# Einstein Chair Mathematics Seminar

**Graduate Center: 365 Fifth Avenue (between 34th and 35th streets) New York, NY 10016**

The Einstein Chair Mathematics Seminar is concentrated on the relationship between algebraic topology and quantum field theory.

This is like the opposite of JFK's famous saying "Ask not what your country can do for you, but rather ask what you can do for your country".

Namely beyond the noble desire of some mathematicians to clarify the foundations of quantum theory by the definitions and methodology of mathematics specifically to illuminate theoretical physics, there is a slightly different opportunity-mining theoretical physics as it is for the sake of mathematics.

Passing over in silence the commonly held conclusion that quantum field theory is physically successful in that it already contains a viable procedure for making predictions that are verifiable in high energy experiments, one can observe the success of the algorithms associated to the action principle, quantum style, in mathematics itself. There are rigorous and famous discussions in mathematics that are separated in mathematics but are actually unified in the minds of theoretical physicists by these algorithms of quantum field theory.

There are also unsolved areas of mathematics that suffer the same technical difficulties as those in quantum field theory; but while the former field outside of applications seems to languish in the math journals the latter field seems to flourish in the theoretical physics journals.

In the first example one can mention the celebrated invariants of differential topology (Donaldson and Vaughn Jones), of symplectic topology and algebraic geometry {Gromov and Witten), and of complex structures (Kodaira Spencer Griffiths...Kontsevich).

While for the second example one can mention intractable nonlinear PDEs like those appearing in 3D fluid dynamics. One may add in the second example the remark that important examples of tractable nonlinear PDEs, the integrable systems or hierarchies, seem to have a deep connection with quantum field theory and conversely they seem to have a deep role to play there.

There may be several veins of precious math material to mine from this opportunity and several methods.

I am particularly interested in the method of algebraic topology which associates linear objects (homology groups) to nonlinear objects with points ( manifolds...) just like quantum theory associates linear spaces of states to classical systems with points. The main character in algebraic topology is the nilpotent operator or boundary operator while in quantum field theory an important role is played by the nilpotent operators called Q and "delta" which encode whatever symmetry is present in the action of the particular theory and measure the obstruction to invariantly assign meaning to the integral over all paths.

In algebraic topology there is a powerful idea, due first to Stasheff but going beyond his famous and elegant concept of an infinitely homotopy associative algebra, which allows one to live with slightly false algebraic identities in a new world where they become effectively true. In quantum field theory the necessity to regularize or cutoff which sometimes destroys, but only slightly, identities expressing various symmetries and structures may provide an opportunity to use this powerful idea from algebraic topology.

Finally algebraic and geometric topology has always directed it efforts towards understanding in an algebraic way geometric objects like manifolds which are the classical models of spacetime, while quantum field theory often begins its specification of a particular theory with the classical action defined on the classical fields spread over spacetime and then proceeds to its algebraic algorithms.

All these connections suggest that one way to enter the mining business in the above sense is to define relevant algebraic structures with a nilpotent operator and formulate mathematically the intuitively clear physical idea of an "effective theory" as a kind of push forward of the entire algebraic structure in the new world or sense created by the idea underlying Stasheff's famous example of an A-infinity algebra.

The format of the seminar is generous regarding time and allows a robust exchange of information between the expositor and the other participants- who usually ask a lot of questions.

**The talks are generally held on TUESDAYS from 5:30p- 6:30pm.**

ZOOM LOGIN INFORMATION:

https://gc-cuny-edu.zoom.us/j/9894931174?pwd=MHprR2Vqa2dvZmVseXAxNzlXZUFhdz09 [gc-cuny-edu.zoom.us]

Meeting ID: 989 493 1174

Passcode: einstein

Videos of Einstein Chair seminars from 1981-2014

Youtube website for current videos: www.youtube.com/channel/UC_jFgn51x3iXh8ljGzWRToA

ZOOM LOGIN INFORMATION:

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**Date: June 1, 2021**

Time: 2:30pm

Speaker: Sasha Migdal, NYU

Title: Confined Vortex Surface and Irreversibility.plus an Exact solution for the flow

Abstract: We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (AM,2021). These surfaces avoid the \KH{} instability by adjusting their discontinuity and shape. The vorticity collapses to the sheet only in an exceptional case considered long ago by Burgers and Townsend, where it decays as a Gaussian on both sides of the sheet. In generic asymmetric vortex sheets (Shariff,2021), vorticity leaks to one side or another, making such sheets inadequate for vortex sheet statistics and anomalous dissipation.

