MenuGeometry Seminar
Spring 2025
| Date: | January 27, 2025 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | JM Landsberg, Texas A&M |
| Title: | The cheapest tensors |
| Abstract: | Motivated by quantum information theory and the complexity of matrix multiplication, one would like to classify tensors of "minimal border rank". This is now understood to be a difficult problem with deep connections to algebraic geometry and commutative algebra. After giving an introduction to the topic with motivation and basic definitions, I will describe recent progress on the question, in particular the introduction of "atomic tensors". This is joint work in progress with J. Jelisiejew and T. Mandziuk. |
| Date: | January 31, 2025 |
| Time: | 5:00pm |
| Location: | BLOC 302 |
| Speaker: | M. Varbaro |
| Title: | Singularities of Herzog varieties |
| Abstract: | Let S be a polynomial ring over a field and I a homogeneous ideal. We say that I as a Herzog ideal if there exists a monomial order < on S such that in_<(I) is squarefree. A projective variety X is a Herzog variety if it admits an embedding in which it is defined by a Herzog ideal. If X is a Herzog variety with respect to a revlex order, with Constantinescu and DeNegri we proved that the smoothness of X forces S/I to be Cohen-Macaulay with negative a-invariant (hence a (F)-rational singularity). We will discuss the problem wether this happens for any Herzog variety (not necessarily w.r.t. a revlex order); this is not even clear when X is a curve. In this case, rephrasing the problem the question is: if X is a Herzog smooth projective curve, does X have genus 0? In this talk we will largely discuss this problem, giving some evidence for it and explaining why it is difficult to show it in general, giving insights on an ongoing work with Amy Huang, Jonah Tarasova and Emily Witt. |
| Date: | February 1, 2025 |
| Time: | 09:00am |
| Location: | BLOC 302 |
| Speaker: | multiple speakers |
| Title: | Symmetries and Singularities in Texas conference |
| Abstract: | See https://people.tamu.edu/~jml//symmetries-texas%203/main.html and please register if you plan to attend any of the talks. The conference will continue Sunday morning as well. |
| Date: | February 10, 2025 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | S. Lovett, UCSD |
| Title: | Corners and arithmetic extensions of Kelley-Meka |
| Abstract: | A classical question in additive combinatorics, dating back for close to 100 years, is what is the densest subset of integers without a 3-term arithmetic progression. In 2023, Kelley and Meka made a huge breakthrough on the problem, proving bounds which are close to the best known constructions. In this talk, I will describe an on-going effort to extend their techniques to more problems in additive combinatorics, and in particular to the "corners" problem, which can be viewed as a 2-dimensional analog of the 3-term arithmetic progression problem, and variants of it. Joint work with Michael Jaber and Anthoni Ostuni. |
| Date: | February 17, 2025 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | K. Sivic |
| Title: | Irreducible components of Hilbert schemes of points |
| Abstract: | Hilbert schemes of points in affine spaces parameterize artinian algebras of given length. In the talk we classify irreducible components of Hilbert schemes of 9 and 10 points in affine spaces of any dimension. The main tool is the connection between Hilbert schemes of points and varieties of commuting matrices. This is joint work with Maciej Gałązka and Hanieh Keneshlou. |
| Date: | February 21, 2025 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Sven Hirsch, Columbia |
| Title: | Gravitational waves and spinors |
| Abstract: | We discuss applications of spin geometry to general relativity and find geometric characterizations of gravitational waves. This is based upon joint work with Yiyue Zhang. |
| Date: | February 24, 2025 |
| Time: | 3:00pm |
| Location: | BLOC 602 |
| Speaker: | Thomas Yahl, University of Wisconsin |
| Title: | Galois Groups of Purely Lacunary Sparse Polynomial Systems |
