Sums of Squares and Symmetric Polynomials

Real algebraic geometry has a long and beautiful history going back to the 1800s. It is the study of real polynomials using algebraic techniques. Convex optimization plays a key role in applied mathematics and engineering, studying the geometric structure of convex sets that arise in optimization problems. Representation theory allows us to understand, study, and exploit the symmetries that naturally arise in many mathematical problems. The work in this thesis lies in the intersection of these fields, studying the symmetries of geometric objects coming from convex algebraic geometry and optimization. In particular, we study the spectrahedra that arise in the theory of symmetric polynomials and sums of squares (SOS) polynomials.Much of this thesis is motivated by polynomial optimization. It is through this lens that we arrive at the study of semidefinite programming, the feasible region of which is called a spectrahedron. A common method for solving a semidefinite program (SDP) is via interior-point methods. Interior-point algorithms cut out a path towards the optimal solution and taking the Zariski closure of this defines the central curve. In this thesis, we discuss the central curve in semidefinite programming, linear programming and quadratic programming. As our first contribution, we prove the degree of the central curve for a generic SDP is equal to the maximum likelihood (ML) degree of a statistical model formulated and discussed in [75]. As such the degree of the central curve of a generic SDP can be computed using complete quadrics [50, 53]. The study of sums of squares and invariant polynomials inevitably leads to the study of invariant semidefinite programs [32]. Such an SDP can be greatly simplified using symmetry reduction techniques. We explore the geometry of the spectrahedra that arise from the invariant SDPs when considering sums of squares and symmetric polynomials. The first spectrahedron we study is the symmetry-adapted PSD cone, denoted PSD_N^{S_n}, a spectrahedral cone that gives representations of symmetric SOS polynomials. We compute its dimension, characterize its extremal rays, and determine that this convex cone is Terracini convex. Furthermore, using tools from representation theory, one can enforce a block-diagonalized structure on the set of matrices in PSD_N^{S_n}. We study the structure of these blocks, each associated to an irreducible representation of the symmetric group. For the trivial irreducible representation, we provide an explicit description of these matrix blocks. The second spectrahedron is the symmetry-adapted Gram spectrahedron. For a given symmetric polynomial f of degree 2d in n variables, the symmetry-adapted Gram spectrahedron of f is the intersection of the affine subspace defined by the coefficients of f and the symmetry-adapted PSD cone. For particular n and d, we describe the facial structure of these spectrahedra, including for binary (n=2), quadratic (d=2), ternary quartic (n=3,d=4), and ternary sextic (n=3,d=6) polynomials. Finally, as an application of the above theory, we find several counterexamples to a 2011 conjecture presented by Cuttler, Greene, and Skandera in [24] pertaining to symmetric mean inequalities.

Advisor: Hosten, Serkan
Wilkening, Jon