We conjecture that the vorticity in a turbulent flow collapses on the special kind of surface. The eigenvalue of the strain tensor along the velocity gap vanishes at this surface. In this case, the Euler vortex sheet solution matches the planar Burgers-Townsend solution of the Navier-Stokes equation in the local boundary layer; otherwise, it matches the asymmetric vortex sheet solution which leaks vorticity.

The most important qualitative observation is that the inequality needed for this solution's stability breaks the Euler dynamics' time reversibility. We interpret this as dynamic irreversibility. We have also represented the enstrophy by a surface integral, conserved in the vortex sheet dynamics in the turbulent limit.

We have found the exact analytic solution for the cylindrical vortex surface for an arbitrary constant background strain with two different eigenvalues. This solution is expressed as a parametric representation using two conformal maps: from the unit circle into the x+iy complex plane a with velocity field v/x + iv/y. We represent these conformal maps as integrals of certain algebraic functions over the unit circle. We have investigated this solution numerically in great detail.

In a coming paper we study turbulent statistics corresponding to this family of steady solutions.

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**Date: May 25, 2021**

Time: 5:30pm

Speaker: Aseel Farhat., Florida State

Title: Geometry of Turbulent Flows and the 3D Navier-Stokes regularity problem

Abstract: We describe several aspects of an analytic/geometric framework for the three-dimensional Navier-Stokes regularity problem, which is directly inspired by the morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of three-dimensional turbulence. Among these, we present our proof that the scaling gap in the 3D Navier-Stokes regularity problem can be reduced by an algebraic factor within an appropriate functional setting incorporating the intermittency of the spatial regions of high vorticity. We will also show that that 3D NSE are regular under some appropriate local condition on the helicity in the the regions of intense vorticity.

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**Date: May 18, 2021**

Time: 5:30pm

Speaker: Theo Drivas, Stony Brook University

Title: Part II On the mystery of large vortex patterns in long time 2D ideal euler fluid motion

Time: 2:30pm

Speaker: Sasha Migdal, NYU

Time: 5:30pm

Speaker: Aseel Farhat., Florida State

Time: 5:30pm

Speaker: Theo Drivas, Stony Brook University

## Abstract: We will discuss some aspects of inviscid, incompressible (perfect) fluid motion in two dimensions.

We describe some old and new results and open questions regarding properties of steady solutions of

the two-dimensional incompressible Euler equations, as well as properties of nearby trajectories. Specifically,

we focus on whether steady states can be isolated, whether, for solutions starting nearby steady states, recurrence can occur and whether singularities must form at long times. Finally, we discuss the infinite-time limit near and far from equilibrium with a focus on the apparent irreversibility/decrease of entropy. See related movies:

https://www.youtube.com/watch?v=25Md9qxIReE [youtube.com]

https://www.youtube.com/watch?v=-YdEYumSSJ0&t=222s [youtube.com]

https://www.youtube.com/watch?v=a3ENdIy1WL0 [youtube.com]

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**Date: May 11, 2021**

Time: 5:30pm

Speaker: Theo Drivas, Stony Brook

Title: Asymptotic pictures of and for 2D incompressible fluid motion without friction

Time: 5:30pm

Speaker: Theo Drivas, Stony Brook

## Abstract: We will discuss some aspects of inviscid, incompressible (perfect) fluid motion in two dimensions.

We describe some old and new results and open questions regarding properties of steady solutions of

the two-dimensional incompressible Euler equations, as well as properties of nearby trajectories. Specifically,

we focus on whether steady states can be isolated, whether, for solutions starting nearby steady states, recurrence can occur and whether singularities must form at long times. Finally, we discuss the infinite-time limit near and far from equilibrium with a focus on the apparent irreversibility/decrease of entropy. See related movies:

https://www.youtube.com/watch?v=25Md9qxIReE [youtube.com]

https://www.youtube.com/watch?v=-YdEYumSSJ0&t=222s [youtube.com]

https://www.youtube.com/watch?v=a3ENdIy1WL0 [youtube.com]

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Date: May 4, 2021

Time: 5:30pm

Speaker: V. Martinez, Hunter College

Title: KDV, uniqueness of invariant measures and beyond

Abstract: We discuss the existence, uniqueness, and regularity of invariant measures for the damped-driven stochastic Korteweg-de Vries equation, where the noise is additive and sufficiently non-degenerate. It is shown that a simple, but versatile control strategy, typically employed to establish exponential mixing for strongly dissipative systems such as the 2D Navier-Stokes equations, can nevertheless be applied in this weakly dissipative setting to establish both unique ergodicity, albeit without mixing rates, as well as regularity of the support of the invariant measure. Under the assumption of large damping, however, exponential mixing can be recovered.

