Georgia Institute of Technology College of Sciences

Power-flow equations model the intricate balancing conditions in electric power grids and are central to analysis and control.  They can be reformulated as Laurent polynomial systems, which makes algebraic and polyhedral techniques applicable.  In this talk, we first explore different ways in which this can be done.

However, certain algebraic formulations may be deficient: the actual number of isolated solutions (counting multiplicity) may fall below the Bernshtein–Kushnirenko–Khovanskii (BKK) bound predicted from Newton polytopes.  By choosing a proper parametrization one uncovers that this deficiency exhibits a certain toric structure.  Recognizing that structure reframes the deficit as a geometric feature rather than a numerical anomaly.  In the second part of this talk, we explore variations of polyhedral homotopy methods designed to respect and exploit this structure.

====(Below is the information on the pre-talk.)====

Title: Mixed volume, mixed cells, and stable self intersections

Abstract: This talk provides an introduction to mixed volume, mixed cells, and their connections to the Bernshtein–Kushnirenko–Khovanskii bound, as well as stable intersections of tropical hypersurfaces.

Giroux and Pardon conjectured that a Lagrangian L in a Weinstein manifold W is regular (that is, compatible with the Weinstein structure in a natural sense) if there is a Lefschetz fibration p: W \to \C such that p(L) is a ray. In this talk, I will discuss forthcoming joint work with A. Roy and L. Wang, which establishes this conjecture. As an application of the proof, we show how all fillings of the rainbow closures of a positive braid can be described by manipulations of arcs in the base of an appropriate Lefschetz fibration.

In this talk, I'll discuss the asymptotics of the cubic nonlinear Schrödinger equation with potential in dimension 1 for small, localized initial data. In the case when the potential is equal to 0, it has been known for some time that solutions exhibit modified scattering. Due to additional complications introduced by the potential, the case with V nonzero has not been addressed until recently. 

Here, we present a method to obtain asymptotics for this problem.  The main ingredients are  (1) a new linear identity, which allows us to relate certain vector field-like quantities for the problem with a potential to those for the problem with no potential, and (2) an adaptation of the method of testing with wave packets introduced by Ifrim and Tataru. Compared to previous results, this method can handle potentials with slower decay at infinity.

We prove that graphs that do not contain a totally odd immersion of $K_t$ are $\mathcal{O}(t)$-colorable. In particular, we show that any graph with no totally odd immersion of $K_t$ is the union of a bipartite graph and a graph which forbids an immersion of $K_{\mathcal{O}(t)}$. Our results are algorithmic, and we give a fixed-parameter tractable algorithm (in $t$) to find such a decomposition.

In order to distinguish Legendrians with the same classical invariants, Chekanov and Eliashberg separately defined the Chekanov-Eliashberg DGA. Chekanov further defined a linearized version. Ekholm, Honda, and Kalman showed an exact Lagrangian cobordism between two Legendrians induces a DGA map on their respective DGAs. We show how to adapt this map to the linearized version. Time permitting, we will use this map to obstruct invertible concordances between negative twist knots. This is joint work with Sierra Knavel.

Average-case reductions establish rigorous connections between different statistical models, allowing us to show that if one problem is computationally hard, then another must be as well. Reductions from the planted clique problem have revealed statistical-to-computational gaps in many statistical problems with combinatorial structure. However, several important models remain beyond the reach of existing reduction techniques—for example, no reduction-based hardness results are currently known for sparse phase retrieval.

In this talk, we introduce a computationally efficient procedure that approximately transforms a single observation from certain source models with continuous-valued sample and parameter spaces into a single observation from a broad class of target models. I will present several such reductions and highlight their applications in computational lower bounds, including universality results and hardness in sparse generalized linear models. I will also discuss a potential application in transforming one differentially private mechanism into another.

This is joint work with Guy Bresler and Ashwin Pananjady. Part of the talk is based on the paper: https://arxiv.org/abs/2402.07717.

We study differential inclusions of the type $A v=0$ and $v \in K$, where $v$ is a vector field satisfying a linear PDE system $A$ and $K$ is a compact set. We are particularly interested in the case when $K$ consists of two vectors (\textit{two-state problem}). We consider Dirichlet boundary conditions for $v$, in which case the differential inclusion typically has no solutions. We study a suitable relaxation of the system, in which we penalize the surface energy required to switch between the two states. We study the asymptotics of the regularized energy integral. We show that the asymptotics depend polynomially on the regularization parameter with a quantification which — somewhat surprisingly — depends on the order of the linear PDE system $A$. Joint work with A. R\”{u}land, C. Tissot, A. Tribuzio.

The classical Wiener-Wintner Theorem says that for all measure preserving systems, and bounded functions f, there is a set of full measure so that the averages below converge for all continuous functions  g from the circle (R/Z)  to the complex numbers.

N^{-1} \sum_{n=1}^N  g( \pi n) f(T^n). 

We extend this result to averages over the prime integers. The proof uses structure of measure preserving systems, higher order Fourier analysis, and the Heath-Brown approximate to the von Mangoldt function.  A key result is a surprisingly small  Gowers norm estimate for the Heath-Brown approximate with fixed height.  

Joint work with  Y. Chen, A. Fragkos,  J. Fornal, B. Krause, and H. Mousavi.  

The rank-$n$ Dowling geometry $Q_n(\Gamma)$ is a matroid associated with a graph edge-labeled by elements of the finite group $\Gamma$. We determine the maximum size of an $N$-free submatroid of $Q_n(\Gamma)$ for various choices of $N$, including subgeometries $Q_m(\Gamma')$, lines $U_{2,\ell}$, and graphic matroids $M(H)$. When the group $\Gamma$ is trivial and $N=M(K_t)$, this problem reduces to Tur\'{a}n's classical result in extremal graph theory. We show that when $\Gamma$ is nontrivial, a complex dependence on $\Gamma$ emerges, even when $N=M(K_4)$.

This is joint work with Rutger Campbell and Jorn van der Pol.