C6

We will talk about our recent results on the uniqueness of regular reflection solutions for the potential flow equation in a natural class of self-similar solutions. The approach is based on a nonlinear version of method of continuity. An important property of solutions for the proof of uniqueness is the convexity of the free boundary.

There was major progress on perfect graphs in the early 2000's: Chudnovsky, Robertson, Thomas and I proved the ``strong perfect graph theorem'' that a graph is perfect if and only if it has no odd hole or odd antihole; and Chudnovsky, Cornuejols, Liu, Vuscovic and I found a polynomial-time algorithm to test whether a graph has an odd hole or odd antihole, and thereby test if it is perfect. (A ``hole'' is an induced cycle of length at least four, and an ``antihole'' is a hole in the complement graph.)

What we couldn't do then was test whether a graph has an odd hole, and this has stayed open for the last fifteen years, despite some intensive effort. I am happy to report that in fact it can be done in poly-time (in time O(|G|^{12}) at the last count), and in this talk we explain how.

Joint work with Maria Chudnovsky, Alex Scott, and Sophie Spirkl.

The Erdos-Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs.

1. A theorem of R\"odl asserts that if an n-vertex graph is non-universal then it contains an almost homogeneous set (i.e one with edge density either very close to 0 or 1) of size \Omega(n). We prove that if a 3-uniform hypergraph is non-universal then it contains an almost homogeneous set of size \Omega(log n). An example of R\"odl from 1986 shows that this bound is tight.

2. Let R_r(t) denote the size of the largest non-universal r-graph G so that neither G nor its complement contain a complete r-partite subgraph with parts of size t. We prove an Erd\H{o}s--Hajnal-type stepping-up lemma, showing how to transform a lower bound for R_r(t) into a lower bound for R_{r+1}(t). As an application of this lemma, we improve a bound of Conlon-Fox-Sudakov by showing that R_3(t) \geq t^{ct).

Joint work with M. Amir and A. Shapira

We solve the construction of the turbulent two point functions in the following manner:

A mathematical theory, based on a few physical laws and principles, determines the construction of the turbulent two point function as the expectation value of a statistically defined random field. The random field is realized via an infinite induction, each step of which is given in closed form.

Some version of such models have been known to physicists for some 25 years. Our improvements are two fold:

1. Some details in the reasoning appear to be missing and are added here.
2. The mathematical nature of the algorithm, difficult to discern within the physics presentation, is more clearly isolated.

Because the construction is complex, usable approximations, known as surrogate models, have also been developed.

The importance of these results lies in the use of the two point function to improve on the subgrid models of Lecture I.

We also explain limitations. For the latter, we look at the deflagration to detonation transition within a type Ia supernova and decide that a completely different methodology is recommended. We propose to embed multifractal ideas within a physics simulation package, rather than attempting to embed the complex formalism of turbulent deflagration into the single fluid incompressible model of the two point function. Thus the physics based simulation model becomes its own surrogate turbulence model.

We discuss three methods for the simulation of turbulent fluids. The issue we address is not the important issue of numerical algorithms, but the even more basic question of the equations to be solved, otherwise known as the turbulence model.  These equations are not simply the Navier-Stokes equations, but have extra, turbulence related terms, related to turbulent viscosity, turbulent diffusion and turbulent thermal conductivity. The extra terms are not “standard textbook” physics, but are hypothesized based on physical reasoning. Here we are concerned with these extra terms.

The many models, divided into broad classes, differ greatly in

Dependence on data
Complexity
Purpose and usage

For this reason, each of the classes of models has its own rationale and domain of usage.

We survey the landscape of turbulence models.

Given a turbulence model, we ask: what is the nature of convergence that a numerical algorithm should strive for? The answer to this question lies in an existence theory for the Euler equation based on the Kolmogorov 1941 turbulent scaling law, taken as a hypothesis (joint work with G-Q Chen).

When studying a systems of conservation laws in several space dimensions, A. Bressan conjectured that the flows $X^n(t)$ generated by a smooth vector fields $\mathbf b^n(t,x)$,
\[
\frac{d}{dt} X^n(t,y) = \mathbf b^n(t,X(t,y)),
\]
are compact in $L^1(I\!\!R^d)$ for all $t \in [0,T]$ if $\mathbf b^n \in L^\infty \cap \mathrm{BV}((0,t) \times I\!\!R^d)$ and they are nearly incompressible, i.e.
\[
\frac{1}{C} \leq \det(\nabla_y X(t,y)) \leq C
\]
for some constant $C$. This conjecture is implied by the uniqueness of the solution to the linear transport equation
\[
\partial_t \rho + \mathrm{div}_x(\rho \mathbf b) = 0, \quad \rho \in L^\infty((0,T) \times I\!\!R^d),
\]
and it is the natural extension of a series of results concerning vector fields $\mathbf b(t,x)$ with Sobolev regularity.

We will give a general framework to approach the uniqueness problem for the linear transport equation and to prove Bressan's conjecture.

Abstract: In 2016 Ayyer, Prasad and Spallone proved that the restriction to 
S_{n-1} of any odd degree irreducible character of S_n has a unique irreducible 
constituent of odd degree.
This result was later generalized by Isaacs, Navarro Olsson and Tiep.
In this talk I will survey some recent developments on this topic.

We consider a vector-valued function $f: \mathbb{R}_+ \to X$ which is locally of bounded variation and give a decay rate for $|A(t)|$ for increasing $t$ under certain conditions on the Laplace-Stieltjes transform $\widehat{dA}$ of $A$. For this, we use a Tauberian condition inspired by the work of Ingham and Karamata and a contour integration method invented by Newman. Our result is a generalisation of already known Tauberian theorems for bounded functions and is applicable to Dirichlet series. We will say something about the connection between the obtained decay rates and number theory.

We consider families of fast-slow skew product maps of the form \begin{align*}x_{n+1}   = x_n+\eps^2 a_\eps(x_n,y_n)+\eps b_\eps(x_n)v_\eps(y_n), \quad

y_{n+1}   = T_\eps y_n, \end{align*} where $T_\eps$ is a family of nonuniformly expanding maps, $v_\eps$ is of mean zero and the slow variables $x_n$ lie in $\R^d$.  Under an exactness assumption on $b_\eps$ (automatically satisfied in the cases $d=1$ and $b_\eps\equiv I_d$), we prove convergence of the slow variables to a limiting stochastic differential equation (SDE) as $\eps\to0$.   Our results include cases where the family of fast dynamical systems

$T_\eps$ consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.Similar results are obtained also for continuous time systems  \begin{align*} \dot x   =  \eps^2 a_\eps(x,y,\eps)+\eps b_\eps(x)v_\eps(y), \quad \dot y   =  g_\eps(y). \end{align*}

Here, as in classical Wong-Zakai approximation, the limiting SDE is of Stratonovich type $dX=\bar a(X)\,dt+b_0(X)\circ\,dW$ where $\bar a$ is the average of $a_0$

and $W$ is a $d$-dimensional Brownian motion.