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**Date: April 27, 2021**

Time: Fluids seminar, 530 to 700

Speaker: Ali Khosronejad

Title: High-fidelity numerical simulation of turbulent fluid flows in arbitrarily complex domains

Abstract: I present in this talk an overview of my research focus on developing three-phase multi-scale multi-physics computational fluid dynamics model for a wide range of applications such as renewable energy, biological flows, and transport processes in the atmosphere and riverine systems. I also present the simulation results obtained from the large-eddy simulation method of turbulent flows across different scales -- from micros (human saliva particles) to mesoscale (atmospheric boundary layer in offshore wind farms).

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Date: April 20, 2021

Time: 5:30pm

Speaker: Open meeting for discussion

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Date: Tuesday, April 13, 2021

Time: 5:30pm

Speaker: Vincent Martinez, Hunter College

Title: More on KdV and invariant measures

Abstract: We discuss the existence, uniqueness, and regularity of invariant measures for the damped-driven stochastic Korteweg-de Vries equation, where the noise is additive and sufficiently non-degenerate. It is shown that a simple, but versatile control strategy, typically employed to establish exponential mixing for strongly dissipative systems such as the 2D Navier-Stokes equations, can nevertheless be applied in this weakly dissipative setting to establish both unique ergodicity, albeit without mixing rates, as well as regularity of the support of the invariant measure. Under the assumption of large damping, however, exponential mixing can be recovered.

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**Date: Tuesday, April 6, 2021**

Time: 5:30pm

Speaker: Jim Glimm, Stony Brook University

Title: Turbulence models and quantum ideas

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* NOTE: THERE ARE 2 TALKS ON MARCH 30, 2021*

**Date: Tuesday, March 30, 2021 **

Time: 1:30p to 2:30p

Speaker: Gregory Falkovich, Weizmann Institute

Title: Fibonacci Turbulence

Abstract: Never is the difference between thermal equilibrium and turbulence so dramatic, as when a quadratic invariant makes the equilibrium statistics exactly Gaussian with independently fluctuating modes. That happens in two very different yet deeply connected classes of systems: incompressible hydrodynamics and resonantly interacting waves. I shall describe an information-theoretic analysis of turbulence in such strongly interacting systems. The analysis involves both energy and entropy and elucidates the fundamental roles of space and time in setting the cascade direction and the changes of the statistics along it. We introduce a beautifully simple yet rich family of discrete models (sets of ODE) with triplet interactions of neighboring modes and show that it has quadratic conservation laws defined by the Fibonacci numbers. Depending on how the interaction time changes with the mode number, three types of turbulence were found: single direct cascade, double cascade, and the first ever case of a single inverse cascade. We describe quantitatively how deviation from thermal equilibrium all the way to turbulent cascades makes statistics increasingly non-Gaussian and find the self-similar form of the one-mode probability distribution. I will end up discussing possible directions of an analytic theory including applications of Renormalization Group to this class of models.

**Time: 5:30p to 6:30p **

Speaker: Vincent Martinez, Hunter College

Title: "On unique ergodicity and mixing for the damped-driven stochastic KdV equation"

Abstract: We discuss the existence, uniqueness, and regularity of invariant measures for the damped-driven stochastic Korteweg-de Vries equation, where the noise is additive and sufficiently non-degenerate. It is shown that a simple, but versatile control strategy, typically employed to establish exponential mixing for strongly dissipative systems such as the 2D Navier-Stokes equations, can nevertheless be applied in this weakly dissipative setting to establish both unique ergodicity, albeit without mixing rates, as well as regularity of the support of the invariant measure. Under the assumption of large damping, however, exponential mixing can be recovered.

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Date: Tuesday, March 16, 2021

Time: 5:30pm

Speaker: Tarek Elgindi, Duke University

Title: "Stationary Incompressible ideal fluid motions in two dimensions"

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Dates: March 8-12, 2021

Einstein Chair seminar replaced by: Virtual Workshop Many faces of renormalization: March 8-12, 2021

http://scgp.stonybrook.edu/archives/30366 [scgp.stonybrook.edu]

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Date: March 2, 2021 (Tuesday)

Time: 5:30-6:30pm

Speaker: Theodore Drivas, Princeton Univ.

