and why two recent Fields medals were awarded for work on them.

I will explain De-Giorgi's proof that solutions elliptic equations with variable coefficients are Holder continuous and explain how this was the last step needed to solve Hilbert's nineteenth problem.

Direct numerical simulation of turbulent incompressible fluid flow is impractical, even on modern supercomputers, due to the need to resolve all scales of the fluid's motion. RANS simulations are much more cost effective, but can only predict mean quantities of the fluid flow (e.g. drag, pressure) with accuracy. Large eddy simulation (LES) attempts to find a compromise between the two approaches. In this talk, we will give an intro to LES and discuss the challenges associated with the LES of flows in the presence of a solid wall.