MenuGeometry Seminar
Fall 2020
| Date: | August 26, 2020 |
| Time: | 10:00am |
| Location: | zoom |
| Speaker: | Anna Seigal, Oxford |
| Title: | Ranks of Cubic Surfaces |
| Abstract: | There are various notions of rank, which measure the complexity of a tensor or polynomial. Cubic surfaces can be viewed as symmetric tensors. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven. We then explore the connection between the rank of a polynomial and the singularities of its vanishing locus, and we find the possible singular loci of a cubic surface of given rank. This talk is based on joint work with Eunice Sukarto. |
| Date: | September 23, 2020 |
| Time: | 11:00am |
| Location: | zoom |
| Speaker: | Filip Rupniewski, IMPAN |
| Title: | Cactus rank and identification of secant inside cactus varieties |
| Abstract: | Every secant variety is contained in the corresponding cactus variety. However, according to our knowledge, there is no explicit equation of the secant variety which does not vanish on the cactus variety. I will present an algorithm for deciding if a given point in the cactus variety belongs to the secant variety in some special cases. I will also show the theorem for calculating the cactus rank of forms divisible by a large power of a linear form which allowed us to design the mentioned algorithm. Based on a joint work with M. Gałązka and T. Mańdziuk. |
| Date: | September 28, 2020 |
| Time: | 3:00pm |
| Location: | zoom |
| Speaker: | Timothy Duff, Georgia Tech |
| Title: | Structured polynomial systems in algebraic vision |
| Abstract: | Minimal problems arise in 3D reconstruction pipelines that attempt to recover the 3D geometry of a scene from data in several images. Solving minimal problems comes down to solving systems of polynomial equations of a very particular structure. Structure can be understood in terms of an associated branched cover and its birational invariants (degree, Galois/monodromy group.) Classical solutions to well-known problems of camera registration, homography estimation, and five-point relative pose implicitly exploit this structure. In work with Kohn, Leykin, and Pajdla, we identify a large zoo of new minimal problems, and in ongoing work with Korotynskiy, Pajdla, and Regan, we identify and further study those problems with special (eg. imprimitive) Galois/monodromy groups. |
| Date: | October 2, 2020 |
| Time: | 4:00pm |
| Location: | zoom |
| Speaker: | David, Sykes, TAMU |
| Title: | Local equivalence problems for 2-nondegenerate, hypersurface-type CR geometry studied via dynamical Legendrian contact structures. |
| Abstract: | The local differential geometry of Levi-nondegenerate CR structures is well understood due in large part to classical results of Cartan, Tanaka, Chern, and Moser, and yet comparatively little is known about other CR structures. There is a natural association between 2-nondegenerate, hypersurface-type CR structures – which are the main focus of this talk – and dynamical Legendrian contact structures, and, moreover, there is a broad class of these CR structures that can be uniquely recovered from their associated dynamical Legendrian contact structure. For these recoverable structures, we construct canonical absolute parallelisms on fiber bundles defined over a manifold with the given CR structure. The construction can be applied to discern local equivalence between CR structures. Other applications that will be discussed include upper bounds for the dimension of a CR manifold’s symmetry group and a characterization of local invariants of certain homogeneous CR manifolds. The latter application, coupled with results by Curtis Porter and Igor Zelenko, enables us to classify the local geometry of homogeneous, 2-nondegenerate, hypersurface-type CR manifolds in low dimensions. |
| Date: | October 7, 2020 |
| Time: | Noon |
| Location: | zoom |
| Speaker: | J. Weyman, U. Krakow |
| Title: | Structure of finite free resolutions. |
| Abstract: | In this talk I will describe the structure of finite free resolutions via so-called generic rings. In the first part I will go through older results of Buchsbaum-Eisenbud and Hochster. Then I will describe a more recent connection to the combinatorics of the root systems of T-shaped diagrams $T_{p,q,r}$. |
| Date: | October 9, 2020 |
| Time: | 4:00pm |
| Location: | zoom |
| Speaker: | Leon Zhang, UC Berkeley |
| Title: | Tropical geometry and applications |
| Abstract: | I will describe results from two recent projects in tropical geometry with relevance in applications. In the first half, I will introduce and give several characterizations for flags of tropical linear spaces, in analogy to Speyer's results for tropical linear spaces. In the second half, I will discuss ongoing work relating tropical fewnomials, vertex bounds of Minkowski sums, and linear regions of maxout neural networks. |
| Date: | October 12, 2020 |
| Time: | 3:00pm |
| Location: | zoom |
| Speaker: | F. Gesmundo, U. Copenhage |
| Title: | Approaching the boundary of tensor network varieties |
| Abstract: | Tensor network states are particular tensors arising via contractions determined by the combinatorics of a weighted graph and are used as ansatz class for a number of problems in applied mathematics. If the graph contains cycles, the corresponding set of tensor network states is (often) not closed in the Zariski topology; its closure is usually referred to as the tensor network variety. There are several tensors of interest lying on the "boundary", that is the difference between the variety and the set itself. In recent work, we introduced sets of tensors, arising in a natural geometric way, which include tensors at the boundary and offer similar properties as the ansatz class of tensor network states. In this seminar, I will introduce the tensor network variety, will show some properties of the boundary and will illustrate how the new ansatz class comes into play. This is based on joint work with M. Christandl, D. Stilck-Franca and A. Werner. |
