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Seminars

Spring 2026: Reading Group on Algorithms in Invariant Theory

  • Time and Place: every Wednesday from 1:00 PM to 2:00 PM in LSK 311

  • Organizers: Maksym Zubkov and Elina Robeva

This reading group focuses on Section 4 of Bernd Sturmfels’ Algorithms in Invariant Theory, which deals with methods for computing the invariants of an arbitrary polynomial representation of the general linear group GLn(C). The goal is to understand how invariant theory can be made explicit and computable. 

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Necessary background from earlier sections (polynomial ideals, Gröbner bases, and basic invariant theory) will be reviewed selectively as needed.

List of topics:

  • Representation theory of the general linear group

  • Cayley’s Omega-process and Hilbert finiteness theorem

  • Invariants and covariants of forms

  • Lie algebra action and the symbolic method

  • Hilbert’s algorithm

Schedule:

  • 01/14/2025 - Section 1.2 What is invariant theory? and Section 3.6 Invariants and covariants of binary forms,

  • 01/21/2025 - Section 4.1 Representation theory of the general linear group,

    • 1st half by Qixia Luo

  • 01/28/2025 - Section 4.1 Representation theory of the general linear group,

    • practice problems by Maksym Zubkov

  • 02/02/2025 - Section 4.1 Representation theory of the general linear group,

    • practice problems by Maksym Zubkov

  • 02/04/2025 -Section 4.1 Representation theory of the general linear group,

    • 2nd half by Qixia Luo

  • 02/11/2025 - Section 4.2 Binary forms revisited

  • 02/18/2025 - Holiday

  • 02/25/2025 - Section 4.3 Cayley’s -process and Hilbert finiteness theorem

  • 03/04/2025 - Section 4.3 Cayley’s -process and Hilbert finiteness theorem

  • 03/11/2025 - Section 4.4 Invariants and covariants of forms

  • 03/18/2025 - Section 4.4 Invariants and covariants of forms

  • 03/25/2025 - Section 4.5 Lie algebra action and the symbolic method

  • 04/01/2025 - Section 4.5 Lie algebra action and the symbolic method

  • 04/08/2025 - Section 4.6 Hilbert’s algorithm

  • 04/15/2025 - Section 4.6 Hilbert’s algorithm

  • 04/22/2025 - Section 4.7 Degree bounds

Fall 2025: Informal Reading Group on Classical Algebraic Geometry

  • Time and Place: every Tuesday at 1:00 PM in Math 126

  • Organizers: Maksym Zubkov and Elina Robeva

The goal of this reading group is to study the first three chapters of Classical Algebraic Geometry: A Modern View by Igor V. Dolgachev.

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There is a beautiful connection between algebraic neural networks and various tensor decompositions, including the Waring problem, Chow varieties, and simultaneous tensor decomposition. Through this reading group, we aim to develop a deeper understanding of the geometry of binary forms, conics, and plane cubics, especially from the perspective of apolar forms and dual varieties.

List of topics:

  • Polar hypersurfaces

  • Dual varieties 

  • Warring problem and polar s-hedra

  • Dual homogeneous forms

  • Binary forms

  • Conics and quadratic Surfaces

  • Plane cubics

Schedule:

  • 09/09/2025 - Introduction: The Interplay of Symmetric Tensors and Neural Networks -- notes

  • 09/16/2025 - Section 1.1.1: Polarization map and kth Polar Hypersurface -- notes

  • 09/23/2025 - Section 1.1.2: First Polars

  • 09/30/2025 - Holiday 

  • 10/07/2025 - Sections 1.1.3-1.1.5 

  • 10/14/2025 - Sections 1.1.5 and 1.1.6

  • 10/21/2025 - Section 1.2.3

  • 10/28/2025 - Sections 1.3.1-1.3.2, 1.3.4-1.3.5

  • 11/04/2025 - Sections 1.4.1-1.4.4

  • 11/11/2025 - Holiday

  • 11/18/2025 - Sections 1.4.5 and 1.5.1-1.5.2

  • 11/25/2025 - Sections 2.1 and 2.3

  • 12/02/2025 - Sections 3.1 and 3.2

  • 12/09/2025 - Sections 3.2 and 3.3

Fall 2025: Learning Seminar on Tensors and Casual Inference

  • Time and Place: IAM, Thursdays from 1:00 PM to 2:00 PM

  • Organizers: Maksym Zubkov and Elina Robeva

This seminar explores the interplay between tensors, causal inference, and modern statistical methods. We will study how tools from multilinear algebra, combinatorics, and algebraic geometry reveal hidden structure in data and enable advances in inference. Topics include tensor decompositions, graphical models, and approaches to understanding complex dependencies in data, with applications ranging from causal discovery to high-dimensional learning.

