- 11:15am - 12:15pm
- in SCGP 313

**Title:** Introduction to Integrable QFT**Speaker:** Sergei Lukyanov

- 11:15am - 12:15pm
- in SCGP 313

**Title:** Introduction to Integrable QFT**Speaker:** Sergei Lukyanov

- 2:00pm - 3:00pm
- in SCGP 313

- 2:05pm - 3:05pm

**Title:** Exotic rotation domains and Herman rings for quadratic H\'enon maps. **Speaker:** Raphael Krikorian [Ecole Polytechnique] **Abstract:** Quadratic H\'enon maps are polynomial automorphism of $\mathbb{C}^2$ of the form $h:(x,y)\mapsto (\lambda^{1/2}(x^2+c)-\lambda y,x)$. They have constant Jacobian equal to $\lambda$ and they admit two fixed points. If $\lambda$ is on the unit circle (one says the map $h$ is conservative) these fixed points can be elliptic or hyperbolic. In the elliptic case, a simple application of Siegel Theorem shows (under a Diophantine assumption) that $h$ admits an open set of quasi-periodic orbits with two frequencies in the neighborhood of its fixed points. Fundamental results by Bedford-Smillie and Barrett-Bedford-Dadok suggest that there could exist more general ``Rotation domains'', i.e. open sets filled with quasi-periodic motions, but without fixed points (the rotation domains are then said to be ``exotic''). In some hyperbolic cases, Shigehiro Ushiki observed numerically some years ago, what seems to be such ``Exotic rotation domains'' (quasi-periodic orbits though no Siegel disks exist). I will give a proof of the existence of such ERDs and provide a mathematical explanation for S. Ushiki's discovery. In the dissipative case ($\lambda$ of module less than 1), the same theoretical framework, predicts, and provides a proof of, the existence of (attracting) Herman rings. These Herman rings, which were not observed before, can be systematically produced in numerical experiments.

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- 4:00pm - 5:00pm

**Title:** Two results on étale fundamental groups in characteristic o **Speaker:** Srinivas Vasudevan [University at Buffalo] **Abstract:** This talk will discuss two results on etale fundamental groups of

varieties over an algebraically closed field of characteristic p > 0, based

on joint work with Helene Esnault and other coauthors. The first,

along with Mark Schusterman, is that the tame fundamental group

is finitely presented for such a variety which is the complement of an

SNC divisor in a smooth projective variety. The second, along with

Jakob Stix, is to give an obstruction for a smooth projective variety

to admit a lifting to characteristic 0, in terms of the structure of its

etale fundamental group as a profinite group.

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- 11:15am - 12:15pm
- in SCGP 313

**Title:** Introduction to Integrable QFT**Speaker:** Sergei Lukyanov

- 3:45pm - 4:45pm
- in 102

**Speaker:** Vlad Vicol[NYU Courant]**Title:** Shock formation and maximal hyperbolic development in multi-D gas dynamics**Abstract:** We consider the Cauchy problem for the multi-dimensional compressible Euler equations, evolving from an open set of compressive and generic smooth initial data. We construct unique solutions to the Euler equations which are as smooth as the initial data, in the maximal spacetime set characterized by: at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time until reaching the Cauchy data prescribed along the initial time-slice. This spacetime is sometimes referred to as the ``maximal globally hyperbolic development'' (MGHD) of the given Cauchy data. We prove that the future temporal boundary of this spacetime region is a singular hypersurface, consisting of the union of three sets: first, a co-dimension-2 surface of ``first singularities" called the pre-shock; second, a downstream co-dimension-1 surface emanating from the pre-shock, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream co-dimension-1 surface consisting of a Cauchy horizon emanating from the pre-shock, which the Euler solution cannot reach. In order to establish this result, we develop a new geometric framework for the description of the acoustic characteristic surfaces, and combine this with a new type of differentiated Riemann-type variables which are linear combinations of gradients of velocity/sound speed and the curvature of the fast acoustic characteristic surfaces. This is a joint work with Prof. Steve Shkoller (University of California at Davis).

- 3:45pm - 4:45pm

**Title:** Shock formation and maximal hyperbolic development in multi-D gas dynamics **Speaker:** Vlad Vicol [NYU Courant] **Abstract:** We consider the Cauchy problem for the multi-dimensional compressible Euler equations, evolving from an open set of compressive and generic smooth initial data.

We construct unique solutions to the Euler equations which are as smooth as the initial data, in the maximal spacetime set characterized by: at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time until reaching the Cauchy data prescribed along the initial time-slice. This spacetime is sometimes referred to as the ``maximal globally hyperbolic development'' (MGHD) of the given Cauchy data.

We prove that the future temporal boundary of this spacetime region is a singular hypersurface, consisting of the union of three sets: first, a co-dimension-2 surface of ``first singularities" called the pre-shock; second, a downstream co-dimension-1 surface emanating from the pre-shock, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream co-dimension-1 surface consisting of a Cauchy horizon emanating from the pre-shock, which the Euler solution cannot reach.

In order to establish this result, we develop a new geometric framework for the description of the acoustic characteristic surfaces, and combine this with a new type of differentiated Riemann-type variables which are linear combinations of gradients of velocity/sound speed and the curvature of the fast acoustic characteristic surfaces.

This is a joint work with Prof. Steve Shkoller (University of California at Davis).

SPECIAL LOCATION: SCGP 102

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- 11:00am - 12:00pm
- in Math 5-127

**Speaker:** Vlad Vicol, NYU Courant**Title:** On anomalous diffusion**Abstract:** Anomalous diffusion is the fundamental ansatz of phenomenological theories of passive scalar turbulence. As with the anomalous dissipation of kinetic energy in a turbulent fluid, the anomalous dissipation of passive scalar variance in a turbulent flow, as the Reynolds and Peclet numbers diverge, has been confirmed numerically and experimentally to an extraordinary extent. A satisfactory theoretical explanation of this phenomenon is however not available.

