Literary Incarnations of (Infinity,1)-Category Theory
Shanna Dobson
Our project lies at the universal intersection of mathematics and the literary arts. The intersectional mathematics herein proposed is the intersection of category theory and literary writing, specifically the universal constructions of n-morphisms in n-categories, in concert with the aims of the Reading Apprenticeship approach to increasing mathematical literacy. The project is inspired by our recent work The Literary Incarnations of Perfectoid Diamonds based on our recent fantastical fiction novel Artemis Blu II: Infinity Diamonds in Infinity Diapsalmata, book two in the series The Artemis Chronicles of Imaginarium.
Aim: Our project invites participants to investigate the universality of category theory by first creating their own literary incarnation of a 2-category in the form of a 1-page short story, and then collectively refining eachother’s short-story incarnation into a collective novella. As a 2-category contains objects, 1-morphism relations between the objects, and 2-morphisms between the 1-morphisms, participants will write their short story establishing characters as objects in a category that interact via 1 and 2-morphism relations, the entirety of which is structure preserving, and then collectively figure out how to universally compose together all the short stories.
Prompt Questions: Potential questions we can explore are: evaluating the correlations between the strength of reader identity and the increase in mathematical literacy; cross-investigating the universality of category theory using constructions of grammatology; cross-investigating the universal construction of mathematical definitions using constructions of grammatology and mathematical logic; evaluating evidenced-based learning at the intersection of mathematics and the arts using category theory (or number theory) as a test case; using the universal structure of category theory to discover the similarities in how mathematical knowledge is constructed and how reader identity is constructed.
Long-Term RCCW: This project is a small-scale prototype of our intended 5-day project apropos for our forthcoming RCCW, wherein participants will create a literary incarnation of the n-morphisms in any n-category, and up to, for advanced and exceptionally curious participants, infinity-morphisms in any infinity-category.
Categorical Dance
Coordinator Claudia Maria Schmidt
Category theory is a major step towards unifying mathematical theory and has applications reaching into the fields of computer science and quantum physics. Categories, such as vector spaces, topological spaces, groups, fields, or logical deductive systems, are defined as sets of objects and three basic relations (morphism - for example group operations, continuous maps etc -, identity, composition) with the properties of associativity and existence of an unique identity morphism. By introducing the notion of functors between categories (that is, structure-preserving maps relating objects to objects and functions to functions), different categories can be related to each other. So the identity functor detects identical objects and morphisms. Moreover, the idea of adjoint functors that reverse the action of a functor reveals when one morphism or object in one category is the inverse of one morphism or object in another category. For example, if a functor related a Hausdorff space to a topological space, the information about separation gets lost, but an adjoint functor such as Stone Czech compactification reverses the process. In this way, several distinct categories form via the functor relating them together a common category of their categories.
In consecutive stages of the project, dancers will, by depicting objects and their relationships by body shapes, spatial relations and their changes between them through movement, visualize fundamental concepts, theorems and selected challenge problems pertinent to the development of the theory. One specific proposal how this might be approached is outlined in Maria Mannone / Luca Turchet 2019: 87 ff, but we want here to be more open to explore rather than to theoretically prescribe how categories, morphisms and functors may be represented through the parameters of dance.
Benefits Enhancing the understanding of the subject in nonverbal imaginative ways. Inspiring mathematical intuition to develop new ideas from creative movement.
Short term goal during this conference Creating a virtual group dance visualizing the categorification of selected systems (in the sense outlined above), including their relationships through functors, and illustrate major theorems, such as the fundamental theorem of category theory stating that each object can be uniquely determined to the other objects in its category.
Longer term goal Visualizing challenges of the theory, such as the infinity categories and cosmoi proposed by Emily Riehl. By developing further the movement created, the resulting dance might inspire mathematical intuition to find further ideas, conjectures and solutions.
Expressing Women's Experiences in Math Through Theatre
Stephanie Lewkiewicz
In this project, we will create an original theatre piece to express the diverse and complex experiences of women working in the field of mathematics. During our first group session (Saturday, October 16th), group members will share thoughts on their experiences, wants, needs, hopes, disappointments, fears, and dreams. Prompts will be given to help motivate our conversation. After our sharing period, participants will be asked to contribute written text reflecting their own and others' experiences and sense of identity, purpose, belonging, etc. We will shape our work into a poetry piece spoken by the members of the group. The poetry piece will be delivered by our group members during the Final Presentation portion of the conference on Sunday evening. Group members can share and participate to the extent to which they feel comfortable!
Those wishing to continue with the project in the long term will develop a woman mathematician character to serve as the lead in our play, and build the arc of her story by developing a setting, supporting characters, and a concrete plot, eventually working up to a full text. This piece will focus on capturing and honoring the experiences of trailblazing individuals working to uplift themselves and those around them in mathematics and the mathematical community. All those interested are welcome to participate in the long term project, even if they did not participate in this project during the conference. (Note: Although the main character will be a woman mathematician, all are welcome and encouraged to participate. We look forward to hearing about a diversity of experiences!)
Emotive Math
Maiko Serizawa
From the beginning of math education at school, we are implicitly taught that math is about intellect and logic and has little to do with our emotions. Instead of sharing the experiences of understanding or solving a problem with others by dialing into our emotions, we are constantly encouraged to dial out of ourselves under the pressure to compete with others. This is also reflected in the languages we hear and use in the space we engage in mathematics: clever, stupid, good, bad, etc. These languages are more about external judgement and less about our internal experiences. However, our personal experiences show us that emotions are what strongly affect our mathematical endeavours. Many of the breakthroughs in our learning and research come with emotions (joy, excitement, etc.) which fuel further work, and many of our mental breakdowns and burnouts stem from and result in other kinds of emotions (depression, anger, loneliness, etc.). There are also different ranges of emotions we have while interacting with others in the work which affect the way we think and feel about ourselves. In this project, we will explicitly talk about and investigate the emotional aspect of our mathematical work based on each individual’s personal experiences. By deepening the understanding of our emotional dimension, we aim to reach a new insight into what experience brings us fulfillment and what not, thereby allowing us to become a more conscious creator of our own mathematical research experiences.
Project End Goal:
The end goal of this project is to collaboratively produce a play performance that describes the internal experience of problem solving in math. In the process, we aim to gain a better understanding of the emotions we experience and the impact they have on us and our work. We will also investigate how our work can be affected by external human factors (both constructive and destructive).
What does it feel like within each of us when we are trying to solve a problem or understand something completely new in mathematics?
What do we experience internally?
Goal for this Conference (1st Stage):
In this October conference, we will share and collect participants' voices. Together in a safe environment we will experiment with expressing our unique internal experiences using words.