| Abstract: | The Galois group of a polynomial system is a group of symmetries of the zeros of the system that reflects its intrinsic structure. These groups were initially studied algebraically by Jordan, and much later Harris interpreted them as geometric monodromy groups. We will consider Galois groups of sparse polynomial systems, systems whose coefficients are general and whose monomial support is fixed. There are two special structures that occur in sparse systems: lacunary systems are those that have been precomposed with a non-invertible monomial map, and triangular systems are those that contain a nontrivial proper subsystem. Galois groups of lacunary systems and triangular systems act imprimitively on the zeros of the system--they are subgroups of a certain wreath product. It is expected that the Galois group is equal to this wreath product, but it is not in many cases. A classification of these Galois groups is currently unknown. We determine the Galois group of a purely lacunary polynomial system--a sparse polynomial system which is lacunary and not triangular. We characterize the Galois groups of purely lacunary systems by showing they satisfy a property analogous to 2-transitivity for imprimitive groups. Further, we show that the Galois group is determined by the automorphism group of a certain variety defined by binomial equations. |
| Date: | February 28, 2025 |
| Time: | 4:00pm |
| Location: | Bloc 302 |
| Speaker: | Frank Sottile, Texas A&M University |
| Title: | Periodic Graph Operators for Algebraic Geometers |
| Abstract: | Understanding the spectrum of the Schröodinger operator in a periodic medium is a fundamental problem in mathematical physics. The discrete version concerns operators on periodic graphs. In this discrete version, the primary objects are real algebraic varieties, and thus algebraic geometry becomes relevant for the study of discrete periodic operators. The purpose of this talk will be to explain some of this to algebraic geometers, and describe some results obtained from this perspective, as well as some computational and combinatorial aspects of this study. |
| Date: | March 3, 2025 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | R. Oliveira, U. Waterloo |
| Title: | Primes via Zeros: interactive proofs for testing primality of natural classes of ideals |
| Abstract: | A central question in mathematics and computer science is the question of determining whether a given ideal I is prime, which geometrically corresponds to the zero set of $I$, denoted $Z(I)$, being irreducible. The current best algorithms for the ideal primality testing problem require, in the worst-case, exponential space (i.e., in EXPSPACE). This state of affairs has prompted intense research on the computational complexity of this problem even for special and natural classes of ideals. Notable classes of ideals are the class of radical ideals, complete intersections (and more generally Cohen-Macaulay ideals). For radical ideals, the current best upper bounds are given by (Buergisser & Scheiblechner, 2009), putting the problem in PSPACE. For complete intersections, the primary decomposition algorithm of (Eisenbud, Huneke, Vasconcelos 1992) coupled with the degree bounds of (Dickenstein et al 1991), puts the ideal primality testing problem in exponential time (EXP). In these situations, the only known complexity-theoretic lower bound for the ideal primality testing problem is that it is coNP-hard for the classes of radical ideals, and equidimensional Cohen-Macaulay ideals. In this work, we significantly reduce the complexity-theoretic gap for the ideal primality testing problem for the important families of ideals (namely, *radical ideals* and *equidimensional Cohen-Macaulay ideals*). For these classes of ideals, assuming the Generalized Riemann Hypothesis, we show that primality testing can be efficiently verified (also by randomized algorithms). This significantly improves the upper bound for these classes, approaching their lower bound, as the primality testing problem is coNP-hard for these classes of ideals. This talked is based on joint work with Abhibhav Garg and Nitin Saxena. |
| Date: | March 7, 2025 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | H. Huang, Texas A&M |
| Title: | Belinson's cohomology of the monad I |
| Abstract: | This will be a special working geometry seminar. |
| Date: | March 21, 2025 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Demetre Kazaras, Michigan State University |
| Title: | Scalar curvature and codimension 2 collapse |
| Abstract: | This talk is about the structure of Riemannian 3-manifolds satisfying a lower bound on their scalar curvature. These manifolds are models for spatial geometry in general relativity. Our motivational question will be "How flat is an isolated gravitational system with very little total mass?" Objects like gravity wells and black holes can distort geometry without accumulating much mass, making this a subtle question. In addition to discussing progress, I will present a "drawstring" construction, which modifies a manifold near a given curve, reducing its length with negligible damage to a scalar curvature lower bound. Unexpected examples are produced with relevance to a few problems. This construction extends ideas of Basilio-Dodziuk-Sormani and Lee-Naber-Neumayer, and is based on joint work with Kai Xu. |
