Events

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. 

Abstract:  Modern secure communication systems, such as iMessage, WhatsApp, and Signal include intricate mechanisms that aim to achieve very strong security properties. These mechanisms typically involve continuously merging fresh secrets into the keying material that is used to encrypt messages during communications. In the literature, these mechanisms have been proven to achieve forms of Post-Compromise Security (PCS): the ability to provide communication security even if the full state of a party was compromised some time in the past. However, recent work has shown these proofs cannot be transferred to the end-user level, possibly because of usability concerns. This has raised the question of whether end-users can actually obtain PCS or not, and under which conditions.

Abstract:  The (parallel) pebbling game is a useful abstraction for analyzing the resources (e.g., space, space-time, amortized space-time) needed to evaluate a function f with a static data-dependency graph G on a (parallel) computer. The underlying question is purely combinatorial: what tradeoffs does a directed acyclic graph (DAG) force between storing intermediate values and recomputing them? This viewpoint has been particularly influential in cryptography, where many “Memory-Hard Function” (MHF) constructions can be modeled by evaluating a fixed DAG.

Abstract: In the space of type A quiver representations, putting rank conditions on the maps cuts out subvarieties called "open quiver loci." These subvarieties are closed under the group action that changes bases in the vector spaces, so their closures define classes in equivariant cohomology, called "quiver polynomials." Knutson, Miller, and Shimozono found a pipe dream formula to compute these polynomials in 2006. To study the geometry of the open quiver loci themselves, we might instead compute "equivariant Chern-Schwartz-MacPherson classes," which interpolate between cohomology classes and Euler characteristic. I will introduce objects called "chained generic pipe dreams" that allow us to compute these CSM classes combinatorially, and along the way give streamlined formulas for quiver polynomials.

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