May 13 (Mell 2550) 
May 14 (Mell 2550) 
May 15 (Mell 2550) 
May 16 (Mell 2550) 
May 17 (Mell 2550)  
8:309:00 
Registration and Early Coffee 
Early Coffee 
Early Coffee 
Early Coffee 
Early Coffee 
9:0010:00 
Colleen Robles Abstract: 👇In this largely expository talk I will explain how Hodge theory is used to study families of nonsingular projective algebraic varieties, and report of the status of an ongoing program to construct completions of period maps. 
Giorgio Ottaviani Abstract: 👇
Let \(f\) be a homogeneous polynomial of even degree \(d\). We study the decompositions \(f=\sum_{i=1}^r f_i^2\) where \(\deg f_i=d/2\). The minimal number of summands \(r\) is called the \(2\)rank of \(f\),
so that the polynomials having \(2\)rank equal to \(1\) are exactly the squares.
Such decompositions are never unique and they are divided into \(O(r)\)orbits, the problem becomes counting how many different \(O(r)\)orbits of decomposition exist.

Nicolas Ressayre (virtual) Abstract: 👇In 2006, BelkaleKumar discovered a new product on the cohomology groups of the flag manifolds (for any reductive group $G$).This product is essential to understand the Horn problem for $G$. In this talk, we present a conjecture about the BelkaleKumar product that states that the product of two Schubert classes can be represented by a parametrized variety. We will also explain the positive answer to this conjecture for the cominuscule case and the case of complete flag manifolds. 
Laurent Manivel (virtual) Abstract: 👇I call tame those tensors that are parametrized by representations appearing in cyclic gradings of simple Lie algebras; they are tame in the sense that their classification is similar to the usual Jordan classification of matrices up to conjugation. I will explain that these tensors are related with several famous chapters of classical algebraic geometry, involving elliptic curves, Kummer surfaces of Coble hypersurfaces. In fact they provide new points of view on these classical topics and allow us to add a few nice pages to the story. 
Virginia Williams Abstract: 👇(TBD) 
10:0010:30 
Coffee 
Coffee 
Coffee 
Coffee 
Coffee 
10:3011:30 
Jarek Buczynski Abstract: 👇
A complex manifold is a contact manifold if there is a distribution in the tangent bundle which is as nonintegrable as possible. I will report on recent progress in classification of projective contact manifolds focusing on the Fano case in (complex) dimensions 7 and 9. Our work implies the classification of quaternionKaehler manifolds of (real) dimensions 12 and 16, a famous problem from Riemannian geometry. The tools we use include representation theory and actions of (complex) reductive groups on manifolds, symplectic geometry, characteristic classes, and equivariant localisation theorems.

Jerzy Weyman (virtual) Abstract: 👇I will discuss the ongoing work with L. Guerrieri and Xianglong Ni on the structure of perfect ideals of codimension 3 and Gorenstein ideals of codimension 4. This extends the classical results of HilbertBurch on perfect ideals of codimension 2 and BuchsbaumEisenbud on Gorenstein ideals of codimension 3. 
Alessandra Bernardi Abstract: 👇Between 13 and 8 years ago, in collaboration with Ranestad, Macias Marques, and Jelisiejew, we devised an approach to bound the dimension of certain Cactus varieties. Subsequently, JM explained the implications regarding the strategies to address the complexity of Matrix Multiplication. Exactly one year ago, the birthday celebrant expressed a keen interest in a renewed detailed exposition of this topic, making today's talk a tailored gift for his interest. 
Tom Ivey Abstract: 👇Chapter 10 in the second edition of Cartan for Beginners is focused on work of Anderson, Fels and Vassiliou on `superposition principles’ for hyperbolic Darbouxintegrable (DI) systems. Through a series of delicate coframe adaptations, they show that each such system is associated to a Lie algebra (the Vessiot algebra) whose infinitesimal action enables one to construct a split integrable extension. In this talk, I report on joint work with Mark Fels on generalizing this construction to the elliptic case. For elliptic DI systems we similarly obtain a canonical integrable extension, but one difference is that existence of the Vessiot algebra depends in an essential way on analytic continuation; another is that the extension doesn't split but is a holomorphic system on product of complex manifolds (one of which is the complexified Vessiot group). In several examples the extension is contactequivalent to a prolongation of the CauchyRiemann equations, leading to solution formulas in terms of arbitrary holomorphic functions and their derivatives. 
Edinah Gnang Abstract: 👇
We propose an algebraic model of computation which formally relates symbolic listings, complexity of Boolean functions, and low depth arithmetic circuit complexity. In this model, algorithms are arithmetic formulae expressing symbolic listings of YES instances of Boolean functions, and computation is executed via partial differential operators. We consider the Chow rank of an arithmetic formula as a measure of complexity and establish the Chow rank of multilinear polynomials with totally nonoverlapping monomial support. We also provide Chow rank nondecreasing transformations from sets of graphs to sets of functional graphs.

