# Einstein Chair Mathematics Seminar

**Graduate Center: 365 Fifth Avenue (between 34th and 35th streets) New York, NY 10016
Math department: Room 4208 Seminar held in: Room 6417**

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 on TUESDAYS from 5:30p- 6:30pm.**

ZOOM LOGIN INFORMATION:

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

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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

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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

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

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

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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**

Title: Euler equations as geodesics on diffeomorphism groups
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

Date: January 18, 2021

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

Speaker: Tarek Elgindi, UC San Diego

Date: January 11, 2021

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

Speaker: Pooja Rao, Stony Brook University

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

Speaker: Abigail Hsu, Stony Brook University

Date: December 28, 2020

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

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

Title:

Date: December 21, 2020

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

Speaker: Dennis Sullivan, Stony Brook Univ/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.