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Computational Geometry |
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Nikolaus school hosted by TU Kaiserslautern and SFB-TRR 195 at Fraunhofer Institute ITWM, Kaiserslautern |
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Schedule |
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Monday (28th November 2022):
Tuesday (29th November 2022):
Wednesday (30th November 2022):
Thursday (1st December 2022):
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Lecture series Lecture series one: Gavin Brown Title: Birational geometry of 3-folds and A_\infty methods. Abstract: I will explain some ideas and results on 3-fold flops. Lecture series two: Alexander M Kasprzyk Title: Towards the landscape of Fano threefolds: computations and classifications. Abstract: The classification of Q-Fano threefolds (Fano varieties with Q-factorial terminal singularities) is a driving open problem in algebraic geometry. Thanks to work by Brown, Reid, and others, the Graded Ring Database contains an over-classification of all possible Hilbert series; what we do not yet know is which of these candidate Hilbert series are realised by a Q-Fano threefold. Recent work on the classification problem draws inspiration from Mirror Symmetry. Although still very conjectural and experimental, a clear description of how to systematically produce Q-Fano threefolds is emerging. This combines ideas from lattice polytopes, huge databases of combinatorial classifications, and vast distributed computer algebra calculations totalling millennia of CPU time. I will explain the ideas behind this work, with an emphasis on the computational challenges we face, and describe some of the many possible directions for collaboration. Lecture series three: Vladimir Lazic Title: An overview of the Minimal Model Program. Abstract: In these mostly theoretical lectures I will give an overview of the Minimal Model Program, which is a still incomplete classification programme in higher dimensional algebraic geometry. I will describe what has been done (=a lot), what remains to be done (=a lot), and present some very recent advances where computer algebra methods have been successfuly applied, as well as open questions which one hopes could be attacked by computer algebra systems such as OSCAR. Lecture series four: Frank-Olaf Schreyer Title: Computer aided constructions and unirational proofs of moduli spaces.Abstract: Computer algebra allows to make complicated constructions
exlicit. A proof of the unirationality of a parameter space, e.g., a
Hilbert scheme or a moduli space, allows in principle to write down
explicitely the universal family. However nowdays computer algebra
systems are usually not strong enough to write down the unversal family
with free parameters.
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Talks Isabel Stenger Title: On the Morrison-Kawamata cone conjecture
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Sabrina Gaube Title: Algorithmic strategies for resolution of determinantal singularities Abstract: Resolution of singularities plays an
important role in Algebraic Geometry. The problem, whether such a
resolution exists, has been solved completely in characteristic 0 by
Hironaka in 1964. This problem is still open in positive
characteristic for dimension greater than 3. |
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Linda Hoyer
Titel: Orthogonal Determinants of unipotent characters of GL_n(q) Abstract: We construct a (potential) method to calculate the square class of the Gram determinant of a GL_n(q)-invariant bilinear form on a representation over the rational numbers affording a given unipotent character of even degree. For this, we use the geometry of flag varieties and the Bruhat order of the symmetric groups. |
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Tobias Metzlaff Title: Orthogonality regions of generalized Chebyshev polynomials as basic semi-algebraic sets Abstract: The theory of crystallographic root systems allows to define a family of orthogonal multivariate Chebyshev polynomials that have connections to Fourier analysis and representations of Lie algebras. Their regions of orthogonality have intriguing shapes and singularities and are the images of generalized cosine functions. In this talk, we shall describe those regions as basic semi-algebraic sets, providing the polynomial inequations that define them as the positivity locus of matrix polynomials. For the families A, B, C, D and G, we give an explicit formula for those matrices in the standard monomial basis and in the Chebyshev basis. Based on joint work with Evelyne Hubert (Inria d'Universite Cote d'Azur) and Cordian Riener (UiT the Arctic University): https://arxiv.org/abs/2203.13152 |
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Rafael Mohr Title: A Simple Recursive Algorithm for Equidimensional Decomposition of Algebraic Sets Abstract: We provide a recursive algorithm that decomposes an algebraic set into pairwise disjoint locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. On a theoretical level, the algorithm only uses elementary properties of saturations of ideals. The standard way of performing these saturations with Gröbner basis computations is then utilized to implement the various operations that we need to perform on equidimensional locally closed sets. Experimental results using an implementation in Oscar indicate that the algorithm is able to tackle polynomial systems which are out of reach of other algorithms for equidimensional decomposition available in state-of-the-art computer algebra systems. |
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Stevell Muller
Title: Automorphisms of double covers of EPW cubes Abstract: In 2015, A. Iliev, G. Kapustka, M. Kapustka and K. Ranestad construct the first non-trivial example of hyperkaehler manifolds of K3^[3]-type, called double cover of EPW cube. Close to K. O'Grady's construction of double EPW sextics, double covers of EPW cubes are obtain by considering Lagrangian degeneracy loci in the Grassmannian $Gr(3, 6)$. This talk will be an overview about this construction and what is known about the automorphisms of such (projective) varieties. |
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Franco Rota Title: Homological mirror symmetry for anticanonical log del Pezzo surfaces Abstract: A classification by Johnson and Kollar produces a series of log terminal, quasi-smooth hypersurfaces in weighted projective space. Hodge-theoretic mirrors to these surfaces have only been recently constructed by G. Gugiatti and A. Corti: I will report on an ongoing collaboration with G. Gugiatti and M. Habermann aiming to lift the mirror statement to the homological level. The simplest case of the Johnson-Kollar series is a smooth del Pezzo surface. For the purpose of the talk, I will focus mainly on this example, since interesting mirror symmetry questions arise already at this level, and some of the technical difficulties are already evident. |
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Johannes Schmitt
Titel: Computing Cox Rings of Linear Quotients in OSCAR. Abstract: By a theorem of Arzhantsev and Gaifullin, the Cox ring of a linear
quotient is isomorphic to a certain invariant ring with a non-standard
grading by a finite abelian group.
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Sebastian Seemann
Title: Positive geometries - examples and open problems Abstract: I will provide some examples of positive geometries. They are frequently arising as the underlying mathematical structures for quantum mechanical observables of many theories in particle physics. In particular, every positive geometry has a so-called canonical form, that encodes its associated physical observables. I will present some examples of positive geometries among polytopes and polypols, and show how to compute their canonical forms. Then, I will outline some computational problems about positive geometries, such as their boundaries, and their canonical forms. |
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