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We compute the degree of Stiefel manifolds, that is, the variety of orthonormal frames in a finite dimensional vector space. Our approach employs techniques from classical algebraic geometry, algebraic combinatorics, and classical invariant theory.
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We give a different and possibly more accessible proof of a general Borsuk--Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of certain $(mathbb{Z}/2)^k$-equivariant maps from a product of $k$ spheres to the unit sphere in a real $(mathbb{Z}/2)^k$-representation of the same dimension. Our proof method allows us to derive Borsuk--Ulam theorems for certain equivariant maps from Stiefel manifolds, from the corresponding results about products of spheres, leading to alternative proofs and extensions of some results of Fadell and Husseini.
We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.
We study the maximum likelihood degree (ML degree) of toric varieties, known as discrete exponential models in statistics. By introducing scaling coefficients to the monomial parameterization of the toric variety, one can change the ML degree. We show that the ML degree is equal to the degree of the toric variety for generic scalings, while it drops if and only if the scaling vector is in the locus of the principal $A$-determinant. We also illustrate how to compute the ML estimate of a toric variety numerically via homotopy continuation from a scaled toric variety with low ML degree. Throughout, we include examples motivated by algebraic geometry and statistics. We compute the ML degree of rational normal scrolls and a large class of Veronese-type varieties. In addition, we investigate the ML degree of scaled Segre varieties, hierarchical loglinear models, and graphical models.
We study which quadratic forms are representable as the local degree of a map $f : A^n to A^n$ with an isolated zero at $0$, following the work of Kass and Wickelgren who established the connection to the quadratic form of Eisenbud, Khimshiashvili, and Levine. Our main observation is that over some base fields $k$, not all quadratic forms are representable as a local degree. Empirically the local degree of a map $f : A^n to A^n$ has many hyperbolic summands, and we prove that in fact this is the case for local degrees of low rank. We establish a complete classification of the quadratic forms of rank at most $7$ that are representable as the local degree of a map over all base fields of characteristic different from $2$. The number of hyperbolic summands was also studied by Eisenbud and Levine, where they establish general bounds on the number of hyperbolic forms that must appear in a quadratic form that is representable as a local degree. Our proof method is elementary and constructive in the case of rank 5 local degrees, while the work of Eisenbud and Levine is more general. We provide further families of examples that verify that the bounds of Eisenbud and Levine are tight in several cases.
In this paper we compute the cohomology of the Fano varieties of $k$-planes in the smooth complete intersection of two quadrics in $mathbb{P}^{2g+1}$, using Springer theory for symmetric spaces.
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