The Santaló point of a convex polytope is the interior point which leads to a polar dual of minimal volume. This dual volume replaces other natural objective functions in convex optimization, such as the logarithmic barrier minimized by the analytic center. When translating the facet hyperplanes, the Santaló point traces out a patchwork of semialgebraic sets. I will describe and compute this geometry using algebraic and numerical techniques. I will also explore connections with statistics, optimization and physics. This is joint work with Dmitrii Pavlov.
One of the most fundamental questions in tropical geometry is: How much topological information does the tropicalization of a variety retain.
Although Viro patchworking was one of the earliest achievements in tropical geometry, the area of real tropical geometry remains comparatively unexplored. In this talk we study real tropicalizations of discriminants, complete intersections, and steady state varieties from reaction networks.
I will present an ongoing work in progress with Simon Telen where we study amplituhedra contained in the Grassmannian Gr(2,4) of lines in projective 3-space. We are particularly interested in geometric descriptions, the algebraic boundary, and the adjoint which should correspond to the numerator of the canonical form in the sense of positive geometry.
The amplituhedron is an object defined by physicists to understand particle scattering, and has garnered much recent attention from physicists and mathematicians alike. Mathematically, it is a linear projection of a positive Grassmannian to a smaller Grassmannian, via a totally positive matrix. Generalizing the amplituhedron, we define a Grasstope to be such a projection by any matrix.
In this talk we ask: can we gain new insights by broadening our horizons, and systematically studying all Grasstopes? I will explain some current work with Yelena Mandelstam and Dmitrii Pavlov in this direction.
The spectrum of the Jacobian matrix G of a dynamical system plays a central role in the stability and bifurcation analysis of equilibria. In particular, a complex pair of purely imaginary eigenvalues of G is a necessary condition for Hopf bifurcation and consequent oscillatory behavior.
Reaction networks give rise to parametric systems of equations, and thus to parametric families of Jacobian matrices. In this talk I share our work-in-progress about characterizing the networks for which there is no choice of parameters such that G possesses purely imaginary eigenvalues.
