Events

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.

Abstract:In a widely circulated manuscript from the 1990s, I.G. Macdonald introduced certain higher-rank analogs of the classical hypergeometric functions $_pF_q$, which are expressed as explicit series in Jack and Macdonald polynomials in one and two sets of variables. For special choices of parameters, these series reduce to the hypergeometric functions of matrix argument introduced earlier by C. Herz and A.T. James, which have numerous applications in number theory, multivariate statistics, signal processing, and random matrix theory.

The classical hypergeometric functions are solutions to the hypergeometric differential equation. Macdonald raised the problem of providing an analogous characterization for higher-rank functions by means of differential equations. Over the years, this problem was solved for a small number of cases where p and q are at most 3. However, as the operators become increasingly complicated, the general problem remained open for 40 years. In this talk, we will present a complete solution. This is joint work with Hong Chen.

There will be a pre-seminar at 1:30pm in MC 6029 in a flipped classroom format based on Macdonald’s manuscript on hypergeometric functions (https://arxiv.org/abs/1309.4568). Participants are expected to read the manuscript in advance, and the session will focus on questions and discussion led by the speaker.

Abstract: A common strategy in many proofs and algorithms is to begin by decomposing a graph into more highly connected pieces. Decomposition is easy when the goal is to obtain connected or 2-connected pieces, and decomposition into 3- or 4-connected pieces is also straightforward in many settings. For higher levels of connectivity, however, no effective and widely applicable notion of decomposition is currently known. To address this, Robertson and Seymour introduced tangles, which capture the k-connected regions of a graph without decomposing. 

Abstract:Motivated by the representation theory of finite-dimensional algebras, we recently investigated the notions of left modularity and extremality in (completely) semidistributive lattices. For lattices of torsion classes, we obtain a simultaneous characterization of left modularity and extremality in terms of the behavior of certain indecomposable modules, called bricks. Our results extend the classical theory beyond the realm of finite lattices, while remaining within the framework of (completely) semidistributive lattices. Time permitting, I will also discuss extensions of these results to arbitrary infinite lattices that are completely semidistributive and weakly atomic. This talk is based on recent joint work with Sota Asai, Osamu Iyama, and Charles Paquette.

There will be a pre-seminar presenting relevant background at beginning graduate level starting at 1:30pm in MC 5417.