In recent years, algebraic geometry (both complex and real) has proven to be useful in numerous applications in optimization, statistics, quantum information, and physics. In this thesis, we concentrate on studying semi-algebraic sets and varieties defined over the real numbers that arise in these applied contexts.
We begin with the study of Gibbs manifolds and Gibbs varieties. Gibbs manifolds are images of affine spaces of symmetric matrices under the matrix exponential map. They appear naturally in the context of entropic regularization for semidefinite programming or entropy maximization in quantum information theory. The Gibbs variety is the zero locus of all polynomials that vanish on the Gibbs manifold. We compute these polynomials and show that the Gibbs variety is low-dimensional. We give an exact formula for this dimension, and an upper bound for the degree of the Gibbs variety. We apply our theory to a range of scenarios: matrix pencils, quantum optimal transport, and sparse matrices.
The role of Gibbs manifolds in quantum information theory leads us to consider the notion of quantum conditional independence from an algebraic perspective. We take inspiration from algebraic statistics, where graphical models encoding conditional independence relations can be described as intersections of an algebraic variety with the probability simplex, and study quantum counterparts of such models. We present several ways to associate an algebraic variety to such a model. We study basic properties of these varieties and provide algorithms to compute their defining equations. We also study toric varieties defined by commuting Hamiltonians arising from a graph in the context of stabilizer codes. We give an efficient algorithm to compute the defining equations of such a toric variety. Moreover, we investigate a quantum analog of maximum likelihood estimation for quantum exponential families, the so-called quantum information projection.
We continue with studying (semi-)algebraic geometry of minimizing dual volumes of polytopes. Similarly to Gibbs manifolds, this appears in the context of regularization of convex optimization problems. The interior point of a convex polytope that leads to a polar dual of minimal volume is called the Santaló point. When translating the facet hyperplanes, the Santaló point traces out a semi-algebraic set called the Santaló patchwork. We describe and compute this set. We then investigate several naturally defined algebraic varieties containing the Santaló patchwork. We continue by treating the question of computing the Santaló points of polytopes numerically. We also explore connections to physics and algebraic statistics.
Finally, we study Grasstopes. This is yet another class of semi-algebraic sets inspired by physics. These are linear projections of the positive Grassmannian. When the linear projection is given by a totally positive matrix, we recover the definition of the amplituhedron, a semi-algebraic set that computes scattering amplitudes in a certain quantum field theory. We concentrate on the case when the image lives in the projective space, and give a combinatorial characterization of such Grasstopes in terms of sign flips, extending the results of Karp and Williams for the amplituhedron.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:93374 |
| Date | 16 August 2024 |
| Creators | Pavlov, Dmitrii |
| Contributors | Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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