| Date: | March 24, 2025 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Tianyi Yu, UQAM |
| Title: | An insertion algorithm for Schubert Cauchy identity via Pieri formula |
| Abstract: | The dual Cauchy identity for Schur polynomials is a fundamental result in symmetric function theory and representation theory. It states that the sum of products of two Schur polynomials indexed by conjugate partitions, in two sets of variables, equals the generating function of binary matrices. Combinatorially, this identity is realized through the dual RSK insertion, which provides a bijection between such matrices and pairs of tableaux. Schubert polynomials, often seen as non-symmetric generalizations of Schur polynomials, satisfy a Cauchy-type formula involving triangular binary matrices. We present an explicit insertion algorithm that establishes a bijection realizing this identity using the Pieri rule. Remarkably, our algorithm retains key features of the classical RSK and naturally involves traversals of increasing binary trees. This talk is based on ongoing joint work with Johnny Gao and Sylvester Zhang. |
| Date: | March 28, 2025 |
| Time: | 4:00pm |
| Location: | Bloc 166 |
| Speaker: | TAGS 28, 29, 30 March |
| Title: | Texas Algebraic Geometry Symposium |
| Abstract: | Speakers: Brendan Hassett Brown University Kimoi Kemboi Princeton University Lucas Mason-Brown University of Texas Joaquin Moraga University of California, Los Angelos Aaron Pixton University of Michigan Padma Srinivasan Boston University Amy Huang Texas A&M University For more information, see the TAGS 2025 Website. |
| Date: | April 11, 2025 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | K. Ganapathy, UCSD |
| Title: | Non-noetherianity of GL-varieties |
| Abstract: | GL-varieties are infinite-dimensional varieties equipped with an action of the infinite general linear group GL, satisfying certain mildness conditions. Draisma proved that the topology of GL-varieties exhibits a striking noetherian property. However, whether a scheme-theoretic analogue of this property holds remains a longstanding open question. In this talk, I will present the first counterexample to this problem over fields of characteristic two. This counterexample is closely connected to recent work aimed at salvaging Weyl's theorem on polarization in positive characteristics, which I will also discuss. |
| Date: | April 14, 2025 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | J. Wilson, Colorado State |
| Title: | DETECTING CLUSTER PATTERNS IN TENSOR DATA USING LIE THEORY |
| Abstract: | I'll introduce a class of cluster patterns for tensor data used in pattern matching, outlier detections, statistics and signal processing. Then I will show they are all shadows of a general pattern detected efficiently by algebra, specifically Lie theory. It is a direction with many open problems, some about theory, others about applied improvements. Reports on joint work with Brooksbank and Kassabov. |
| Date: | April 28, 2025 |
| Time: | 3:00pm |
| Location: | Bloc 302 |
| Speaker: | Emily McMillon, Rice University |
| Title: | Building Codes from Group Testing Matrices |
| Abstract: | Parity-check codes are a class of linear codes defined by their parity-check matrices. Disjunct matrices are used extensively in combinatorial group testing but were not previously well-studied as parity-check matrices for binary linear codes. We will give results on code parameters including rate, distance, and girth for both codes from general disjunct matrices and codes obtained via the Kautz-Singleton construction for superimposed codes. The talk will be accessible to those without previous knowledge of parity-check codes and combinatorial group testing. |
| Date: | May 2, 2025 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | JM Landsberg |
| Title: | AG homework |
| Date: | May 12, 2025 |
| Time: | 3:00pm |
| Location: | BLOC302 |
| Speaker: | JM Landsberg, Texas A&M |
| Title: | Border rank lower bounds for tensors |
| Abstract: | I will survey techniques for proving border rank lower bounds for tensors with an emphasis on recent developments. |
| Date: | May 19, 2025 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | JM Landsberg, Texas A&M |
| Title: | Tensors of minimal border rank |
