Georgia Institute of Technology College of Sciences

Batchelor's law describes the power law spectrum of the turbulent regime of passive scalars (e.g., temperature or a dilute concentration of some tracer chemical) advected by an incompressible fluid (e.g., the Navier-Stokes equations at fixed Reynolds number), in the limit of vanishingly low molecular diffusivity. Predicted in 1959, it has been confirmed empirically in a variety of experiments, e.g. salinity concentrations among ocean currents. On the other hand, as with many turbulent regimes in physics, a true predictive theory from first principles has been missing (even a non-rigorous one), and there has been some controversy regarding the extent to which Batchelor's law is universal. 

In this talk, I will present a program of research, joint with Jacob Bedrossian (UMD) and Sam Punshon-Smith (Brown), which has rigorously proved Batchelor's law for passive scalars advected by the Navier-Stokes equations on the periodic box subjected to Sobolev regular, white-in-time body forces. The proof is a synthesis of techniques from dynamical systems and smooth ergodic theory, stochastics/probability, and fluid mechanics. To our knowledge, this work constitutes the first mathematically rigorous proof of a turbulent power law spectrum. It also establishes a template for predictive theories of passive scalar turbulence in more general settings, providing a strong argument for the universality of Batchelor's law. 

String topology studies various algebraic structures given by intersecting loops in a manifold, as well as those on the Hochschild chains or homology of an algebra. In this preparatory talk, we survey a collection of such structures and their relationships with one another.

It's know that when discretizing stochastic ordinary equation with non-globally Lipschitz coefficient, the traditional numerical method, like
Euler method, may be divergent and not converge in strong or weak sense. For stochastic partial different equation with non-globally Lipschitz
coefficient, there exists fewer result on the strong and weak convergence results of numerical methods. In this talk, we will discuss several numerical schemes approximating stochastic Schrodinger Equation.  Under certain condition, we show that the exponential integrability preserving schemes are strongly and weakly convergent with positive orders.

Spaces of fatgraphs have long been used to study a variety of topics in math and physics. In this talk, we introduce two spaces of fatgraphs arising in string topology—one which parameterizes operations on chains of the free loop space of a manifold and one which parametrizes operations on Hochschild cochains of a “V-infinity” algebra. We present a conjecture relating these two spaces to one another and to the moduli space of Riemann surfaces. We also introduce polyhedra called “assocoipahedra” which generalize Stasheff’s associahedra to algebras with a compatible co-inner product. Assocoipahedra are used to prove that the dioperad governing V-infinity algebras satisfies certain algebraic properties. 

A combinatorial neural code is convex if it arises as the intersection pattern of convex open subsets of Euclidean space. We relate the emerging theory of convex neural codes to the established theory of oriented matroids, both categorically and with respect to feasibility and complexity. By way of this connection, we prove that all convex codes are related to some representable oriented matroid, and we show that deciding whether a neural code is convex is NP-hard.

Noetherian operators are a set of differential operators that encode the scheme structure of a primary ideal. We propose a framework for studying primary ideals numerically by using a combination of witness sets and Noetherian operators. We will also present a method for computing Noetherian operators using numerical data. 

The first step in the theory of Noetherian operators are the Macaulay dual spaces. Indeed, for an ideal that is primary over a maximal ideal corresponding to a rational point, the generators of the dual space are a valid set of Noetherian operators. We will start by presenting basic ideas, results and algorithms in the classical dual space theory, and then revisit some of these ideas in the context of Noetherian operators.

Descriptive combinatorics studies the interaction between classical combinatorial concepts, such as graph colorings and matchings, and notions from measure theory and topology. Results in this area enable one to apply combinatorial techniques to problems in other (seemingly unrelated) branches of mathematics, such as the study of dynamical systems. In this talk I will give an introduction to descriptive combinatorics and discuss some recent progress concerning a particular family of combinatorial tools---the probabilistic method---and its applications in the descriptive setting.

An embedding of a manifold into a trivial disc bundle over another manifold is called braided if projection onto the first factor gives a branched cover. This notion generalizes closed braids in the solid torus, and gives an explicit way to construct many embeddings in higher dimensions. In this talk, we will discuss when a covering map of surfaces lift to a braided embedding.

Let $N_n$ be an $n\times n$ matrix whose entries are i.i.d. copies of a random variable $\zeta=\xi+i\xi'$, where $\xi,\xi'$ are i.i.d., mean zero, variance one, subgaussian random variables. We will present a result of Luh, according to which the probability that $N_n$ has a real eigenvalue is exponentially small in $n$. An interesting part of the proof is a small ball probability estimate for the smallest singular value of a complex perturbation $M_n=M+N_n$ of the original matrix.

I will present some recent results, obtained with D. Bresch and Z. Wang, on large stochastic many-particle or multi-agent systems. Because such systems are conceptually simple but exhibit a wide range of emerging macroscopic behaviors, they are now employed in a large variety of applications from Physics (plasmas, galaxy formation...) to the Biosciences, Economy, Social Sciences...

