Courant Institute

ife forms have devised impressive and subtle ways to exploit fluid flows. I’ll talk about birds as flying machines whose behaviors can give surprising insights into flow physics. One story explains how flocking interactions can help to bring flapping flyers into orderly formations. A second story involves the more subtle role of aerodynamics in the highly efficient breathing of birds, which is thought to be critical to their ability to fly.

This is joint work with Paul Bourgade and Benjamin McKenna (Courant Institute, NYU).
The elastic manifold is a paradigmatic representative of the class of disordered elastic systems. These models describe random surfaces with rugged shapes resulting from a competition between random spatial impurities (preferring disordered configurations), on the one hand, and elastic self-interactions (preferring ordered configurations), on the other. The elastic manifold model is interesting because it displays a depinning phase transition and has a long history as a testing ground for new approaches in statistical physics of disordered media, for example for fixed dimension by Fisher (1986) using functional renormalization group methods, and in the high-dimensional limit by Mézard and Parisi (1992) using the replica method.
We study the topology of the energy landscape of this model in the Mézard-Parisi setting, and compute the (annealed) topological complexity both of total critical points and of local minima. Our main result confirms the recent formulas by Fyodorov and Le Doussal (2020) and allows to identify the boundary between simple and glassy phases. The core argument relies on the analysis of the asymptotic behavior of large random determinants in the exponential scale.

Dobrushin (1972) showed that, at low enough temperatures, the interface of the 3D Ising model - the random surface separating the plus and minus phases above and below the $xy$-plane - is localized: it has $O(1)$ height fluctuations above a fixed point, and its maximum height $M_n$ on a box of side length $n$ is $O_P(\log n)$. We study this interface and derive a shape theorem for its "pillars" conditionally on reaching an atypically large height. We use this to analyze the maximum height $M_n$ of the interface, and prove that at low temperature $M_n/\log n$ converges to $c\beta$ in probability. Furthermore, the sequence $(M_n - E[M_n])_{n\geq 1}$ is tight, and even though this sequence does not converge, its subsequential limits satisfy uniform Gumbel tails bounds.
Joint work with Reza Gheissari.

We explore a specific system in which geometry and loading conspire to generate fine-scale wrinkling. This system -- a twisted ribbon held with small tension -- was examined experimentally by Chopin and Kudrolli 
[Phys Rev Lett 111, 174302, 2013].

There is a regime where the ribbon wrinkles near its center. A recent paper by Chopin, D\'{e}mery, and Davidovitch models this regime using a von-K\'{a}rm\'{a}n-like 
variational framework [J Elasticity 119, 137-189, 2015]. Our contribution is to give upper and lower bounds for the minimum energy as the thickness tends to zero. Since the bounds differ by a thickness-independent prefactor, we have determined how the minimum energy scales with thickness. Along the way we find estimates on Sobolev norms of the minimizers, which provide some information on the character of the wrinkling. This is a joint work with  Robert V. Kohn in Courant Institute, NYU.

Some results of R.Harvey and B.Lawson on the Dirichlet problem for a class of fully nonlinear elliptic equations will be presented.

No background is required; the talk will be expository.