
Date Time 
Location  Speaker 
Title – click for abstract 

08/26 10:00am 
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Anna Seigal Oxford 
Ranks of Cubic Surfaces
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 nonsymmetric 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. 

09/23 11:00am 
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Filip Rupniewski IMPAN 
Cactus rank and identification of secant inside cactus varieties
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. 

09/28 3:00pm 
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Timothy Duff Georgia Tech 
Structured polynomial systems in algebraic vision
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 wellknown problems of camera registration, homography estimation, and fivepoint 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. 

10/02 4:00pm 
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David, Sykes TAMU 
Local equivalence problems for 2nondegenerate, hypersurfacetype CR geometry studied via dynamical Legendrian contact structures.
The local differential geometry of Levinondegenerate 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 2nondegenerate, hypersurfacetype 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, 2nondegenerate, hypersurfacetype CR manifolds in low dimensions.


10/07 Noon 
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J. Weyman U. Krakow 
Structure of finite free resolutions.
In this talk I will describe the structure of finite free resolutions via socalled generic rings.
In the first part I will go through older results of BuchsbaumEisenbud and Hochster.
Then I will describe a more recent connection to the combinatorics of the root systems of Tshaped diagrams $T_{p,q,r}$. 

10/09 4:00pm 
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Leon Zhang UC Berkeley 
Tropical geometry and applications
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. 

10/12 3:00pm 
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F. Gesmundo U. Copenhage 
Approaching the boundary of tensor network varieties
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. StilckFranca and A. Werner. 

10/14 11:00am 
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Markus Blaeser 
Irreversibility of tensors of minimal border rank and barriers for fast matrix multiplication
Determining the asymptotic algebraic complexity of matrix multiplication is a central problem in algebraic complexity theory. The best upper bounds on the socalled 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 socalled 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. 

10/19 3:00pm 
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Guangbo Xu Texas A&M University 
Fukaya categories and blowups
Under Kontsevich's homological mirror symmetry conjecture, the Fukaya category is the openstring invariant on the symplectic side (Amodel), as opposed to the derived category of coherent sheaves as the invariant on the complex side (Bmodel). In this semiexpository talk, I will first give a handwaiving 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. 

11/13 4:00pm 
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Joe Kileel University of Texas Austin 
Fast symmetric tensor decomposition
Tensors are higherorder 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 lowrank symmetric tensor decomposition, building on the usual power method and ideas from classical algebraic geometry. The approach achieves a speedup over the stateoftheart 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 higherorder moments, analogously to matrixfree techniques in linear algebra. Finally, we will make some quantitative statements about the nonconvex optimization landscape underlying our method.
This talk is based on joint works with Joao Pereira, Tammy Kolda and Timo Klock. 

11/16 3:00pm 
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H. Derksen Northeastern U. 
The GStable Rank for Tensors
A tensor of order d is a ddimensional 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 Gstable rank, and compare it to other
rank notions such as the tensor rank and the slice rank. The Gstable rank is related to the notion of stability in
Geometric Invariant Theory. As an application of the Gstable rank, we find better upper bound for the Cap Set Problem. 

11/23 3:00pm 
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Tom Gannon University of Texas 
Recovering Lie(G)Modules from the Weyl Group Action
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 themespecifically, finite dimensional Lie(G) representations and the study of the BGG category Oand 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. 

12/04 4:00pm 
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A. Pal TAMU 
Tensors of minimal border rank 

12/07 3:00pm 
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R. Geng TAMU 
On the geometry of geometric rank 