Sep 13 - Ali Çataltepe (Northeastern)
Title: Time complexity for deterministic string machines
Abstract: Algorithms that learn environments represented by automata in the past have had complexity scaling with the number of states in the automaton, which can be exponentially large even for automata recognizing regular expressions with a small description length. We thus formalize a compositional language that can construct automata as transformations between certain types of categories, representable as string diagrams, which better reflects the description complexity of various automata. We define complexity constraints on this framework by having them operate on categories enriched over filtered sets, and using these constraints, we prove elementary results on the runtime and expressivity of a subset of these transformations which operate deterministically on finite state spaces. These string diagrams, or "string machines," are themselves morphisms in a category, so string machines can create other string machines in runtime to model computations that take more than constant memory. We prove sufficient conditions for polynomial runtime guarantees on these, which can help develop complexity constraints on string machines that also encapsulate runtime complexity.
Sep 27 - Alon Duvall (Northeastern)
Title: Omega limit sets of nonexpansive vector fields
Abstract: In this talk, we discuss systems of time-invariant ordinary differential equations whose flows are non-expansive with respect to a norm, meaning that the distance between solutions may not increase. Since non-expansiveness (and contractivity) are norm-dependent notions, the topology of $\omega$-limit sets of solutions may depend on the norm. For example, and at least for systems defined by real-analytic vector fields, the only possible $\omega$-limit sets of systems that are non-expansive with respect to polyhedral norms (such as $\ell^p$ norms with $p =1$ or $p=\infty$) are equilibria. In contrast, for non-expansive systems with respect to Euclidean ($\ell^2$) norm, other limit sets may arise (such as multi-dimensional tori): for example linear harmonic oscillators are non-expansive (and even isometric) flows, yet have periodic orbits as $\omega$-limit sets. This talk discusses how the Euclidean linear case is what can be expected in general: for flows that are non-expansive with respect to any strictly convex norm (such as $\ell^p$ for any $p\not=1,\infty$), and if there is at least one bounded solution, then the $\omega$-limit set of every trajectory is also an $\omega$-limit set of a linear time-invariant system.
Oct 11 - Tanishq Bhatia (Northeastern)
Title: Topological Autoencoders via Mapping Cylinders and Algebraic Losses
Abstract: Autoencoder neural networks are valuable tools for nonlinear dimensionality reduction. These networks usually produce a low dimension representation, called the latent code, through an encoder network and a subsequent reconstruction of the input through a decoder network. Despite the pervasiveness of autoencoders, the standard regularization terms used for training cannot guarantee the preservation of the global shape of the input point cloud. To address this limitation, a new body of research applies topological optimization and persistent homology to develop shape-preserving autoencoders. This approach usually consists of comparing the persistence diagrams of the latent point cloud to those of the input point cloud through a metric or a matching.
In this talk, we will present a new outlook for comparing the persistent homology of the latent and original point cloud by juxtaposing the two induced Vietoris-Rips filtrations via the mapping cylinder construction. Next, we will show an application of this construction to train better autoencoders through two novel differentiable topology-based regularization losses based on well-known algebraic certificates for isomorphism: relative homology diagrams and kernel/cokernel diagrams. Finally, we will discuss a speed up for the used topological optimization schemes through the recently introduced critical set method.
Oct 25 - Forrest Miller (Northeastern)
Title: Applied Global Polynomial Optimization
Abstract: In this talk, I will introduce the global optimization of polynomials. I will discuss its applications, the benefits of solving these problems algorithmically, and the challenges this area currently faces. I will then go into specific work I am conducting on redundant constraint generation to improve the exactness of the convex (semidefinite) relaxations of these polynomial problems.
Nov 8 - Erika Bessera (Northeastern)
Title: KLR Algebras
Abstract: KLR Algebras are a family of algebras with connections to representation theory and low-dimensional topology. Their construction is diagrammatic, with elements being collections of oriented strings that are allowed to cross and have formally placed dots. The diagrammatic description allows for topological, combinatorial, and graph theoretic techniques to study these algebras. I will discuss some of the structure and properties of KLR algebras, highlighting the techniques mentioned above, along with applications to representation theory and knot invariants.
Nov 22 - Aaron Agulnick (Northeastern)
Title: Higher-Order Autocorrelations and Signal Reconstruction on Number Fields
Abstract: From a signal on a locally compact abelian group, one can extract so-called autocorrelation data of order n for any natural number n. For applications such as X-ray crystallography's phase problem, it would be useful to understand when finitely many orders of autocorrelation data suffice to reconstruct a function, and even more useful to be able to perform this reconstruction algorithmically. In this talk, I will describe when and how we know that this is or is not possible, and how we are beginning to turn nonconstructive results in this area into algorithms.
Nov 26 - Dezhou Li (Northeastern) [544 Nightingale at 4.30 pm]
Title: The Cartan—Leray Spectral Sequence of the Braid Group
Abstract: In Cohen's famous calculation of the mod p cohomology of configuration spaces, the key ingredient was a complete description of the Cartan—Leray spectral sequence for the configuration space of k = p points. In this talk, I will discuss this aimed at giving a complete description of this spectral sequence for arbitrary k. This work not only provides a geometric way to prove the Arone–Mahowald theorem and Kjaer’s theorem, but also gives the potential to determine the ring structure of cohomology of unordered configuration spaces.
Dec 6 - Jacob Zelko (Northeastern)
Title: An Introduction To Compositional Public Health
Abstract: Compositional public health is an emerging research field that exists to address the complexity in public health responses. The field lies at the intersection of category theory, epidemiology, and systems theory and utilizes tools from applied category theory for public health applications. This talk will present the motivation of this field, an overview of the mathematics involved in its approaches, current state of the art, live demonstrations, and future research directions as well as how to get involved.