Topics in Algebraic Geometry - Classical and Applied

Date: November 9th (Mon) - 13th (Fri), 2026

Place: DGIST, E7 building

Invited Speakers

• Fyodor Zak (Russan Academy of Sciences)

• Yeongrak Kim (Pusan National University)

• Jong In Han (KIAS)

• Maksym Zubkov (DGIST BRL)

• Derek Wu (DGIST BRL)

Title and Abstracts

• Lecture series by F. Zak  
  Title: Multiruled Varieties
  Abstract: see the pdf file attached
  
  

• Yeongrak Kim (Pusan National University)
  Title: On the Hessian matrix factorizations
  Abstract: Let F be a polynomial. A pair of matrices (A, B) with polynomial entries is called a matrix factorization of F if AB=BA=F I_n for some n where I_n denotes the n×n identity matrix. Theory of matrix factorization, introduced by Eisenbud to study homological algebra over a hypersurface ring, becomes popular thanks to its various connections between (non-)commutative algebra, ACM/Ulrich sheaves, complexity theory, singularity theory, and mathematical physics. It is practically hard to achieve a matrix factorization of a given polynomial, and it seems to be extremely difficult to classify these matrix factorizations. One naive idea to obtain a sqaure matrix from a given polynomial F is by taking the Hessian matrix H of F, and in a very few cases H provides a matrix factorization of F itself, which I will call it a Hessian matrix factorization.  The simplest nontrival case is when F=xyz defined in three variables. In this case, the gradient of F defines so-called the standard Cremona transformation of the projective plane. I will talk about the complete classification on homogeneous cubic forms F having graded Hessian matrix factorizations. The key ingredients are the classfication of the EKP-homaloidal cubics and the XJC-correspondence introduced by Pirio and Russo. If time permits, I will show you a few examples on quartic forms having Hessian matrix factorizations.
  
  

• Jong In Han (KIAS)
  Title: Linear syzygies and linear subspaces whose lines are multisecant
  Abstract: The locus $S_d(X)$ of $d$-secant lines to a projective variety $X$ plays an important role in the study of projective varieties via projections (Lazarsfeld 1987, Kwak 1998, Beheshti-Eisenbud 2010). For a projective scheme $X \subseteq \mathbb{P}^r$ satisfying $\textbf{N}_{d,2}$, we show that $S_d(X)\cup X$ is cut out set-theoretically by the $r$-minors of a matrix $M$ constructed from $d$-forms scheme-theoretically defining $X$ and their linear syzygies. In particular, $S_d(X)=\mathbb{P}^r$ if and only if every $r$-minor of $M$ vanishes. Using this, we find a singular threefold $X\subseteq\mathbb{P}^5$ with $S_4(X)\ne \mathbb{P}^5$ and $(I_X)_3=0$, and a normal fourfold $X\subseteq\mathbb{P}^7$ with $S_3(X)\ne \mathbb{P}^7$ and $(I_X)_2=0$. The nonexistence of such varieties in the smooth case is an open question raised by the second author and by Gruson--Peskine. Next, we consider the locus $S_{k,d}(X)$ of $k$-planes $L$ such that $L\cap X$ contains a hypersurface of degree $\ge d$ in $L$ so that $S_{1,d}(X)=S_d(X)$. The locus $S_{k,d}(X)\cup X$ is set-theoretically defined by the $(r+1-k)$-minors of $M$. As a result, we obtain determinantal equations vanishing on the $(q+1)$-secant variety $\sigma_{q+1}X$ from the equations of $\sigma_qX$ and their linear syzygies when $\sigma_qX$ satisfies $\textbf{N}_{q+1,2}$. Finally, we extend the $d$-secant lemma for lines to the locus $S_{k,d}(X)$ of $k$-planes. This yields a lower bound on the number of linear syzygies.
  
  

• Maksym Zubkov (DGIST BRL)
  Title: The Geometry and Optimization of Algebraic Neural Networks
  Abstract: In this talk, we will explore the geometry of the function spaces represented by fixed neural-network architectures. We will focus on algebraic feedforward neural networks with polynomial, rational, and ReLU activation functions. For a fixed architecture, we will associate a geometric object called a neuromanifold: the image of its parameter space in an ambient space of functions. Taking the Zariski closure of the neuromanifold allows us to associate an algebraic variety with the architecture. I will explain how geometric invariants of this variety, such as dimension, degree, defining equations, and singularities, can reveal structural properties of neural networks and shed light on their black-box behavior.
  
  

• Derek Wu (DGIST BRL)
  Title: Lower bounds of tensor border rank via Young flattenings
  Abstract: Given an arbitrary bilinear map, we wish to quantify its complexity by how many scalar multiplications we require to compute it. To better use the tools of algebraic geometry, we can reformulate the problem using tensors, where complexity is measured by (border) rank, defined by inclusion in secant varieties of Segre embeddings. Computing, or even bounding, the border rank of an arbitrary tensor remains an open problem. Landsberg-Ottaviani introduced Young flattenings as a method for tensor border rank lower bounds in some cases and used them for symmetric tensors. We present the first application of Young flattenings for general tensors, where we obtain two new examples of "hay in a haystack".

Schedule

• Nov. 9th (Mon) 13:30 - 14:30, E7-G01
  F. Zak, "Multiruled Varieties I"
  
  

• Nov. 10th (Tue) 16:30 - 17:30, E7-G02
  Y. Kim, "On the Hessian matrix factorizations"
  
  

• Nov. 11th (Wed) 10:30 - 11:30, E7-235
  J. I. Han, "Linear syzygies and linear subspaces whose lines are multisecant"

• Nov. 11th (Wed) 13:30 - 14:30, E7-G01
  F. Zak, "Multiruled Varieties II"
  
  

• Nov. 12th (Thu) 16:00 - 17:00, E7-G02
  M. Zubkov, "The Geometry and Optimization of Algebraic Neural Networks"

• Nov. 12th (Thu) 17:30 - 18:30, E7-G02
  D. Wu, "Lower bounds of tensor border rank via Young flattenings"
  
  

• Nov. 13th (Fri) 11:00 - 12:00, E7-G02
  F. Zak, "Multiruled Varieties III"