## Fall 2023

Time & Location: Typically talks will be in Gibson Hall 325 at 3:00 PM on a Friday.

Organizers: Punshon-Smith, Samuel and Buvoli, Tommaso

**September 8****Title:** *Navigating from statistical mechanics to random matrix theory: exploring the realm of integrability*

**Guido Mazzuca - Tulane University**

**Abstract: **This presentation delves into the fascinating relationship between integrable systems theory and random matrix theory. After a general introduction explaining how these two fields are connected, we delve into a more concrete example. In particular, we show a method to describe the eigenvalue density of the Ablowitz-Ladik lattice with random initial data sampled from a Generalized Gibbs ensemble. This characterization is achieved in two ways: by the transfer operator approach, and by employing a large deviation principle (LDP). Additionally, based on these characterizations, we establish a connection between the Ablowitz–Ladik lattice and the Circular beta ensemble in the high-temperature regime. As a result, we can explicitly compute the eigenvalue density of the Ablowitz-Ladik lattice using the density of the random matrix ensemble. This talk is mainly based on separate works with R. Memin and T. Grava.

**Time: **3:00pm**Location:** Gibson Hall 325

**September 15****Title:** *Computational modeling of small-scale ballistics*

**Christina Hamlet - Bucknell University**

**Abstract: **The small, stinging organelles (nematocysts) found in Cnidarians as well as dinoflagellates are the fastest-known accelerating structures in the animal world, with rates over 5 million times the acceleration due to gravity. For the nematocyst's barb-like projectile to penetrate its prey, high accelerations facilitate the transition from a viscous regime to one where inertial forces dominate. We construct and implement a fluid-structure interaction model to numerically simulate the dynamics of a barb-like structure accelerating towards stationary, passively suspended prey. These studies help shed light on predatory and defensive strategies in small-scale interactions among microorganisms. Our results indicate that transitioning to higher Reynolds numbers is necessary to overcome the significant boundary layer interactions between the structures at low to zero Reynolds numbers usually associated with typical cellular-level interactions.

**Time:** 3:00**Location: ** 325

**September 22****Title:**** TBA**

**Speaker | Tulane University**

**Abstract: **

**Time:** 3:00**Location:** TBA

**September 29****Title:*** Solitons and inverse scattering transform: from the first soliton to ultra-short pulses*

**Katerina Gkogkou - Tulane University**

**Abstract: **An exciting and extremely active area of research investigation is the study of solitons and the nonlinear partial differential equations that describe them. In this talk, we will discuss what solitons are, and what makes them so special. We will see when the first solitons were observed, and when the first math that describe them were derived. We will introduce ourselves to integrability and the inverse scattering transform by presenting the complex coupled short-pulse equation (ccSPE), a complex vector generalization of the short pulse equation, a nonlinear partial differential equation that describes the propagation of ultra-short pulses in optics. If time permits, we will discuss soliton solutions and their interactions in the ccSPE.

**Time: **3:00**Location: **Gibson Hall 325

**October 6**

Fall Break

**November 3**

**Title: **Stochastic homogenization with space-time ergodic divergence-free drift

**Ben Fehrman - LSU**

**Abstract: **We will discuss the large-scale behavior of diffusions with space-time stationary and ergodic, divergence-free drift. Such processes provide a simple approximation of transport in rough, incompressible flows and have been used to describe passive advected quantities such as temperature in several contexts. In the large-scale homogenization limit, the evolution is characterized by the solution to a deterministic stochastic partial differential equation with Stratonovich transport noise. In the absence of spatial ergodicity, the drift is only partially absorbed into the skew-symmetric part of the flux through the use of an appropriately defined stream matrix. This leaves a time-dependent, spatially-homogenous transport which, for mildly decorrelating fields, converges to a Brownian noise with deterministic covariance in the limit. The results apply to uniformly elliptic, stationary and ergodic environments in which the drift admits a suitably defined stationary and square-integrable stream matrix.

Time: 3:00

Location: Gibson Hall 325

**November 10****Title: **Asymptotic properties and separation rates for local energy solutions to the Navier-Stokes equations

**Patrick Phelps - Temple University**

**Abstract: **We present recent results on spatial decay and properties of non-uniqueness for the 3D Navier-Stokes equations. We show asymptotics for the ‘non-linear’ part of scaling invariant flows with data in subcritical classes. Motivated by recent work on non-uniqueness, we investigate how non-uniqueness of the velocity field would evolve in time in the local energy class. Specifically, by extending our subcritical asymptotics to approximations by Picard iterates, we may bound the rate at which two solutions, evolving from the same data, may separate pointwise. We conclude by extending this separation rate to solutions with no scaling assumption. Joint work with Zachary Bradshaw.

**Time:** 3:00**Location: ** Gibson Hall 325

**November 17****Title: ***Stability of traveling fronts in a model arising from nanoscale pattern formation*

**Gregory Lyng **

**Abstract: ** We examine the stability of traveling-front solutions of a modified Kuramoto–Sivashinsky equation derived to describe ripple formation in physical experiments featuring the bombardment of a nominally flat surface by a broad ion beam at an oblique angle of incidence. Structurally, the linearized operators associated with these fronts have unstable essential spectrum, which corresponds to instability of the spatially asymptotic states, and stable point spectrum, corresponding to stability of the transition layer itself. We first show that these waves are linearly orbitally asymptotically stable in appropriate exponentially weighted spaces. Second, we seek to better understand the long-time pattern formation that is observed experimentally and in numerical simulations. Specifically, we consider a periodic array of unstable front and back solutions. While not an exact solution of the governing equation, this periodic pattern mimics experimentally observed phenomena. Our numerical experiments suggest that the convecting instabilities associated with each individual wave are damped as they pass through transition layers and that this stabilization mechanism underlies the pattern formation seen in experiments. This is joint work with Mat Johnson and Connor Smith.

**Time:** 3:00**Location: ** Gibson Hall 325

**November 24****Title:**** TBA**

**Speaker | Tulane University**

**Abstract: **TBA

Time: 3:00

Location: Gibson Hall 325

**December 1****Title:**** TBA**

**Speaker - University **

**Abstract: **TBA

Time: 3:00

Location: Gibson Hall 325

**December 8****Title:**** TBA**

**Speaker | Tulane University**

**Abstract: **TBA

Time: 3:00

Location: Gibson Hall 325