11:3012:00 
Questions/ Break 
Questions/ Break 
Questions/ Break 
Questions/ Break 
Questions/ Break 
12:002:00 
Lunch (Mell 4546) 
Lunch (Mell 4546) 
Lunch (Mell 4546) 
Lunch (at The Edge) 
Lunch (at The Edge) 
2:003:00 
Frédéric Holweck
Abstract: 👇
In the early 2000's Joseph M. Landsberg and Laurent Manivel wrote a series of papers using algebraic geometry to solve problems in representation theory. In one of them they investigated the geometry of the Freudenthal Magic square. Ten years later these constructions and the corresponding geometry were investigated in the quantum physics literature. It is now well understood that the orbit structure of the third row of the magic square describes entanglement patterns for threepartite quantum systems in different Hilbert spaces (bosonic qubit, qubits, fermions, fermionic fock spaces). 
JongIn Han
Abstract: 👇We investigate new lower bounds on the tensor ranks of the determinant and the permanent tensors via recursive usage of the Koszul flattening method introduced by LandsbergOttaviani and HauensteinOedingOttavianiSommese. Our lower bounds on the ranks of determinant tensors completely separate the determinant and the permanent tensors by their tensor ranks. Furthermore, we determine the exact tensor rank of the determinant tensor of order 4 as 12 and the permanent tensor of order 4 as 8 over any field of characteristic not equal to 2.
Paul Simanjuntak Abstract: 👇The fundamental result in geometry analysis, arguably, is the isoperimetric inequality: the fact that the extremal case of certain geometric functionals is the Euclidean ball. For example, the classical isoperimetric inequality states that the disc possesses the smallest volume among all convex sets of the same perimeter on the plane. Such inequalities often rely explicitly on the convexity property of the class in question. Indeed, the classical result above will be false if the class is expanded to include nonconvex cases. In this talk, we will discuss a way to sidestep convexity by constructing associated randomized bodies that converge in some sense to the wanted nonconvex body. Surprisingly, even in such cases we often see that the extremal case is unchanged. Moreover, the method used may be used to bodies with certain product structure, which may open path to studying isoperimetrictype inequalities for random tensors. Based on works with R. Adamczak, G. Paouris, and P. Pivovarov. 
Austin Conner Abstract: 👇I describe two ideal generation problems which may be resolved using essentially the same technique. First, determine the ideal of polynomial relations among the coefficients of n degree d binary forms equivalent to there being a simultaneous solution (i.e., generalizing the Sylvester resultant when \(n=2\)). Second, determine the ideal of the set of inverses of a certain linear subspace of symmetric matrices, the Gaussian graphical model of the cycle. In both cases, one can choose a term order and show that a conjectured set of polynomials forms a Gröbner basis for the answer by studying the geometry of the initial ideal. If it is squarefree (equivalently, radical) and if its algebraic set consists of equidimensional coordinate subspaces and has the correct dimension and degree, then one can conclude. In both cases, establishing the necessary facts becomes a combinatorial problem on monomial ideals. 
Luca Chiantini Abstract: 👇I will introduce the notion of Weddle loci, which are important in the study of configurations of points and in computer vision, and I will discuss some recent application of the loci to the description of spaces of tensors. I will mainly focus on a work in progress related the use of Weddle loci for determining the rank of linear systems of quadrics, and related partially symmetric tensors. 
Hang (Amy) Huang Abstract: 👇
A linear subspace of the space of \(b\times c\) matrices is of bounded rank \(r\) if no matrix in the space has rank greater than \(r\). Classifications of them are an interesting and important problem with potential application to Strassen's Laser method as well as quantum information theory. The only case known is when \(r \leq 3\) with two proofs given by Atkinson via the study of Atkinson normal form and EisenbudHarris via the study of the first Chern class of some sheaf associated with the space. I will discuss a connection between these two approaches as well as the recent progress in the classification of basic space of matrices of bounded rank 4. ChiaYu Chang Generic Border Subrank Abstract: 👇Subrank and border subrank were introduced by Strassen. The rank and border rank of a generic tensor are the same and equal to the maximal border rank. However, we know less about the behavior of the subrank and border subrank. In this talk, we will introduce subrank and border subrank. I will state the main result that the growth rate of the generic subrank is the same as the growth rate of the generic border subrank. Then I will give an idea of proof of the main result. 
