# The Adjoint School 2025

The 2025 Adjoint School features projects mentored by Robin Kaarsgaard Sales, Laura Scull, Justin Hsu, and Georgios Bakirtzis. It is organized by Ari Rosenfield, Elena Dimitriadis Bermejo, Innocent Obi, and Drew McNeely. The research week will be held at the University of Florida.

##### Important Dates

*Dec 1, 2024*. Application Due*Jan - May, 2025*. Learning Seminar*May 26 - 30, 2025*. Research Week at the University of Florida

### Research Projects

##### Homotopy of Graphs

*Mentor:* Laura Scull

*TA: * TBD

*Students:* TBD

Graphs are discrete structures made of vertices connected by edges, making
examples easily accessible. We take a categorical approach to these, and work in
the category of graphs and graph homomorphisms between them. Even though
many standard graph theory ideas can be phrased in these terms, this area
remains relatively undeveloped.

This project will consider discrete homotopy theory, where we define the
notion of homotopy between graph morphisms by adapting definitions from
topological spaces. In particular, we will look at the theory of ×-homotopy as
developed by Dochtermann and Chih-Scull. The resulting theory has some but
not all of the formal properties of classical homotopy of spaces, and diverges in
some interesting ways.

Our project will start with learning about the basic category of graphs and
graph homomorphisms, and understanding categorical concepts such as limits,
colimits and expnentials in this world. This offers an opportunity to play with
concrete examples of abstract universal properties. We will then consider the
following question: do homotopy limits and colimits exist for graphs? If so,
what do they look like? This specific question will be our entry into the larger
inquiries around what sort of structure is present in homotopy of graphs, and
how it compares to the classical homotopy theory of topological spaces. We will
develop this theme further in directions that most interest our group.

Readings

*On the Concrete Categories of Graphs*, G. McRae, D. Plessas, and L. Rafferty-
*A Homotopy Category for Graphs*, T. Chih and L. Scull

##### Compositional Generalization in Reinforcement Learning

*Mentor:* Georgios Bakirtzis

*TA: * TBD

*Students:* TBD

Reinforcement learning is a form of semi-supervised learning.
In reinforcement learning we have an environment,
an agent that acts on this environment through actions,
and a reward signal.
It is the reward signal that makes reinforcement learning a powerful technique
in the control of autonomous systems, but it is also the sparcity
of this reward structure that engenders issues.
Compositional methods decompose reinforcement learning to parts
that are tractable. Categories provide a nice framework
to think about compositional reinforcement learning.

An important open problem in reinforcement learning is /compositional generalization.
This project will tackle the problem
of compositional generalization in reinforcement learning
in a category-theoretic computational framework in Julia.
Expected outcomes are of this project are category theory derived algorithms and concrete experiments.
Participants will be expected to be strong computationally,
but not necessarily have experience in reinforcement learning.

Readings

*Structure in Deep Reinforcement Learning: A Survey and Open Problems*, A. Mohan, A. Zhang, and M. Lindauer*Categorical semantics of compositional reinforcement learning*, G. Bakirtzis, M. Savvas, and U. Topcu.

##### Categorical Metric Structures for Numerical Analysis

*Mentor:* Justin Hsu

*TA: * TBD

*Students:* TBD

Numerical analysis studies computations that use finite approximations to continuous data, e.g., finite precision floating point numbers instead of the reals. A core challenge is to bound the amount of error incurred. Recent work develops several type systems to reason about roundoff error, supported by semantics in categories of metric spaces. This project will focus on categorical structures uncovered by these works, seeking to understand and generalize them.

More specifically, the first strand of work will investigate the neighborhood monad, a novel graded monad in the category of (pseudo)metric spaces. This monad supports the forward rounding error analysis in the NumFuzz type system. There are several known extensions incorporating particular computational effects (e.g., failure, non-determinism, randomization) but a more general picture is currently lacking.

The second strand of work will investigate backwards error lenses, a lens-like structure on metric spaces that supports the backward error analysis in the Bean type system. The construction resembles concepts from the lens literature, but a precise connection is not known. Connecting these lenses to known constructions could enable backwards error analysis for more complex programs.

Readings

*Toward a Formal Theory of Graded Monads*, S. Fujii, S. Katsumata, P-A. Melliès*The Dialectica Categories*, V. de Paiva.