| Date: | October 14, 2020 |
| Time: | 11:00am |
| Location: | zoom |
| Speaker: | Markus Blaeser |
| Title: | Irreversibility of tensors of minimal border rank and barriers for fast matrix multiplication |
| Abstract: | Determining the asymptotic algebraic complexity of matrix multiplication is a central problem in algebraic complexity theory. The best upper bounds on the so-called exponent of matrix multiplication if obtained by starting with an "efficient" tensor, taking a high power and degenerating a matrix multiplication out of it. In the recent years, several so-called barrier results have been established. A barrier result shows a lower bound on the best upper bound for the exponent of matrix multiplication that can be obtained by a certain restriction starting with a certain tensor. We prove the following barrier over the complex numbers: Starting with a tensor of minimal border rank satisfying a certain genericity condition, except for the diagonal tensor, it is impossible to prove ω = 2 using arbitrary restrictions. This is astonishing since the tensors of minimal border rank look like the most natural candidates for designing fast matrix multiplication algorithms. We prove this by showing that all of these tensors are irreversible, using a structural characterisation of these tensors. Joint work with Vladimir Lysikov. |
| Date: | October 19, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Guangbo Xu, Texas A&M University |
| Title: | Fukaya categories and blowups |
| Abstract: | Under Kontsevich's homological mirror symmetry conjecture, the Fukaya category is the open-string invariant on the symplectic side (A-model), as opposed to the derived category of coherent sheaves as the invariant on the complex side (B-model). In this semi-expository talk, I will first give a hand-waiving introduction to the Fukaya category. Then I will talk about a recent result joint with Sushmita Venugopalan and Chris Woodward. We showed that under a point blowup, the Fukaya category "grows" in a similar way as the change of the cohomology group. |
| Date: | November 13, 2020 |
| Time: | 4:00pm |
| Location: | zoom |
| Speaker: | Joe Kileel , University of Texas Austin |
| Title: | Fast symmetric tensor decomposition |
| Abstract: | Tensors are higher-order matrices, and decomposing tensors can reveal structure in datasets. In recent years, tensor decomposition has found applications in statistics, computational imaging, signal processing, and quantum chemistry. In this talk, we will present a new numerical method for low-rank symmetric tensor decomposition, building on the usual power method and ideas from classical algebraic geometry. The approach achieves a speed-up over the state-of-the-art by roughly one order of magnitude. We will also discuss an “implicit” variant of the algorithm for the case of moment tensors which avoids the explicit formation of higher-order moments, analogously to matrix-free techniques in linear algebra. Finally, we will make some quantitative statements about the non-convex optimization landscape underlying our method. This talk is based on joint works with Joao Pereira, Tammy Kolda and Timo Klock. |
| Date: | November 16, 2020 |
| Time: | 3:00pm |
| Location: | zoom |
| Speaker: | H. Derksen, Northeastern U. |
| Title: | The G-Stable Rank for Tensors |
| Abstract: | A tensor of order d is a d-dimensional array. There are various generalizations of the rank of a matrix to tensors of order 3 or more. I will introduce one such generalization, the G-stable rank, and compare it to other rank notions such as the tensor rank and the slice rank. The G-stable rank is related to the notion of stability in Geometric Invariant Theory. As an application of the G-stable rank, we find better upper bound for the Cap Set Problem. |
| Date: | November 23, 2020 |
| Time: | 3:00pm |
| Location: | zoom |
| Speaker: | Tom Gannon, University of Texas |
| Title: | Recovering Lie(G)-Modules from the Weyl Group Action |
| Abstract: | Let G be a semisimple group, for example, G = SL_n. One pervasive theme in representation theory is recovering information about representations of Lie(G) from a maximal torus T in G (for example, T may be identified with the diagonal matrices of SL_n) and its natural action by the Weyl group W := N_G(T)/T. In this talk, we will explore historical incarnations of this theme--specifically, finite dimensional Lie(G) representations and the study of the BGG category O--and then discuss a recent theorem which identifies a "varying central character" version of category O with sheaves on a space determined by the action of W on T. No prior knowledge of representation theory will be assumed. |
| Date: | December 4, 2020 |
| Time: | 4:00pm |
| Location: | zoom |
| Speaker: | A. Pal, TAMU |
| Title: | Tensors of minimal border rank |
| Abstract: | We know 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 the classical question, and recent progress towards classifying minimal border rank tensors. This is joint work with JM Landsberg and Joachim Jelisiejew. |
| Date: | December 7, 2020 |
| Time: | 3:00pm |
| Location: | zoom |
| Speaker: | R. 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 based on joint work with J.M. Landsberg, including classification of tensors with geometric rank two. |
