The Absolute Line Quadric and Camera Autocalibration

We introduce a geometrical object providing the same information as the absolute conic: the absolute line quadric (ALQ). After the introduction of the necessary exterior algebra and Grassmannian geometry tools, we analyze the Grassmannian of lines of P^3 from both the projective and Euclidean points of view. The exterior algebra setting allows then to introduce the ALQ as a quadric arising very naturally from the dual absolute quadric. We fully characterize the ALQ and provide clean relationships to solve the inverse problem, i.e., recovering the Euclidean structure of space from the ALQ. Finally we show how the ALQ turns out to be particularly suitable to address the Euclidean autocalibration of a set of cameras with square pixels and otherwise varying intrinsic parameters, providing new linear and non-linear algorithms for this problem. We also provide experimental results showing the good performance of the techniques

Fano varieties of K3 type and IHS manifolds

We construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure and we link some of them to projective families of irreducible holomorphic symplectic manifolds

Geometric, Algebraic, and Topological Combinatorics

The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics" was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) Karim Adiprasito presented his very recent proof of the $g$-conjecture for spheres (as a talk and as a "Q\&A" evening session) (2) Federico Ardila gave an overview on "The geometry of matroids", including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz

Algebraic Groups

The workshop dealt with a broad range of topics from the structure theory and the representation theory of algebraic groups (in the widest sense). There was emphasis on the following areas: • classical and quantum cohomology of homogeneous varieties, • representation theory and its connections to orbits and flag varieties

The Springer Morphism, Polynomial Representation Rings, and the Cohomology Ring of Grassmannians

To any almost faithful representation of a complex, connected, reductive algebraic group $G$ of highest weight $\\lambda$ one can associate a dominant morphism from the group to its Lie algebra $\\fg$. This map enjoys many nice properties. In particular, when restricted to a maximal torus it maps to the Cartan subalgebra. This map can be used to give a natural definition of polynomial representations for the classical groups of types B, C, and D. Given a parabolic subgroup $P\\subset G$, Kumar showed there is a surjective algebra homomorphism from the polynomial representations of a Levi subgroup of P to the cohomology of G/P which extends a classical result relating the polynomial representations of GL(r) and the cohomology of the Grassmannian of r-planes in n-space $H^*(Gr(r,n))$. In this work we give an explicit determination of the map \\theta_\\lambda for simple groups and consider Kumar's map for types B, C, and G.Doctor of Philosoph