Georgia Institute of Technology College of Sciences

A graph is chordal if each of its cycles of length at least four has a chord. Chordal graphs occupy an extreme end of a trade-off between structure and generalization: they have strong structure and admit many interesting characterizations, but this strong structure makes them a special case, representative of only a few graphs. In this talk, I'll discuss a new class of graphs called locally chordal graphs. Locally chordal graphs are more general than chordal graphs, but still have enough structure to admit interesting characterizations. In particular, most of the major characterizations of chordal graphs generalize to locally chordal graphs in natural and powerful ways. In addition to explaining these characterizations, I’ll discuss some ideas about how locally chordal graphs relate to new width parameters and to results in structural sparsity. This talk is based on joint work with Paul Knappe and Jonas Kobler. 

Algebraic torsion is a means of understanding the topological complexity of certain homomorphic curves counted in some Floer theories of contact manifolds.  This talk focuses on algebraic torsion and the contact invariant in embedded contact homology, useful for obstructing symplectic fillability and overtwistedness of the contact 3-manifold, but mostly left unexplored. We discuss results for concave linear plumbings of symplectic disk bundles over spheres admitting a concave contact boundary, whose boundaries are contact lens spaces.  We explain our curve counting methods in terms of the Reeb dynamics and their parallel with the topological contact toric description of these lens spaces. This talk is based on joint work with Aleksandra Marinkovic, Ana Rechtman, Laura Starkston, Shira Tanny, and Luya Wang.  Time permitting, we will discuss exploration of our methods to find nonfillable tight contact 3-manifolds obtained from more general plumbings.

A new strategy is presented for the statistical forecasts of multiscale nonlinear systems involving non-Gaussian probability distributions. The capability of using reduced-order models to capture key statistical features is investigated. A closed stochastic-statistical modeling framework is proposed using a high-order statistical closure enabling accurate prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. A new efficient ensemble forecast algorithm is developed dealing with the nonlinear multiscale coupling mechanism as a characteristic feature in high-dimensional turbulent systems. To address challenges associated with closely coupled spatio-temporal scales in turbulent states and expensive large ensemble simulation for high-dimensional complex systems, we introduce efficient computational strategies using the random batch method. Effective nonlinear ensemble filters are developed based on the nonlinear coupling structures of the explicit stochastic and statistical equations, which satisfy an infinite-dimensional Kalman-Bucy filter with conditional Gaussian dynamics. It is demonstrated that crucial principal statistical quantities in the most important large scales can be captured efficiently with accuracy using the new reduced-order model in various dynamical regimes of the flow field with distinct statistical structures.

The permutation removal lemma was first proved by Klimosová and Král’, and later reproved by Fox and Wei in the context of permutation property testing. In this talk, we study a local version of the permutation removal problem. We show that for any permutation σ not equal to 12, 21, 132, 231, 213, or 312, there exists ε(σ) > 0 such that for any sufficiently large integer N, there is a permutation π of length N that is ε-far from being σ-free with respect to the ρ∞ distance, yet contains only a single copy of σ. Here, the ρ∞ distance is defined as an L∞-variant of the Earth Mover’s Distance between two permutations. We will also discuss our result on the local induced graph removal problem. This is joint work with Fan Wei.