Title: A Lagrangian perspective on anomalous dissipation.

**Abstract**: We will review some observations and results that connect (near)

random motion of tracer particles in (near) rough velocities fields. In fluid turbulence,

this connection has been known to exist since the pioneering work of L.F. Richardson.

For passive scalar fields (non-reactive dye), it was observed by Bernard-Gawedzki-Kupiainen

in the context of the Kraichnan model that the stochastic behavior of particles is explicitly

connected to anomalous dissipation of scalar fluctuations in the non-diffusive limit. A similar

statement can be made for a pressureless fluid in one-dimension which loses energy at shocks.

We will end with some speculative remarks about incompressible Navier-Stokes turbulence.

**ZOOM LOGIN INFORMATION: **

https://gc-cuny-edu.zoom.us/j/9894931174?pwd=MHprR2Vqa2dvZmVseXAxNzlXZUFhdz09 [gc-cuny-edu.zoom.us]

Meeting ID: 989 493 1174

Passcode: einstein

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Date: February 8, 2021

Time: 2-3:30pm; discussion until 4pm

Speaker: Theodore Drivas, Princeton university

Title: Remarks on mathematical foundations of Kolmogorov's 1941 turbulence theory

Abstract: In this talk, we review certain aspects of three-dimensional incompressible

fluid turbulence, focusing on the phenomenon of anomalous dissipation. We will describe

in detail Duchon-Robert's (2000) contribution to establishing a rigorous framework with which to

discuss anomalous energy flux due to possible singularities in Leray solutions and, in the inviscid limit,

singular weak solutions of the Euler equations in the inviscid limit as envisioned by Onsager.

**ZOOM LOGIN INFORMATION: **

https://gc-cuny-edu.zoom.us/j/9894931174?pwd=MHprR2Vqa2dvZmVseXAxNzlXZUFhdz09 [gc-cuny-edu.zoom.us]

Meeting ID: 989 493 1174

Passcode: einstein

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Date: February 1, 2021

Time: 2-3:30pm; discussion until 4pm

Speaker: Daniel An, SUNY Maritime College

Title: A report on lattice fluid model simulations

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Date: January 25, 2021

Time: 2-3:30p; discussion until 4pm

Speaker: Tarek Elgindi, Theo Drivas and Dennis Sullivan

Informal discussion

Date: May 4, 2021

Time: 5:30pm

Speaker: V. Martinez, Hunter College

Time: Fluids seminar, 530 to 700

Speaker: Ali Khosronejad

Date: April 20, 2021

Time: 5:30pm

Speaker: Open meeting for discussion

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Date: Tuesday, April 13, 2021

Time: 5:30pm

Speaker: Vincent Martinez, Hunter College

Time: 5:30pm

Speaker: Jim Glimm, Stony Brook University

*NOTE: THERE ARE 2 TALKS ON MARCH 30, 2021*

Time: 1:30p to 2:30p

Speaker: Gregory Falkovich, Weizmann Institute

Speaker: Vincent Martinez, Hunter College

Date: Tuesday, March 16, 2021

Time: 5:30pm

Speaker: Tarek Elgindi, Duke University

Dates: March 8-12, 2021

Date: March 2, 2021 (Tuesday)

Time: 5:30-6:30pm

Speaker: Theodore Drivas, Princeton Univ.

Title

Date: February 8, 2021

Time: 2-3:30pm; discussion until 4pm

Speaker: Theodore Drivas, Princeton university

Date: February 1, 2021

Time: 2-3:30pm; discussion until 4pm

Speaker: Daniel An, SUNY Maritime College

Date: January 25, 2021

Time: 2-3:30p; discussion until 4pm

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Date: January 18, 2021
Time: 2-3:30p; discussion until 4pm
Speaker: Tarek Elgindi, UC San Diego**

Title: Holder continuous initial vorticity developing singulatity in finite time for Euler

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Date: January 11, 2021

Time: 2-3:30p; discussion until 4pm

Speaker: Pooja Rao, Stony Brook University

Date: January 11, 2021

Time: 2-3:30p; discussion until 4pm

Speaker: Pooja Rao, Stony Brook University

Title: Part II of layer instabilities and front tracking

(these two lectures from PhD work with J.Glimm)

