Spring 2024

Jan 26 - Tony Xiao (Northeastern)

Title: Extracting Eilenberg-MacLane Coordinates via Principal Bundles

Abstract: I am going to present an application of the theory of principal bundles to the problem of nonlinear dimensionality reduction in data analysis. More explicitly, given data with nontrivial underlying topology, we may derive coordinates for data points on an Eilenberg-MacLane space K(G,q) from persistent cohomology computation on dimension q with coefficients in G.

In the presentation, I will first give a short introduction on how the general framework of topological data analysis works, showing that the long-term goal being to recover the data up to homotopy equivalences. The second part explains how to derive a circular coordinate from data with nontrivial Cech Cohomology on the first dimension, by principal bundles and Milnor construction. The final part explains how to reduce Cech Cohomology on higher dimensions into the first, by connecting homomorphisms and bar construction.

Feb 9 - Yunmeng Wu (Northeastern)

Title: Noncommutative Gröbner Bases and Anick-Green Resolution

Abstract: Introduced by Bruno Buchberger in the 1960s, a Gröbner basis for an ideal in a polynomial ring is a specific set of generators that simplifies the process of solving polynomial systems. In this talk, we will first go over Gröbner basis and Buchberger's algorithm, and then explore how these ideas extended to the noncommutative algebras case. In addition, we'll touch briefly on computation of projective resolution for simple modules over homomorphic images of path algebras.

Feb 16 - Forrest Miller (Northeastern)

Title: Certifiable Estimation Through Semidefinite Programming

Abstract: It is critical that the fundamental tasks in machine intelligence - control, perception, and planning - can be effectively and efficiently executed by autonomous systems that operate in real world environments. Unfortunately, many of these problem formulations are NP hard in general, and thus are intractable to solve in total generality. Remarkably, recent work has shown that real world problems can be solved efficiently through semidefinite programming and convex relaxations. In this talk, I will provide an overview of certifiable estimation, how relaxations are designed, and techniques to solve these problems efficiently. I will use the example of point cloud registration to highlight the theoretical developments that have recently been made in this field.

Feb 23 - Tanishq Bhatia (Northeastern)

Title: Introduction to Topological Optimization

Abstract: Topological Optimization is a novel application of persistent homology used to minimize loss functions defined on the points of a persistence diagram through automatic differentiation. This optimization scheme can help accomplish several topology-dependent tasks like denoising signals and constructing shape-preserving autoencoders. This presentation will provide an overview of Topological Optimization, strategies for computational efficiency, and its applications to machine learning.

Mar 15 - Anadil Rao (Northeastern) 

Title: Supersymmetry and Cohomological Field Theories

Abstract: Supersymmetry has been a major theme in speculative Particle Physics theory, to construct models of Unification that go beyond the Standard Model of Particle Physics. Since, its inception in the 70s Supersymmetry has birthed powerful analytic tools to probe low dimensional geometry and topology. This talk will revolve around three themes:

1. Derivation of Supersymmetry algebras from Clifford algebras, which are themselves Quantizations of Exterior algebras.

2. Review of the formalism of Cohomological Field theory and argue the Partition function of such a theory can compute Topological quantities.

3. Demonstrate, Supersymmetrization of an existing Field theory amounts to "perturbing" the original Field theory by a Cohomological Field theory, which then admits an exact computation of the Partition function, via localization methods.

Time permitting, I will try to tie this story with another interesting emerging story in Mathematical Physics: Yang-Mills theory generalizes the study of Electromagnetism. A special class of solutions of Yang-Mills theory in 4d are called Instantons, which satisfy Dual/Anti Self-dual equations. Dimensional reduction of these equations to 3d yield the Bogomolny equations, whose solutions are called Monopoles. I will briefly review how studying Knots in R^3 via a Supersymmetric Chern-Simons Yang-Mills theory amounts to studying the Moduli space of solutions of a certain Bogomolny equation.

Mar 22 - Matt Piekenbrock (Northeastern)

Title: The Rank Invariant Perspective of Persistence & its Applications

Abstract: In this presentation, I will briefly introduce the rank invariant perspective of persistent homology and discuss recent advances in its computation over zero-characteristic fields. The first part of the presentation introduces persistence from the perspective of size functions, approachable to any who have taken an undergraduate calculus course. The second part of the talk will introduce a spectral relaxation of the rank invariant using the theory of persistent measures, a viewpoint which unveils surprising connections to other areas of applied mathematics, including Hodge theory, the “Laplacian Paradigm,” and even the theory of compressed sensing. Recent advances in implicit trace estimation and its connections to matrix function approximation will also discussed.

Mar 29 - Brad Turow (Northeastern)

Title: Analysis Of Optical Flow Data Using Bundle-Theoretic Techniques

Abstract: Video recordings project our 3D world onto a 2D screen.  Optical flow refers to the apparent motion of pixels from frame to frame, and statistical estimation of optical flow is a key step in many computer vision tasks.  In 2020, Adams et al. proposed a torus model for the space of high-contrast optical flow patches, using persistent and zigzag homology computations for validation. 

In this talk, I will describe the proposed torus model and discuss weaknesses in the evidence which bring the model into question.  I will then outline a ‘novel’ approach to analyzing the dataset using ideas and techniques from the theory of fiber bundles.  As a result of my investigation using this approach, I propose an improved model for high-contrast optical flow patches as a line bundle over a torus, which includes the original torus model as the zero section.  In addition to describing the model, I will present some important lessons learned from the analysis and the potential benefits of generalizing this ‘fiberwise TDA’ approach for modeling other datasets. 

Apr 5 - Vitor Gulisz (Northeastern)

Title: A Recipe to Get Correspondences and Examples of Categories

Abstract: A remarkable result in representation theory is the Auslander Correspondence, which describes a correspondence between Artin algebras of finite type and Auslander algebras. In this talk, we discuss how this result is part of a much more general phenomenon, and we explain how we can establish similar correspondences. Furthermore, we show how these correspondences can be used to produce examples of categories that satisfy specific properties. For instance, we explain how Rump has used these ideas to obtain one of the first examples of a category that is semi-abelian but not quasi-abelian, which gives a negative answer to a conjecture that had been open for more than 30 years.

Apr 12 - Arturo Ortiz San Miguel (Northeastern)

Title: What is The d-Regular Graph on n Vertices With The Most k-Cycles?

Abstract: We construct the d-regular graph G on n vertices with the maximum number of k-cycles for the cases where k = 5 and k = 6. We use a Mobius inversion relation between graph homomorphism numbers and injective homomorphism numbers to reframe the problem as a continuous optimization problem on the eigenvalues of G by leveraging the fact that the number of closed walks of length k is tr(A^k).

For k = 5 and d > 3, we show G can be represented as a collection of disjoint K_{d+1} graphs. For d = 3, we have disjoint Petersen graphs. For k = 6 with sufficiently large d, G consists of copies of K_{d,d}. Additionally, we derive new inequalities between homomorphism numbers of graphs and introduce and give formulas for non-backtracking homomorphism numbers. Moreover, we find the d-regular graph on n vertices with the most non-backtracking closed walks of length k by considering an optimization problem on the non-backtracking spectrum of G.

We conjecture that for odd k and sufficiently large d, the optimal G is a collection of K_{d+1}, while for even k with sufficiently large d, the optimal G consists of copies of K_{d,d}.