3:003:30 
Coffee 
Coffee 
Coffee 
Coffee 
Coffee 
3:304:30 
Cosimo Flavi Abstract: 👇An isotropic form is a linear form whose coefficients correspond to an isotropic point. For any natural numbers \(n\) and \(d\), the space of homogeneous harmonic polynomial of degree \(d\) in \(n\) variables is generated by the \(d\)th powers of isotropic forms. This allows us to define the isotropic rank of a homogeneous polynomial \(h\) as the minimum natural number \(r\) such that \(h\) can be written as a linear combination of the \(d\)th powers of \(r\) isotropic forms. Using secant varieties and Terracini's Lemma we determine the generic isotropic rank for any value of \(n\) and \(d\). This is joint work with Cristiano Bocci and Enrico Carlini. Colm Kelleher
Abstract: 👇Quantum contextuality is a phenomenon of quantum systems, less celebrated than other features such as entanglement and superposition. It shows that quantum systems can display seemingly contradictory behaviour when combined with assumptions from classical mechanics. This can be shown visually through quantum games, which provide a testing ground for experimentally ruling out such classical assumptions. In this talk we will cover the most basic and celebrated quantum game  the Mermin Game  and talk through how to play it on a quantum computer. We will also examine the geometry underpinning this game and a Cayley invariant of degree 3 connecting related geometries, their games, and quantum contextuality. 
Alessandro Danelon Abstract: 👇
Loosely speaking, \(GL\)varieties are affine schemes whose coordinate rings are polynomial representations. Their closed points can be obtained as the inverse limits of \(\textrm{Vec}\)varieties, functors from the category of finite dimensional vector spaces to affine varieties.
Sara Marziali
Abstract: 👇I will give an introduction to the BhattacharyaMesner product for tensors of order \(n\). This product is particularly suited to the study of neural networks described by directed acyclic graphs in which each node decides its own activation through tensors of choice. The final state of the network can be determined by the product of a standard modification (blow etc.) of node tensors. Moreover, the case of small networks under the general Markov hypothesis is analyzed, showing that the algebraic tools introduced make it possible to separate the various classes of networks, with interpretations in both phylogenetics and quantum information. 
Ian Tan Abstract: 👇We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of almost all \(n\)qubit pure states under the action of the local unitary group. Taking advantage of the double cover \(\operatorname{SL}_2(\mathbb{C})\times \operatorname{SL}_2(\mathbb{C})\to\operatorname{SO}_4(\mathbb{C})\), we produce similar algorithms (distinguished by the parity of \(n\)) that compute normal forms for almost all \(n\)qubit pure states under the action of the SLOCC group. Vincenzo Galgano
Abstract: 👇Secant varieties are among the main protagonists in tensor decomposition, whose study involves both pure and applied mathematical areas. Any generalised Grassmannian \(G/P\) is a set of rank1 "tensors" in the corresponding (minimal) irreducible representation, like Grassmannians for skewsymmetric tensors. Despite the geometry of secant varieties to \(G/P\) is in general not completely understood, several contributions appear in the literature, among which many works by J.M. Landsberg and L. Manivel on the geometry of the Freudenthal magic square. In this talk we discuss the secant variety of lines to a generalised Grassmannian, and we give results on the identifiability and singularity in the case of Grassmannians. Some results are from a joint work with Reynaldo Staffolani. 
Maciej Galazka Abstract: 👇It is known that for \(r \leq 13\) and any natural numbers \(n, d\), the \(r\)th cactus variety of \(v_d(\mathbb{P}^n)\) is equal to the \(r\)th secant variety. In the first extremal case, we know that \(\kappa_{14}(v_d(\mathbb{P}^n))\) has two irreducible components for \(d\geq 5, n\geq 6\). We look at the Grassmann secant variety of lines \(\kappa_{r,2}(v_d(\mathbb{P}^n))\) and investigate what is the least \(r\) such that this variety is reducible for almost all \(n\). In order to do this, we analyze the locus of socle dimension at most two in the Hilbert scheme of \(r\) points on \(\mathbb{P}^n\) for some small \(r\). Nathaniel Collins
Abstract: 👇Tensor rank, identifiability, and the number of orbits has been studied extensively in many different contexts. In the case of \(2\times \cdots \times 2\) tensors, the maximum rank over \(\mathbb{F}_2\) is known[SL2020]. We survey results on the distribution of ranks and identifiability for \(2\times 2\times 2\) and \(2\times 2\times 2 \times 2\) tensors over \(\mathbb{F}_2\). We investigate the distribution of tensor ranks, identifiability, and orbits for symmetric and partially symmetric tensors over \(\mathbb{F}_2\) and conclude with future directions. 
Robert Bryant (virtual) Abstract: 👇
There are various methods known now for constructing moreorless explicit metrics with special holonomy; most of these rely on assumptions of symmetry and/or reduction. Another promising method for constructing special solutions is provided by the strategy of looking for metrics that satisfy algebraic curvature conditions. This method often leads to a study of structure equations that satisfy an overdetermined system of partial differential equations, sometimes involutive sometimes not, and the theory of exterior differential systems is particularly wellsuited for analyzing these problems.

4:305:30 
Problem Session 
Problem Session 
Problem Session 
Open Problems 
Open Problems 
6:00— 
Welcome Drinks The Collegiate Rooftop 
Conference Banquet 
Social Event Red Clay Brewery in Opelika 
Social Event and Poster Session 
Closing Drinks 