Stony Brook University

We prove a homological mirror symmetry result for a one-parameter family of genus 2 curves (https://arxiv.org/abs/1908.04227), and then mention current joint work with H. Azam, H. Lee, and C.-C. M. Liu on generalizing this to the 6-parameter family of all genus 2 curves.

First we describe the B-model genus 2 curve in a 4-torus and the geometric construction of the generalized SYZ mirror. Then we set up the Fukaya-Seidel category on the mirror. Finally we will see the main algebraic HMS result on homogenous coordinate rings, which is at the level of cohomology. The method involves first considering mirror symmetry for the 4-torus, then restricting to the hypersurface genus 2 curve and extending to a mirror Landau-Ginzburg model with fiber the mirror 4-torus. 

The classic Fatou lemma states that the lower limit of expectations is greater or equal than the expectation of the lower limit for a sequence of nonnegative random variables. This talk describes several generalizations of this fact including generalizations to converging sequences of probability measures. The three types of convergence of probability measures are considered in this talk: weak convergence, setwise convergence, and convergence in total variation. The talk also describes the Uniform Fatou Lemma (UFL) for sequences of probabilities converging in total variation. The UFL states the necessary and sufficient conditions for the validity of the stronger inequality than the inequality in Fatou's lemma. We shall also discuss applications of these results to sequential optimization problems with completely and partially observable state spaces. In particular, the UFL is useful for proving weak continuity of transition probabilities for posterior state distributions of stochastic sequences with incomplete state observations known under the name of Partially Observable Markov Decision Processes. These transition probabilities are implicitly defined by Bayes' formula, and general method for proving their continuity properties have not been available for long time. This talk is based on joint papers with Pavlo Kasyanov, Yan Liang, Michael Zgurovsky, and Nina Zadoianchuk.

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).

Fibrillation is a chaotic, turbulent state for the electrical signal fronts in the heart. In the ventricle it is fatal if not treated promptly. The standard treatment is by an electrical shock to reset the cardiac state to a normal one and allow resumption of a normal heart beat.

The fibrillation wave fronts are organized into scroll waves, more or less analogous to a vortex tube in fluid turbulence. The centerline of this 3D rotating object is called a filament, and it is the organizing center of the scroll wave.

The electrical shock, when turned on or off, creates charges at the conductivity discontinuities of the cardiac tissue. These charges are called virtual electrodes. They charge the region near the discontinuity, and give rise to wave fronts that grow through the heart, to effect the defibrillation. There are many theories, or proposed mechanisms, to specify the details of this process. The main experimental data is through signals on the outer surface of the heart, so that simulations are important to attempt to reconstruct the electrical dynamics within the interior of the heart tissue. The primary electrical conduction discontinuities are at the cardiac surface. Secondary discontinuities, and the source of some differences of opinion, are conduction discontinuities at blood vessel walls.

In this lecture, we will present causal mechanisms for the success of the virtual electrodes, partially overlapping, together with simulation and biological evidence for or against some of these.

The role of small blood vessels has been one area of disagreement. To assess the role of small blood vessels accurately, many details of the modeling have been emphasized, including the thickness and electrical properties of the blood vessel walls, the accuracy of the biological data on the vessels, and their distribution though the heart. While all of these factors do contribute to the answer, our main conclusion is that the concentration of the blood vessels on the exterior surface of the heart and their relative wide separation within the interior of the heart is the factor most strongly limiting the significant participation of small blood vessels in the defibrillation process.

Since the pioneering work of Hodgkin and Huxley , we know that electrical signals propagate along a nerve fiber via ions that flow in and out of the fiber, generating a current. The voltages these currents generate are subject to a diffusion equation, which is a reduced form of the Maxwell equation. The result is a reaction (electrical currents specified by an ODE) coupled to a diffusion equation, hence the term reaction diffusion equation.

The heart is composed of nerve fibers, wound in an ascending spiral fashion along the heart chamber. Modeling not individual nerve fibers, but many within a single mesh block, leads to partial differential equation coupled to the reaction ODE.

As with the nerve fiber equation, these cardiac electrical equations allow a propagating wave front, which normally moves from the bottom to the top of the heart, giving rise to contractions and a normal heart beat, to accomplish the pumping of blood.

The equations are only borderline stable and also allow a chaotic, turbulent type wave front motion called fibrillation.

In this lecture, we will explain the 1D traveling wave solution, the 3D normal wave front motion and the chaotic state.

The chaotic state is easiest to understand in 2D, where it consists of spiral waves rotating about a center. The 3D version of this wave motion is called a scroll wave, resembling a fluid vortex tube.

In simplified models of reaction diffusion equations, we can explain much of this phenomena in an analytically understandable fashion, as a sequence of period doubling transitions along the path to chaos, reminiscent of the laminar to turbulent transition.

I'll discuss the problem of controlling energy concentration in YM flow over a four-manifold. Based on a study of the rotationally symmetric case, it was conjectured in 1997 that bubbling can only occur at infinite time. My thesis contained some strong elementary results on this problem, which I've now solved in full generality by a more involved method.

It is by now well established that loading conditions with sufficiently large triaxialities can induce the sudden appearance of internal cavities within elastomeric (and other soft) solids. The occurrence of such instabilities, commonly referred to as cavitation, can be attributed to the growth of pre-existing defects into finite sizes.

In this talk, I will present a new theory to study the phenomenon of cavitation in soft solids that, contrary to existing approaches,

simultaneously: (i) allows to consider general 3D loading conditions with arbitrary triaxiality, (ii)  applies to large (including compressible and anisotropic) classes of nonlinear elastic solids, and

(iii) incorporates direct information on the initial shape, spatial distribution, and mechanical properties of the underlying defects at which cavitation can initiate. The basic idea is to first cast cavitation in elastomeric solids as the homogenization problem of nonlinear elastic materials containing random distributions of zero-volume cavities, or defects. Then, by means of a novel iterated homogenization procedure, exact solutions are constructed for such a problem. These include solutions for the change in size of the underlying cavities as a function of the applied loading conditions, from which the onset of cavitation - corresponding to the event when the initially infinitesimal cavities suddenly grow into finite sizes - can be readily determined. In spite of the generality of the proposed approach, the relevant calculations amount to solving tractable Hamilton-Jacobi equations, in which the initial size of the cavities plays the role of "time" and the applied load plays the role of "space".

An application of the theory to the case of Ne-Hookean solids containing a random isotropic distribution of vacuous defects will be presented.