In this talk, I will discuss a joint work with Scott Armstrong (NYU) in which we construct a class of incompressible vector fields that have many of the properties observed in a fully turbulent velocity field, and for which the associated scalar advection-diffusion equation generically displays anomalous diffusion. We also propose an analytical framework in which to study anomalous diffusion, via a backward cascade of renormalized eddy viscosities. Our proof is by "fractal" homogenization, that is, we perform a cascade of homogenizations across arbitrarily many length scales.

- 11:00am - 12:00pm

**Title:** On anomalous diffusion **Speaker:** Vlad Vicol [NYU Courant] **Abstract:** Anomalous diffusion is the fundamental ansatz of phenomenological theories of passive scalar turbulence. As with the anomalous dissipation of kinetic energy in a turbulent fluid, the anomalous dissipation of passive scalar variance in a turbulent flow, as the Reynolds and Peclet numbers diverge, has been confirmed numerically and experimentally to an extraordinary extent. A satisfactory theoretical explanation of this phenomenon is however not available.

In this talk, I will discuss a joint work with Scott Armstrong (NYU) in which we construct a class of incompressible vector fields that have many of the properties observed in a fully turbulent velocity field, and for which the associated scalar advection-diffusion equation generically displays anomalous diffusion. We also propose an analytical framework in which to study anomalous diffusion, via a backward cascade of renormalized eddy viscosities. Our proof is by "fractal" homogenization, that is, we perform a cascade of homogenizations across arbitrarily many length scales.

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- 1:30pm - 2:30pm
- in 102

A discussion about the *Oppenheimer* film led by Gregory Falkovich, Pollack Professorial Chair in Physics, Weizmann Institute of Science

**Friday, October 18, 2024**

Dr. Falkovich will give a popular explanation of the basic physics which let us better see the human drama of war, peace and unpredictable internal logic of research.

Specific issues to be discussed

1. Why did the same people stand at the end of the energy epoch and the beginning of the information epoch?

2. Oppenheimer and Rabi; the role of orange in the movie.

3. Who asked the question that Oppenheimer discussed with Einstein? What were the consequences?

4. Why the movie is more about thermonuclear than nuclear weapons? What is the simple graph that physicists see behind that.

6. How did Germans fail, and Russians succeed?

The discussion will be illustrated with stills from the movie and historical pictures.

- 2:05pm - 3:05pm

**Title:** Exotic rotation domains and Herman rings for quadratic H\'enon maps. **Speaker:** Raphael Krikorian [Ecole Polytechnique] **Abstract:** Quadratic H'enon maps are polynomial automorphism of $mathbb{C}^2$ of the form $h:(x,y)mapsto (lambda^{1/2}(x^2+c)-lambda y,x)$. They have constant Jacobian equal to $lambda$ and they admit two fixed points. If $lambda$ is on the unit circle (one says the map $h$ is conservative) these fixed points can be elliptic or hyperbolic. In the elliptic case, a simple application of Siegel Theorem shows (under a Diophantine assumption) that $h$ admits an open set of quasi-periodic orbits with two frequencies in the neighborhood of its fixed points. Fundamental results by Bedford-Smillie and Barrett-Bedford-Dadok suggest that there could exist more general ``Rotation domains'', i.e. open sets filled with quasi-periodic motions, but without fixed points (the rotation domains are then said to be ``exotic''). In some hyperbolic cases, Shigehiro Ushiki observed numerically some years ago, what seems to be such ``Exotic rotation domains'' (quasi-periodic orbits though no Siegel disks exist). I will give a proof of the existence of such ERDs and provide a mathematical explanation for S. Ushiki's discovery. In the dissipative case ($lambda$ of module less than 1), the same theoretical framework, predicts, and provides a proof of, the existence of (attracting) Herman rings. These Herman rings, which were not observed before, can be systematically produced in numerical experiments.

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- in SCGP 102

For more information, please visit: https://scgp.stonybrook.edu/archives/43112

- 12:30pm - 1:55pm
- in Math P-131

**Title:** Plumber's PROP action on symplectic cohomology **Speaker:** Yash Deshmukh [IAS] **Abstract:** I will introduce a new structure on (relative) symplectic cohomology defined in terms of a PROP called the “Plumber’s PROP.” This PROP consists of nodal Riemann surfaces, of all genera and with multiple inputs and outputs, satisfying a condition that ensures the existence of positive Floer data on the surfaces. This action is defined on the chain-level and generalizes the work of Abouzaid–Groman–Varolgunes. I will discuss the relationship of this structure to cohomological field theories, with potential applications to curve counts, as well as algebraic structures defined on variants of symplectic cohomology such as Rabinowitz Floer cohomology.

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- 4:00pm - 5:15pm
- in P-131

**Title:** On Warped QAC Calabi-Yau Manifolds **Speaker:** Dashen Yan [Stony Brook University] **Abstract:** Warped quasi-asymptotically conical (QAC) Calabi-Yau manifolds are examples of complete non-compact Calabi-Yau manifolds with maximal volume growth. These manifolds roughly admit holomorphic fibrations over the complex line, with asymptotically conical Calabi-Yau manifolds as generic fibers; they are modeled on a warped product of a flat metric on the base-space with an asymptotically conical metric on the fibers. In this talk, I will explain my work on a gluing construction for families of warped QAC Calabi-Yau metrics that converge to the product of the complex line and a Calabi-Yau cone, thereby confirming a conjecture of Yang Li. If time permits, I will also discuss bubbling phenomena for suitable sequences of collapsing metrics.

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