# Next Talk

Speaker: Eyal Lubetzky Random walks on the random graph April 24, 1:00 p.m. (light refreshments at 12:45 p.m.) WWH 1302

#### Abstract

We study random walks on the giant component of the Erdős-Rényi random graph $G(n,p)$ where $p= lambda / n$ for $lambda > 1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(log n)$ and concentrates it (the cutoff phenomenon occurs). Joint work with N. Berestycki, Y. Peres and A. Sly.

This seminar is meant to benefit young mathematicians, particularly graduate students and postdocs.
It aims to accomplish the following:
• provide a venue for talks that young mathematicians will understand
• expose students to areas of research at the Courant Institute
The research talks should be fairly introductory and accessible to students and non-specialists in the audience.

# Schedule Spring 2015

## February 20

Speaker: Lise-Marie Imbert-Gérard The hydrid resonance of Maxwell's equations in slab geometry: bessel functions and general dissipation tensor Abstract The hybrid resonance corresponds to an infinite limit of the local wavenumber in the so-called cold plasma model, modeling wave propagation in a plasma. An appropriate limit absortpion principle is a powerful tool to understand the singular behavior of waves at the resonance. This talk will focus on two aspects of the mathematical analysis of such a regularization process: the representation of solutions by means of Bessel functions on the one hand, the fact that the resulting heating term is independent of the regulatization term on the other hand.

## February 27

Speaker: Miles Wheeler Steady water waves Abstract Having calculated the first five terms in a small-amplitude expansion of periodic traveling water waves, Stokes conjectured in 1880 that such waves could be 'pushed to the extent of yielding crests with sharp edges'. In this mostly expository talk, I will introduce the mathematical theory of steady traveling water waves, in particular the proof of Stokes' conjecture in the eighties using bifurcation theory techniques. Local bifurcation theory allows us to solve equations near points where the implicit function theorem fails, while global bifurcation theory tells us about the possible limiting behavior along branches of solutions; both will be illustrated with simple finite-dimensional examples. I will also review classical results on the symmetry, elevation, and speed of solitary waves, as well as the surprisingly useful reformulation of the steady water wave problem as an ill-posed dynamical system (with the horizontal variable playing the role of time).

## March 6

Speaker: Peter Nandori Dispersing billiards and the heat equation Abstract Dispersing billiard or Sinai billiard is one of the very few physical systems where chaos is well understood (see the laudation for Sinai's Abel prize in 2014). In the talk, we will briefly review the fascinating results of dispersing billiards. Then I will report on some endeavors for deriving heat equation from large billiard type systems in two toy models: for the distribution of mass when energy is fixed (joint work with D. Dolgopyat) and the evolution of the energy when mass is fixed (joint work with P. Balint, I.P. Toth, D. Szasz).

## March 27

Speaker: Aleksandar Donev Simulating passive and active Brownian suspensions of particles Abstract I will describe how to model Brownian suspensions of passive or active particles and rigid bodies at zero Reynolds number. I will first explain an immersed boundary approach where spherical particles are represented as regularized Stokeslets or 'blobs'. More complex rigid bodies suspended in fluid can be represented with different degrees of fidelity by enforcing a rigidity constraint on a collection of blobs resolving the body to some degree of fidelity. Thermal fluctuations and thus Brownian motion can be consistently modeled by including a fluctuating (random) stress in the momentum equation, as dictated by fluctuating hydrodynamics. Implementing all of this in practice requires some sophisticated numerical linear algebra which I will discuss in some detail.

## April 3

Speaker: Sophie Marques Group scheme theory Abstract We will give a quick introduction of group scheme theory. In particular, we will try to understand how far we get analogies with usual group theory, which will be in reality a particular case of group scheme theory. We will see that group schemes are hardly understood and full of mysteries.

## April 17

Speaker: Abtin Rahimian High-order Boundary Integral Algorithms for Soft Particles in Stokesian Flow Abstract I will present fast schemes for the simulation of suspensions of deformable particles in Newtonian fluids at vanishing Reynolds number. Soft particles are modeled as vesicles, which are locally-inextensible membranes that resist bending. Vesicles are used as basic models of cell membranes, intracellular organelles, and viral particles. The vesicles' shapes are evolved as a result of interplay between bending and tensile interfacial forces, hydrodynamic interaction between vesicles, as well as the fluid flow, which is governed by Stokes equation. Simulations of such flows require algorithms for highly stiff, nonlocal, and nonlinear interfacial forces. Semi-implicit time stepping schemes coupled with spectral representation of the surface, reparametrization scheme to avoid extreme mesh distortion, and anti-aliasing techniques enables us to overcome numerical instabilities. We will review these computational components that are essential for simulating large number of particles in dense suspensions. This is joint work with Bryan Quaife (ICES, University of Texas), Libin Lu (CIMS), Shravan K. Veerapaneni (University of Michigan), George Biros (ICES, University of Texas), and Denis Zorin (CIMS)

## April 24

Speaker: Eyal Lubetzky Random walks on the random graph Abstract We study random walks on the giant component of the Erdős-Rényi random graph $G(n,p)$ where $p= lambda / n$ for $lambda > 1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(log n)$ and concentrates it (the cutoff phenomenon occurs). Joint work with N. Berestycki, Y. Peres and A. Sly.

## May 1

Speaker: David Belius TBA Abstract TBA

# Contact Info

Aukosh Jagannathaukosh [at] cims [dot] nyu [dot] edu
Manas Rachhrachh [at] cims [dot] nyu [dot] edu
Klaus Widmayerklaus [at] cims [dot] nyu [dot] edu

## Previous semesters

### Spring 2011 schedule

Descriptions of earlier talks are here.

Department of Mathematics
Courant Institute of Mathematical Sciences
New York University
251 Mercer St.
New York, NY 10012