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**Date: January 4, 2021**

Time: 2-3:30p; discussion until 4pm

Speaker: Abigail Hsu, Stony Brook University

Time: 2-3:30p; discussion until 4pm

Speaker: Abigail Hsu, Stony Brook University

Title: Probing the statistical structure using direct numerical simulation of turbulence

Abstract: Examine the intermittency phenomenon that signifies coexistence of turbulent regions with regions is still an open question in modern turbulence theory. One way to examine this phenomenon, is to study statistical properties of the scalar increment, i.e. the difference of the scalar field at two points separated by a distance l. We are interested in developing a universal scaling law applicable to velocity increments and velocity gradient as these quantities vary across length scales. In the localized ranges of length scales in which the turbulence is only partially developed, we propose multifractal scaling laws with scaling exponents modified from the classical inertial range values. We have discovered the deviation from the classical theories for the energy dissipation rate in the inertial range. The scaling exponent of the velocity increment zeta_p and the exponent of the energy dissipation rate tau_p for the p-th moment structure functions are linear in log length in the dissipation range. New parameterized models have been proposed with explicit formulas that characterize this relation for high order moment statistics. DNS data verifications subsume and extend the existing models of inertial range turbulence and are especially focused on the dissipation range for small length scales. This study finds quantitative corrections to the theory of fully developed turbulence which describes partially developed turbulent flows.

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Date: December 28, 2020

Time: 2-3:30p; discussion until 4pm

Speaker: Dennis Sullivan, Stony Brook Univ/CUNY Grad Center

Title:Perturbative expansion of RHS of effective fluid ODEs based on Algebra of NaCl lattice coarse graining of the continuum

Date: December 28, 2020

Time: 2-3:30p; discussion until 4pm

Speaker: Dennis Sullivan, Stony Brook Univ/CUNY Grad Center

Title:

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Date: December 21, 2020

Time: 2-3:30p; discussion until 4pm

Speaker: Dennis Sullivan, Stony Brook Univ/CUNY Grad Center

Date: December 21, 2020

Time: 2-3:30p; discussion until 4pm

Speaker: Dennis Sullivan, Stony Brook Univ/CUNY Grad Center

Title: Perturbative expansion of RHS of effective fluid ODEs based on Algebra of NaCl lattice coarse graining of the continuum

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**Date: December 14, 2020**

**Time: 2-3:30pm; discussion until 4pm**

**Speaker: Stephen Preston, Brooklyn college/ CUNY Grad center**

Abstract: I will discuss Arnold’s perspective on the Euler equations as geodesics on the group of volume-preserving diffeomorphisms, with an eye to generalizations on other diffeomorphism groups. I’ll talk about the general notion of vorticity conservation and what it looks like in 2D, 3D, and in the 3D axisymmetric case. I will also describe how this infinite-dimensional geometry approach allows for simpler proofs of the local existence result. In addition we’ll look at curvature computations and what they say about stability of fluids, along with conjugate points along geodesics and how they differ in 2D and 3D.

In the second half of the talk I will describe other PDEs that can be described as geodesics on diffeomorphism groups, including 1D models that can be understood more easily, including the breakdown mechanisms and the global existence of weak solutions.

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Date: December 7, 2020**

**Time: 2-3:30p; Discussion until 4pm**

Speaker: Pooja Rao, Stony Brook University & Theo Drivas, Princeton University

Speaker: Pooja Rao, Stony Brook University & Theo Drivas, Princeton University

Title: Modeling mixing in interfacial instabilities

Abstract: Turbulent mixing from hydrodynamic instabilities, such as the Richtmyer-Meshkov (RMI) and Rayleigh-Taylor (RTI) instabilities, plays a critical role in numerous applications ranging from performance degradation in inertial confinement fusion capsules to supernova explosions. To accurately model these flow regimes requires special treatment of the interface between the two fluids. In this talk, we introduce one such numerical approach, Front-tracking, which has shown great success in modeling interfacial instabilities. To build up this approach, we first introduce the basics of these interfacial instabilities and discuss the main theoretical ideas behind a Riemann problem and the Front-tracking algorithm in 1-dimension. Next, we build on these ideas and show some comparisons between Front-tracking and other approaches. This is followed by reviewing results from simulations of the Richtmyer-Meshkov instability in a 2-dimensional setting using our proposed framework.