Schedule:

  • 09/04/2025 - Covering numbers of regular sets with applications -- Sharvaj Kubal

    • ​Abstract: Covering numbers of sets play an important role in learning theory, approximation theory and the analysis of random matrices. I will discuss the covering number bounds of Zhang and Kileel (2025) which are particularly suited to polynomially defined sets such as real varieties, polynomial and rational images, and semialgebraic sets. Bounds on the approximation capability of low CP rank tensors, and the Rademacher complexity of classes of neural networks follow as corollaries of their main results.

  • 09/11/2025 - Inferring stochastic potential-driven dynamics from population snapshots -- Vincent Guan

    • Abstract: In many scientific applications, observations are limited to population snapshots, or temporal marginals. For example, in single cell biology, scRNA measurement destroys cells, and in hydrology, geochemical sensors can only track plume migrations rather than individual particles. We consider the case where the dynamics are modeled by the gradient of a potential, while also featuring stochasticity from Brownian motion, i.e. the overdamped Langevin SDE. We overview basic properties of this model, identifiability theory for the conditions under which a unique SDE is compatible with the observed snapshots, and methods for inferring the SDE in practice. Finally, we discuss some open questions.​

  • 09/18/2025 - The Linear Targeting Problem -- Kyle Bierly

    • ​Abstract: For given real or complex m x n data matrices X, Y, we investigate when there is a matrix A such that AX = Y, and A is invertible, Hermitian, positive (semi)definite, unitary, an orthogonal projection, a reflection, complex symmetric, or normal.

  • 09/25/2025 - An Introduction to Free Probability and Random Matrix Theory -- Qixia Luo

    • ​Abstract: This talk is an introduction to the theory of free probability and its applications to random matrix theory. Free probability offers a set of tools and concepts for the study of non-commutative random variables. I will introduce the free analogues of familiar classical probabilistic concepts such as non-commutative probability spaces, random variables, moments, distributions, convergence in distribution, free product of non-commutative probability spaces, free independence, free cumulants, and the free central limit theorem. I will emphasize the central role the lattice of non-crossing partitions plays in the combinatorial theory of free probability. Finally, I will discuss how freeness naturally emerges in the large-dimension limit of common random matrix ensembles and explain the consequences of this phenomenon for their asymptotic eigenvalue distributions.

  • 10/30/2025 - Basics of Macaulay2 -- Elina Robeva

    • ​Abstract: This will be a mini workshop. Please make sure that you will have an installed Macaulay2 on your laptops. Use this [link] for guidance.

  • 11/20/2025 - Unraveling the Sudoku Mystery: a Grobner Basis Perspective -- Emmanuel Akhalu

    • ​Abstract: This project looks at Sudoku through an algebraic lens by turning the puzzle’s rules into a system of polynomial equations and solving them with Gröbner bases. Once the ideal for a given Sudoku is set up, its Gröbner basis makes the solution easier to extract, often reducing the system to a more manageable form. The method is compared with familiar solving techniques, and the role of monomial orderings in the computation is also examined. The work offers a different way to think about Sudoku and points to how similar algebraic tools might be used for other combinatorial problems.

  • 11/27/2025 - Systems of polynomial equations, higher-order tensor decompositions, and multidimensional harmonic retrieval -- Matija Tomic

    • ​​Abstract: I will present a tensor-based method for solving systems of polynomial equations with simple roots, presented in [1]. The method is based on the construction of the Macaulay matrix from the coefficients of the polynomials. The key idea is that the null space of the Macaulay matrix is spanned by a highly structured set of multivariate Vandermonde vectors corresponding to the system’s roots. By exploiting shift-invariance relations in these vectors as in the case of multidimensional harmonic retrieval problem, one can construct a third-order tensor whose Canonical Polyadic Decomposition (CPD) reveals all affine roots and roots at infinity.

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