The number of agents or particles is typically quite large, with 10^20-10^25 particles in many Physics settings for example and just as many equations. Analytical or numerical studies of such systems are potentially very complex  leading to the key question as to whether it is possible to reduce this complexity, notably thanks to the notion of propagation of chaos (agents remaining almost uncorrelated).

To derive this propagation of chaos, we have introduced a novel analytical method, which led to the resolution of two long-standing conjectures:
        _The quantitative derivation of the 2-dimensional incompressible Navier-Stokes system from the point vortices dynamics;
       _The derivation of the mean-field limit for attractive singular interactions such as in the Keller-Segel model for chemotaxis and some Coulomb gases.

Given a vertex-weighted graph G= (V, E), the MaximumWeight Internal Spanning Tree (MWIST) problem is to find a spanning tree T of G such that the total weight of internal vertices in T is maximized. The unweighted version of this problem, known as Maxi-mum Internal Spanning Tree (MIST) problem, is a generalization of the Hamiltonian path problem, and hence, it is NP-hard. In the literature lot of research has been done on designing approximation algorithms to achieve an approximation ratio close to 1. The best known approximation algorithm achieves an approximation ratio of 17/13 for the MIST problem for general graphs. For the MWIST problem, the current best approximation algorithm achieves an approximation ratio of 2 for general graphs. Researchers have also tried to design exact/approximation algorithms for some special classes of graphs. The MIST problem parameterized by the number of internal vertices k, and its special cases and variants, have also been extensively studied in the literature. The best known kernel for the general problem has size 2k, which leads to the fastest known exact algorithm with running time O(4^kn^{O(1)}). In this talk, we will talk about some selected recent results on the MWIST problem.

In critical Bernoulli percolation on $\mathbb{Z}^d$ for $d$ large, it is known that there are a.s. no infinite open clusters. In particular, for n large, every path from the origin to the boundary of $[-n, n]^d$ must contain some closed edges. Let $T_n$ be the (random) minimal number of closed edges in such a path. How does $T_n$ grow with $n$? We present results showing that for d larger than the upper critical dimension for Bernoulli percolation ($d > 6$), $T_n$ is typically of the order $\log \log n$. This is in contrast with the $d = 2$ case, where $T_n$ grows logarithmically. Perhaps surprisingly, the model exhibits another major change in behavior depending on whether $d > 8$.

This is the open forum for Pierre-Emmanuel   Jabin (https://home.cscamm.umd.edu/~jabin/)

as a candidate for Elaine M. Hubbard Chair in Mathematics.

In the $k$-cut problem, we are given an edge-weighted graph and want to find the least-weight set of edges whose deletion breaks the graph into $k$ connected components. Algorithms due to Karger-Stein and Thorup showed how to find such a minimum $k$-cut in time approximately $O(n^{2k-2})$. The best lower bounds come from conjectures about the solvability of the $k$-clique problem and a reduction from $k$-clique to $k$-cut, and show that solving $k$-cut is likely to require time $\Omega(n^k)$. Our recent results have given special-purpose algorithms that solve the problem in time $n^{1.98k + O(1)}$, and ones that have better performance for special classes of graphs (e.g., for small integer weights).

In this work, we resolve the problem for general graphs, by showing that for any fixed $k \geq 2$, the Karger-Stein algorithm outputs any fixed minimum $k$-cut with probability at least $\widehat{O}(n^{-k})$, where $\widehat{O}(\cdot)$ hides a $2^{O(\ln \ln n)^2}$ factor. This also gives an extremal bound of $\widehat{O}(n^k)$ on the number of minimum $k$-cuts in an $n$-vertex graph and an algorithm to compute a minimum $k$-cut in similar runtime. Both are tight up to $\widehat{O}(1)$ factors.

The first main ingredient in our result is a fine-grained analysis of how the graph shrinks---and how the average degree evolves---under the Karger-Stein process. The second ingredient is an extremal result bounding the number of cuts of size at most $(2-\delta) OPT/k$, using the Sunflower lemma.

The most popular model for Image Denoising is without any doubt the ROF (for Rudin-OsherFatemi) model. However, since the ROF approach has some drawbacks (the stair-case effect being one of them) practitioners have been looking for alternatives. One of them is the Elastica model, relying on the minimization in an appropriate functional space of the energy functional $J$ defined by

$$ J(v)=\varepsilon \int_{\Omega} \left[ a+b\left| \nabla\cdot \frac{\nabla v}{|\nabla v|}\right|^2 \right]|\nabla v| d\mathbf{x} + \frac{1}{2}\int_{\Omega} |f-v|^2d\mathbf{x} $$

where in $J(v)$: (i) $\Omega$ is typically a rectangular region of $R^2$ and $d\mathbf{x}=dx_1dx_2$. (ii) $\varepsilon, a$ and $b$ are positive parameters. (iii) function $f$ represents the image one intends to denoise.

Minimizing functional $J$ is a non-smooth, non-convex bi-harmonic problem from Calculus of  Variations. Its numerical solution is a relatively complicated issue. However, one can achieve this task rather easily by combining operator-splitting and finite element approximations. The main goal of this lecture is to describe such a methodology and to present the results of numerical experiments which validate it.