| Date: | September 16, 2022 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Thomas Yahl, Texas A&M University |
| Title: | Computing Galois groups of Fano problems |
| Abstract: | A Fano problem consists of enumerating linear spaces of a fixed dimension on a variety, generalizing the classical problem of the 27 lines on a smooth cubic surface. Those Fano problems with finitely many linear spaces have an associated Galois group that acts on these linear spaces and controls the complexity of computing them in coordinates via radicals. Galois groups of Fano problems have been studied both classically and modernly and have been determined in some special cases. We use computational tools to prove that several Fano problems of moderate size have Galois group equal to the full symmetric group, all of which were previously unknown. |
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| Date: | September 19, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Chia-Yu Chang, TAMU |
| Title: | Maximal border subrank tensors |
| Abstract: | The motivation for this work was the study of the complexity of matrix multiplication. In this talk, I will introduce subrank and border subrank. Unlike their cousins, rank and border rank, almost nothing is known about them. I will prove a lower bound on the dimension of the set of maximal border subrank tensors. This is the first such result on maximal border subrank tensors. |
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| Date: | September 23, 2022 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Runshi Geng, TAMU |
| Title: | On the geometry of geometric rank |
| Abstract: | Geometric Rank of tensors was introduced by Kopparty et al. as a useful tool to study algebraic complexity theory, extremal combinatorics and quantum information theory. In this talk I will introduce Geometric Rank and results from their paper, in particular showing the relation between geometric rank and other ranks of tensors. Then I will present recent results of geometric rank, including and the connections between geometric rank and spaces of matrices of bounded rank, and classifications of tensors with geometric rank one, two and three.
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| Date: | September 30, 2022 |
| Time: | 5:00pm |
| Location: | BLOC 302 |
| Speaker: | TAGS conference |
| Title: | |
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| Date: | October 7, 2022 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Arpan Pal, TAMU |
| Title: | Concise tensors of minimal border rank |
| Abstract: | We know that if a collection of square matrices are simultaneously diagonalizable then they commute, however the converse does not hold. It has been a classical problem in linear algebra to classify the closure of the space of simultaneously diagonalizable matrices. This problem is closely related to a problem regarding tensors. In this talk I shall describe the problem, the relation to classical question, and recent progress towards classifying minimal border rank tensors. This is joint work with JM Landsberg and Joachim Jelisiejew. |
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| Date: | October 17, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Luigi Ferraro, Texas Tech |
| Title: | The Improved New Intersection Theorem Revisited |
| Abstract: | The origins of the Improved New Intersection Theorem can be traced back to the following linear algebra exercise: Let U, V, and W be vector spaces over a field with U and V subspaces of W. Then the dimension of the intersection of U and V is at least dim U + dim V - dim W. In their most modern forms, the intersection theorems are concerned with bounding the length of finite free complexes over local rings. In this talk, we will explore the history of these theorems, culminating in a result due to L. Christensen and me. |
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| Date: | November 18, 2022 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Amy (Hang) Huang, Auburn |
| Title: | Vanishing Hessian and Wild Polynomials |
| Abstract: | Notions of ranks and border rank abounds in the literature. Polynomials with vanishing hessian and their classification is also a classical problem. Motivated by an observation of Ottaviani, we will discuss why when looking at concise polynomials of minimal border rank, being wild, i.e. their smoothable rank is strictly larger than their border rank, is the same as having vanishing Hessian. The main tool we are using here is the recent work of Buczynska and Buczynski relating the border rank of polynomials and tensors to the multi-graded Hilbert scheme. From here, we found two infinite series of wild polynomials and we will try to describe their border varieties of sums of powers, which is an analog of the variety of sums of powers. |
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| Date: | December 2, 2022 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Daniel Erman, U. Wisc. |
| Title: | The geometry of weighted syzygies |
| Abstract: | In the 1980s, Mark Green gave a new perspective on the classical correspondence between geometry and polynomial equations. He showed that the defining equations of a curve become increasingly rigid as the degree increases, and that this rigidity could be precisely measured by syzygies. I will discuss how these ideas extend to new contexts like weighted projective spaces, and how this is a part of overarching effort to extend work on syzygies to toric varieties. |
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| Date: | December 5, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Bernhard Hanke, U. Augsburg |
| Title: | Lipschitz rigidity for scalar curvature |
| Abstract: | Lower scalar curvature bounds on spin Riemannian manifolds exhibit remarkable rigidity properties determined by spectral properties of Dirac operators. For instance, a fundamental result of Llarull states that there is no smooth Riemannian metric on the n sphere which dominates the round metric and whose scalar curvature is greater than or equal to the scalar curvature of the round metric, except the round metric itself. A similar result holds for smooth comparison maps from spin Riemannian manifolds to round spheres.
In a joint work with Simone Cecchini and Thomas Schick, we generalize this result to Riemannian metrics with regularity less than C^1 and Lipschitz comparison maps, answering a question of Gromov in his "Four Lectures". To this end, we rely on a notion of scalar curvature in the distributional sense introduced by Lee-LeFloch and on spectral properties of Lipschitz Dirac operators. It turns out that the existence of a nonzero harmonic spinor field - guaranteed by the Atiyah-Singer index theorem - forces the given comparison map to be quasiregular in the sense of Reshetnyak. Thus we build an unexpected bridge from spin geometry to the theory of quasiconformal mappings. |
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