Vorträge in der Woche 29.01.2024 bis 04.02.2024
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Dienstag, 30.01.2024: Tropical invariants of binary quintics and reduction types of Picard curves.
Yassine El-Maazouz (Aachen)
We give a general framework for tropical invariants associated with group actions on arbitrary varieties. This is then applied to find tropical invariants for binary forms by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification of M_{0,n}. This allows us to express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. This is joint work with Paul Helminck and Enis Kaya.
| Uhrzeit: | 14:15 |
| Ort: | C4H33 |
| Gruppe: | Oberseminar Kombinatorische Algebraische Geometrie |
| Einladender: | Felix Röhrle |
Mittwoch, 31.01.2024: Recursive Koszul flattening and tensor ranks of determinant/permanent tensors
Yeongrak Kim (Busan)
The rank of a tensor T is the minimum number of decomposable tensors whose sum equals to T which extends the notion of the matrix rank. Understanding the rank of a given tensor has great theoretical and practical applications, however, the rank of a tensor of high order is very hard to determine in most cases. For instance, Strassen's algorithm for matrix multiplication tells us that we only need 7 multiplications (not 8) when we multiply two 2 by 2 matrices, in other words, the 2 by 2 matrix multiplication tensor has rank 7. Usually, the study of rank complexities of a tensor is based on a flattening method that derives a certain matrix from the given tensor. The Koszul flattening method, introduced by Landsberg and Ottaviani, is a simple and powerful method that works for a tensor of order 3 using the exterior product. It has several applications in the study of lower bounds of tensor ranks and Waring ranks for various tensors (of order 3) appearing in algebra and geometry, including the matrix multiplication tensor and the determinant/permanent polynomial for 3 by 3 matrices. Motivated by their observations, I will introduce a recursive Koszul flattening method, a successive usage of Koszul flattening for tensors of higher orders. As applications, I will discuss some observations on the lower bounds on tensor ranks of the determinant and permanent as tensors of order n. These results greatly improve lower bounds on the border ranks of those tensors for n at least 4. This is a joint work in progress with Jong In Han and Jeong-Hoon Ju.
| Uhrzeit: | 10:00 - 11:00 |
| Ort: | S08 |
| Gruppe: | Oberseminar kombinatorische algebraische Geometrie |
| Einladender: | Daniele Agostini, Hannah Markwig |
Mittwoch, 31.01.2024: Category of matroids with coefficients
Manoel Jarra (Groningen)
Matroids are combinatorial abstractions of the concept of independence in linear algebra. There is a way back: when representing a matroid over a field we get a linear subspace. Another algebraic object for which we can represent matroids is the semifield of tropical numbers, which gives us valuated matroids. In this talk we introduce Baker-Bowler's theory of matroids with coefficients, which recovers both classical and valuated matroids, as well linear subspaces, and we show how to give a categorical treatment to these objects that respects matroidal constructions, as minors and duality. This is a joint work with Oliver Lorscheid and Eduardo Vital.
| Uhrzeit: | 11:00 - 12:00 |
| Ort: | S08 |
| Gruppe: | Oberseminar kombinatorische algebraische Geometrie |
| Einladender: | Daniele Agostini, Hannah Markwig |
Donnerstag, 01.02.2024: Interior-boundary conditions for the Dirac equation in naked Reissner-Nordström spacetimes
Bipul Poudyal (Tübingen)
Hamiltonians involving particle creation and annihilation in quantum field theory (QFT) are often ultraviolet divergent. To circumvent this problem, Teufel and Tumulka (2015) introduced a new way of obtaining well defined and ultraviolet finite Hamiltonians using what is called an Interior Boundary Condition (IBC). In this approach, particle creation and annihilation is modeled using boundary conditions that relate wave functions on n and n+1 particle sectors of Fock space. This method has been applied to various non-relativistic models and recently to the Dirac Hamiltonian in Minkowski spacetime. In this talk, we discuss the extension of this method to the case of Dirac Hamiltonians in the naked Reissner-Nordström background. In particular, we construct a self-adjoint IBC Hamiltonian involving creation and annihilation of Dirac particles at the singularity of naked Reissner-Nordström spacetime.
| Uhrzeit: | 14:30 |
| Ort: | C3N14 |
| Gruppe: | Oberseminar Mathematical Physics |
| Einladender: | Capel, Keppeler, Lemm, Pickl, Teufel, Tumulka |