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.

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.

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.

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.

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.

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.