Events

Title:Erdős-Pósa theorem for matroids

Abstract: We will look at an analogue theorem of the classical Erdős-Pósa Theorem. We prove a $GF(q)$-representable matroid analogue of Robertson and Seymour's theorem that planar graphs have an Erdős-Pósa property. Given a matroid $N$, we prove that for every matroid $M$ with bounded branch width, $M$ either contains $r$ skew copies of $N$, or there is a small perturbation of $M$ that doesn't contain $N$ as a minor.

Title:PRISM: Simple And Compact Identification and Signatures From Large Prime Degree Isogenies

Abstract: The problem of computing an isogeny of large prime degree from a supersingular elliptic curve of unknown endomorphism ring is assumed to be hard both for classical as well as quantum computers. 

Title: The degrees of Stiefel Manifolds

Abstract:

The set of orthonormal bases for k-planes in R^n is cut out by the equations X*X^T = I
where X is a k x n matrix of variables and I is k x k identity. This space, known as the Stiefel manifold St(k,n), generalizes the orthogonal group and can be realized as the homogeneous space O(n)/O(n-k). Its algebraic closure
gives a complex affine variety, and thus, it has a degree.

I will discuss our derivation of these degrees. Extending 2017 work on the degrees of special orthogonal groups, joint work with Fulvio Gesmundo gives a combinatorial formula in terms of non-intersecting lattice paths.
This result relies on representation theory, commutative algebra, Ehrhart theory, polyhedral geometry, and enumerative combinatorics.

I will conclude with some open problems inspired by these objects.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm.

Abstract: In this talk, we consider a linear algebraic object known as a tridiagonal pair which arises naturally in the context of Q-polynomial distance-regular graphs. We will focus on a special class of tridiagonal pairs said to have Racah type. Given a tridiagonal pair of Racah type, we associate with it several linear transformations which act on the underlying vector space in an attractive manner and discuss their relationships with one another. In an earlier work, we introduced the double lowering operator Ψ for a tridiagonal pair. In this talk, we will explore this double lowering map further under the assumption that our tridiagonal pair has Racah type and will use the double lowering map to obtain new relations involving the operators associated with two oriented versions of our tridiagonal pair.

Abstract: We give an overview of a new Legendre-based signature scheme called Loquat that is friendly for SNARK-based applications. While we primarily focus on the constructive applications, we also discuss the historical context, security and cryptanalysis of the Legendre PRF. The primary content of the talk is based on ia.cr/2024/868.

Abstract:  Discretizable Distance Geometry Problems (DDGP) consist in a subclass of Distance Geometry Problems (DGP) where the search space can be discretized and reduced to a binary tree. Such problems can be tackled by applying a Branch-and- Prune algorithm (BP), which is able to perform an exhaustive enumeration of the solution set. 

Abstract:  Exceptional collections are a powerful tool for understanding the derived category of coherent sheaves on algebraic varieties, with applications in commutative algebra, birational geometry, and mirror symmetry. While the existence of exceptional collections is known for classical varieties such as Grassmannians and flag varieties, constructing explicit collections for toric varieties presents challenges in combinatorial algebraic geometry. In this talk I will describe a computational approach to constructing full strong exceptional collections consisting of complexes of line bundles for toric varieties. No background in derived categories is assumed.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm.

Abstract:  The study of matroid tensor products dates back to the 1970s, extending the tensor operation from linear algebra to the combinatorial setting. While any two matroids representable over the same field admit a tensor product via the Kronecker product of matrices, Las Vergnas showed that such products do not exist for matroids in general, leaving the area underexplored. In this work, we utilize this operation to study skew-representability — representation over division rings that need not be commutative — by proving that a matroid is skew-representable if and only if it admits iterated tensor products with specific test matroids. A key consequence is the existence of algorithmic certificates for non-representability. We further show that every rank-3 matroid admits a tensor product with any uniform matroid, constructing the unique freest such product. Finally, we demonstrate the power of this framework by deriving the first known linear rank inequality for (folded skew-)representable matroids that is independent of the common information property.