Home  Program 
The program consists of 2 parts, both consisting of online virtual talks that are streamed on Zoom, recorded, with additional discussions happening on the Zulip chat.
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A more irregular schedule of deep dives into specific topics of Category Theory, taught by invited experts in the area, some already showing applications to Machine Learning and some which have not been applied yet.
The lectures are finished for the moment, but you can still check out their recordings!
We had weekly introductory lectures, where we
taught the basics of category theory with a focus on applications to Machine Learning.
November 14  Neural network layers as parametric spans  Recording link and Slides 
Pietro Vertechi  
Properties such as composability and automatic differentiation made artificial neural networks a pervasive tool in applications. Tackling more challenging problems caused neural networks to progressively become more complex and thus difficult to define from a mathematical perspective. In this talk, we will discuss a general definition of linear layer arising from a categorical framework based on the notions of integration theory and parametric spans. This definition generalizes and encompasses classical layers (e.g., dense, convolutional), while guaranteeing existence and computability of the layer's derivatives for backpropagation. 
November 21  Causal Model Abstraction & Grounding via Category Theory  Recording link and Slides 
Taco Cohen  
Causal models are used in many areas of science to describe data generating processes and reason about the effect of changes to these processes (interventions). Causal models are typically highly abstracted representations of the underlying process, consisting of only a few carefully selected variables, and the causal mechanisms between them. This simplifies causal reasoning, but the relation between the model and the underlying system is never described in mathematical terms, and this has led to considerable philosophical confusions. Furthermore, it has made it hard to understand how causal modeling relates to other fields such as physics (where systems are described by dynamical laws without reference to causes), dynamical systems, and agentcentric frameworks such as Markov Decision Processes (MDPs). In this talk we study this idea of abstraction from a categorical perspective, focussing on two questions in particular:

December 12  Category Theory Inspired by LLMs  Recording link and Slides 
TaiDanae Bradley  
The success of today's large language models (LLMs) is striking, especially given that the training data consists of raw, unstructured text. In this talk, we'll see that category theory can provide a natural framework for investigating this passage from texts—and probability distributions on them—to a more semantically meaningful space. To motivate the mathematics involved, we will open with a basic, yet curious, analogy between linear algebra and category theory. We will then define a category of expressions in language enriched over the unit interval and afterwards pass to enriched copresheaves on that category. We will see that the latter setting has rich mathematical structure and comes with readymade tools to begin exploring that structure. 
March 20  Introduction to Categorical Cybernetics  Recording link and Slides 
Jules Hedge  
Categorical cybernetics is based on two things: (1) the abstract theory of categories of optics and related things, and (2) a whole bunch of specific examples.
These tend to arise in topics that historically were called "cybernetics" (before that term drifted beyond recognition)  AI, control theory, game theory, systems theory.
Specific examples of "things that compose optically" are derivatives (well known as backprop), exact and approximate Bayesian inverses, payoffs in game theory, values in control theory and reinforcement learning, updates of data (the original setting for lenses), and updates of state machines.
I'll do a gentle tour through these, emphasising their shared structure and the field we're developing to study it. The talk will cover material related to the paper Towards Foundations of Categorical Cybernetics 
March 27  Dynamic organizational systems: from deep learning to prediction markets  Recording link and Slides 
David Spivak  
In training artificial neural networks (ANNs), both neurons and arbitrary populations of neurons can be seen to perform the same type of task. Indeed, at any given moment they provide a function A>B, and given any input from A and loss signal on B, they do two things: provide an updated function A>B and backpropagate a loss signal on A. Populations of neurons, which we called "Learners", can be put together in series or in parallel, forming a symmetric monoidal category. However, ANNs satisfy an additional property: there is a consistent method by which the functions update and errors backpropagate; namely, they all use gradient descent. The chain rule implies that the composite of gradient descenders is again a gradient descender. In this talk I will discuss a generalization called "dynamic organizational systems", which includes ANNs, prediction markets, Hebbian learning, and strategic games. It is founded on the category Poly of polynomial functors, which generalizes Lens. I will review the relevant background on Poly and then explain dynamic organizational systems as coherent procedures by which a network of component systems can rewire its network structure in response to the data flowing through it. I'll explain the ANN case, and possibly the prediction market case, time permitting. The talk will cover material related to the paper Dynamic categories, dynamic operads: From deep learning to prediction markets 
May 29  Sheaves for AI  Recording link and Slides 
Thomas Gebhart  
Many datagenerating systems studied within machine learning derive global semantics from a collection of complex, local interactions among subsets of the system’s atomic elements. In order to properly learn representations of such systems, a machine learning algorithm must have the capacity to faithfully model these local interactions while also ensuring the resulting representations fuse properly into a consistent whole. In this talk, we will see that cellular sheaf theory offers an ideal algebrotopological framework for both reasoning about and implementing machine learning models on data which are subject to such localtoglobal constraints over a topological space. We will introduce cellular sheaves from a categorical perspective before turning to a discussion of sheaf (co)homology as a semicomputable tool for implementing these categorical concepts. Finally, we will observe two practical applications of these ideas in the form of sheaf neural networks, a generalization of graph neural networks for processing sheafvalued signals; and knowledge sheaves, a sheaftheoretic reformulation of knowledge graph embedding. 
Week of October 10  Week 1: Why Category Theory?  Recording link and Slides 
Bruno Gavranović  
By the end of this week you will:

Week of October 17  Week 2: Essential building blocks: Categories and Functors  Recording link and Slides 
Petar Veličković  
By the end of this week you will:

Week of October 24  Week 3: Categorical Dataflow: Optics and Lenses as data structures for backpropagation  Recording link and Slides 
Bruno Gavranović  
By the end of this week you will:

Week of October 31  Week 4: Geometric Deep Learning & Naturality  Recording link and Slides 
Pim de Haan  
By the end of this week you will:

Week of November 7  Week 5: Monoids, Monads, Mappings, and lstMs  Recording link and Slides 
Andrew Dudzik  
By the end of this week you will:

Design by Mike Pierce 