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.

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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.

Margulis inequalities and Margulis functions (a.k.a Foster-Lyapunov stability) have played a major role in modern dynamics, in particular in the fields of homogeneous dynamics and Teichmuller dynamics.
Moreover recent exciting developments in the field of random walks over manifolds give rise to related notions and questions in a much larger geometrical content, largely motivated by recent work of Brown-Eskin-Filip-Rodriguez Hertz.

I will explain what are Margulis functions and Margulis inequalities and describe the main lemma due to Eskin-Margulis (“uniform expansion”) that allows one to prove such an inequality. I will also try to sketch some interesting applications.

No prior knowledge is needed, the talk will be self-contained and accessible.

Zoom Link: https://gatech.zoom.us/j/94689623118?pwd=Ie8Ir2bExulIP4joQbcmZiwsxpIq75.1 Meeting ID: 946 8962 3118 Passcode: 910355

We consider Klein-Gordon equations with an external potential and cubic nonlinearity in three spatial dimensions. It is assumed that the linear operator has internal modes, and hence the unperturbed linear equation has multiple time-periodic solutions known as bound states. In 1999, Soffer and Weinstein treated the case when the linear operator has one large eigenvalue and proved the decay of the solution. In 2022, we solved the general one eigenvalue case. In our recent work, we solved the multiple internal modes case: the operator can has multiple and possibly degenerate eigenvalues. Indeed, we determined the sharp decay rate of the overall solution, as well as distinct decay rates for different modes of the solution. This is a joint work with Prof. Zhen Lei and Dr. Jie Liu.

Statistical mechanics explains that systems in thermal equilibrium spend a greater fraction of their time in states with apparent order because these states have lower energy. This explanation is remarkable, and powerful, because energy is a "local" property of states. While non-equilibrium steady states can similarly exhibit order, there can be no local property analogous to energy that explains why, as Landauer argued 50 years ago. However, recent experiments suggest that a local property called “rattling” predicts which states are favored, at least for a broad class of non-equilibrium systems.

I will present a Markov chain theory that explains when and why rattling predicts non-equilibrium order. In brief, it "works" when the correlation between a Markov chain's effective potential and the logarithm of its exit rates is high. It is therefore important to estimate this correlation in different classes of Markov chains. As an example, I will discuss estimates of the correlation exhibited by reaction kinetics on disordered energy landscapes, including dynamics of the random energy model and the Sherrington–Kirkpatrick spin glass. (Joint work with Dana Randall.)