Georgia Institute of Technology College of Sciences

A result by Ozsvath and Szabo states that the knot Floer complex of an L-space knot is a staircase. In this talk, we will discuss a similar result for two-component L-space links: the link Floer complex of such links can be thought of as an array of staircases. We will describe an algorithm to extract this array directly from the H-function of the link. As an application, we will discuss how to use this and the link surgery formula to compute the knot Floer complex and the tau-invariant of a certain class of satellite knots. This is joint work with Ian Zemke and Hugo Zhou.

One essential challenge in the computational modeling of multiscale systems is the availability of reliable and interpretable closures that faithfully encode the micro-dynamics. For systems without clear scale separation, there generally exists no such a simple set of macro-scale field variables that allow us to project and predict the dynamics in a self-determined way. We introduce a machine-learning (ML) based approach that enables us to reduce high-dimensional multi-scale systems to reliable macro-scale models with low-dimensional variational structures that preserve canonical degeneracies and symmetry constraints. The non-Newtonian hydrodynamics of polymeric fluids is used as an example to illustrate the essential idea. Unlike our conventional wisdom about ML modeling that focuses on learning the PDE form, the present approach directly learns the energy variational structure from the micro-model through an end-to-end process via the joint learning of a set of micro-macro encoder functions. The final model, named the deep non-Newtonian model (DeePN2), retains a multi-scale nature with clear physical interpretation and strictly preserves the frame-indifference constraints. We show that DeePN2 can capture the broadly overlooked viscoelastic differences arising from the specific molecular structural mechanics without human intervention.

The forced 2D Euler equations exhibit non-unique solutions with vorticity in $L^p$, $p > 1$, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit $\nu \to 0^+$ from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of ``resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity $\nu$ and consider $O(\varepsilon)$-size perturbations of his initial datum. We discover a uniqueness threshold $\varepsilon \sim \nu^{\kappa_{\rm c}}$, below which the vanishing viscosity solution is unique and radial, and at which certain vanishing viscosity solutions converge to non-unique, non-radial solutions. Joint work with Maria Colombo and Giulia Mescolini (EPFL).

The Ceresa cycle and the Gross—Kudla—Schoen modified diagonal cycle are algebraic $1$-cycles associated to a smooth algebraic curve. They are algebraically trivial for a hyperelliptic curve and non-trivial for a very general complex curve of genus $>2$. Given an algebraic curve, it is an interesting question to study whether the Ceresa and GKS cycles associated to it are rationally or algebraically trivial. In this talk, I will discuss some methods and tools to study this problem

While the research on water waves modeled by Euler's equations has a long history, mainly in the last two decades traveling periodic rotational waves have been constructed rigorously by means of bifurcation theorems. After introducing the problem, I will present a new reformulation in two dimensions in the pure-gravity case, where the problem is equivalently cast into the form "identity plus compact," which is amenable to Rabinowitz's global bifurcation theorem. The main advantages (and the novelty) of this new reformulation are that no simplifying restrictions on the geometry of the surface profile and no simplifying assumptions on the vorticity distribution (and thus no assumptions regarding the absence of stagnation points or critical layers) have to be made. Within the scope of this new formulation, global families of solutions, bifurcating from laminar flows with a flat surface, are constructed. Moreover, I will discuss the possible alternatives for the global set of solutions, as well as their nodal properties. This is joint work with Erik Wahlén.

Beta-ensembles generalize the eigenvalue distributions of self-adjoint
real, complex, and quaternion matrices for beta=1,2, and 4,
respectively. These ensembles naturally extend to two dimensions by
introducing operations such as corner truncation, addition, or
multiplication of matrices. In this talk, we will explore the edge
asymptotics of the resulting two-dimensional ensembles. I will present
the Airy-beta line ensemble, a universal object that governs the
asymptotics of time-evolving largest eigenvalues. This ensemble
consists of an infinite collection of continuous random curves,
parameterized by beta. I will share recent progress in developing a
framework to describe this remarkable structure.

The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function in the presence of the drift term. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains and discuss how the introduction of the transport term influences the regularity of the solutions.

https://gatech.zoom.us/j/94295986362?pwd=8euEJ3ojkWl5c3Y3hLyXTiKBts3Rrq.1

Given an $r$-uniform hypergraph $H$, the random Turán number $\mathrm{ex}(G^r_{n,p},H)$ is the maximum number of edges in an $H$-free subgraph of $G^r_{n,p}$, where $G^r_{n,p}$ is the Erdős-Rényi random hypergraph with parameter $p$. In the case when $H$ is not r-partite, the problem has been essentially solved independently by Conlon-Gowers and Schacht. In the case when $H$ is $r$-partite, the degenerate case, only some sporadic results are known.

The Sidorenko conjecture is a notorious problem in extremal combinatorics. It is known that its hypergraph analog is not true. Recently, Conlon, Lee, and Sidorenko discovered a relation between the Sidorenko conjecture and the Turán problem. 

 In this talk, we introduce some recent results on the degenerate random Turan problem and its relation to the hypergraph analog of the Sidorenko conjecture.

Nonlinear functionals of Gaussian fields are ubiquitous in probability theory and PDEs.  In work in progress with Robert Chang (Rhodes College), we introduce a family of random curves in the plane which encode the random values of certain nonlinear functionals of fractional Brownian motions on a circle with positive Hurst index s -1/2.  For a special Cameron-Martin shift, the low variance limit of the fractional Brownian motion induces a LLN and CLT for the associated random curves that is nearly identical to the global behavior of Plancherel measures on large Young diagrams.  The limit shape is independent of s and is that of Vershik-Kerov-Logan-Shepp.  The global Gaussian fluctuations depend on s and, if we continue s to negative values, coincides with the process in Kerov's CLT for s = - 1/2.  Although it might be possible to give a direct explanation for this coincidence by regularization, in this talk we give an indirect dynamical explanation by combining (i) results of Eliashberg and Dubrovin for a specific Hamiltonian QFT and (ii) the fact that in Hamiltonian systems, at short time scales, the quantum evolution of pure Gaussian wavepacket initial data agrees statistically with the classical evolution of mixed Gaussian random initial data.