Events

Abstract:We define the codegree of a given graph as the maximum number of neighbors that any two distinct vertices have in common.

Abstract: Webs are graphical objects that give a tangible, combinatorial way to compute and classify tensor invariants. Recently, Gaetz, Pechenik, Pfannerer, Striker, and Swanson (arXiv:2306.12501) found a rotation-invariant web basis for SL₄, as well as its quantum deformation U_q(sl₄), and a bijection between move equivalence classes of SL₄-webs and fluctuating tableaux such that web rotation corresponds to tableau promotion. They also found a bijection between the set of plane partitions in an a×b×c box and a benzene move equivalence class of SL₄-webs by determining the corresponding oscillating tableau. In this talk, I will similarly find the oscillating tableaux corresponding to plane partitions in certain symmetry classes by characterizing them via certain lattice words. A dynamical action on tableaux, called promotion, corresponds to rotation of SL₄-webs. I will show how promotion of certain subtableaux aligns with rotation of their respective webs. I will also show that this correspondence maps through a projection to either SL₂ or SL₃ webs. Moreover, this projection is exactly a partial evaluation of webs. This talk will be given through the lens of the combinatorics of webs and tableaux. Some of this work is joint with Jessica Striker.

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

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: Tensor decompositions are fundamental tools in scientific computing and data analysis. In many applications — such as simulation data on irregular grids, surrogate modeling for parameterized PDEs, or spectroscopic measurements — the data has both discrete and continuous structure, and may only be observed at scattered sample points. The CP-HIFI (hybrid infinite-finite) decomposition generalizes the Canonical Polyadic (CP) tensor decomposition to settings where some factors are finite-dimensional vectors and others are functions drawn from infinite-dimensional spaces — a natural framework when the underlying data has continuous structure. The decomposition can be applied to a fully observed tensor (aligned) or, when only scattered observations are available, to a sparsely sampled tensor (unaligned). Current methods compute CP-HIFI factors by solving a sequence of dense linear systems arising from regularized least-squares problems, but these direct solves become computationally prohibitive as problem size grows. We propose new algorithms that achieve the same accuracy while being orders of magnitude faster. For aligned tensors, we exploit the Kronecker structure of the system to efficiently compute its eigendecomposition without ever forming the full system, reducing the solve to independent scalar equations. For unaligned tensors, we introduce a preconditioned conjugate gradient method applied to a reformulated system with favorable spectral properties. We analyze the computational complexity and memory requirements of the new methods and demonstrate their effectiveness on problems with smooth functional modes. I will also discuss the “First Proof” project, which aims to understand the capabilities of AI systems on problems that come up in math research, and the role that results from that experiment played in this project.

Abstract: A cosmological correlator is an Euler integral, associated to a graph G, which encodes information about the state of the early universe. Evaluation of these integrals is extremely challenging, even in simple cases. However, it turns out the integrand can be identified with the so-called canonical form of the cosmological polytope, revealing a rich combinatorial structure and allowing the application of techniques from commutative algebra. I'll sketch my contribution to this story and advertise the fledgling field of positive geometry, which seeks to generalize the notion of canonical forms to geometric objects more exotic than polytopes.

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

Abstract: The set of dimers (aka perfect matchings) of a connected bipartite plane graph G is a distributive lattice, as shown by Propp. The order relation on this lattice comes from the "height" of a dimer, which is a vector of nonnegative integers. In this talk, I'll focus on the dimer face polynomial of G, which is the height generating function of all dimers of G. This polynomial has close connections to knot invariants on the one hand, and cluster algebras on the other. I'll discuss joint work with Mészáros, Musiker and Vidinas in which we explore these connections. No knowledge of knot theory or cluster algebras will be assumed.

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