#### Participants

#### Publications and Presentations

In Summer 2023, the Foundations Group has received a 5-year, $1.35 million dollar NSF grant to form a Vertical Research Training Group (RTG) in Logic and its Application. This RTG will conduct foundational aspects of classical and non-classical logics, and their application to computer science and quantum computing. This grant will form a small community consisting of five undergraduate students, two graduate students, a postdoctoral fellow and several faculty. The group will conduct research and educational activities with each level supporting the others. The research conducted will be internationally recognized and the group will contribute to a highly qualified workforce in areas of critical need. The structure of the group and its activities will serve as a model for research groups supporting undergraduate and graduate education at minority serving institutions.

The RTG is constructed around "thematic problems." Each year a topic will be chosen that naturally lends itself to stratification at the research, graduate, and undergraduate levels. At the research level, this will be a problem of interest to the PI's and postdoc. The aim is to publish high-quality novel work in the area. The graduate-level version of the problem will be related to the students' coursework, and can be presented at a regional conference. At the undergraduate level, one can find a simplified version of the problem so that it is within reach from topics in undergraduate coursework. If possible, aspects of the research level problem could be asked of the undergraduates in this simplified setting. At the end, the undergraduates will be able to place their work in relation to the other levels and gain an appreciation of research. The "thematic problems" will be chosen from the PI's shared research interests and will reinforce one another from year to year.

Each year, the RTG will support a cohort of five undergraduate students, two graduate students, and a postdoc (in years 2-4). The undergraduate students will be supported during the academic year by the RTG and could gain additional funding as peer learning assistants in lower-level undergraduate classes. In the summer, the undergraduate students will either be placed into internships or will receive stipends to join in research seminars and projects. The seminars and projects will be designed and overseen by the project personnel and led by the supported graduate students and postdoc. During the academic year, the postdoc will teach one graduate-level course in one semester, and an undergraduate level course the other. Weekly seminars will be held by the group focusing on thematic research problems of the group that will be continued into the summer.

The RTG in Logic and Its Applications will be novel in its scope and organization. A centered research group in a discipline not overpopulated in the U.S. allows for an internationally recognized faculty that can attract strong graduate and postdoctoral students, while located in the heart of one of the poorest minority/majority areas of the country. The vertical structure of the group is designed to lower the barriers faced by talented students in imagining and realizing a life in research mathematics. We hope that this can become an economically viable model for minority serving institutions to move students to top levels of mathematical research.

**Thematic Problems**

To achieve our goal of a vertically integrated group, we aim for all levels to be working on problems with a shared theme. This makes for natural mentorship with graduate students serving as mentors to undergraduates, the postdoc as a mentor to both graduate students and undergraduates, and the faculty as mentors to all. The emphasis will be on providing an environment that fosters collaboration. The topics our cohort will work on will be grouped into what we term "thematic problems," different versions of which are meaningful at all levels.

A different thematic problem will be assigned each year. They are briefly outlined below. The collection of thematic problems will be independent of one another, in that none will be required to progress on another, but they will have substantial overlap and reinforce the others. Ideally, a graduate student could work in an area of a thematic problem for their thesis, and will find other thematic problems related to the problem area chosen for their thesis. As a whole, these thematic problems will paint the view of our group.

**Thematic Problem: Modal Logic**

Models of modal logic are labeled directed graphs. Such structures can represent spatial, temporal, physical or geometric configurations. They can also be used to represent behavior of computational processes (transition systems). In spite of such wide range of applications, these structures are simple enough to be easily understood by undergraduates. Yet, there are many difficult problems in working with these structures that remain open to this day. Some of these belong to logic (completeness and axiomatization problems), some relate to other branches of mathematics (topology, algebra, category theory), and some are algorithmic (decidability and computational complexity). While these have wide range, they are all closely related.

Vertical integration in this setting can be achieved by having our undergrads study the basics of these structures, having our graduate students do more advanced work with these structures, while mentoring our undergrads, and finally having our postdoc address some open problems related to these structures. This has potential to provide a fruitful environment at all levels, resulting in scientific camaraderie. It will also provide structured goals, which could motivate and excite our cohort at all levels.

**Thematic problem: Duality Theory**

Starting from Stone's groundbreaking work in 1930s, duality theory had a major influence on different areas of mathematics, including Foundations. It provides a bridge between algebra and geometry, using the language of category theory. On the one hand, duality theory uses geometric intuition to provide convenient representation of algebraic structures, thus helping to understand them better. On the other hand, it allows us to use algebraic tools to perform complicated calculations with geometric objects. As a result, this enhances dramatically our understanding of the mathematical structures of interest.

Vertical integration in this setting could have undergraduates provide representation of Boolean algebras and distributive lattices as fields and rings of sets, respectively. Special attention could be paid to the finite case, where the representations are especially easy to reconstruct. Graduate students can then work on fully understanding Stone and Priestley dualities, as well as their generalizations involving additional operations on Boolean algebras and distributive lattices. Finally, the postdoc can address some open problems in this direction as these relate to current work of our faculty.

**Thematic problem: Decidability and complexity**

After the first undecidability results in the 1930s by Church and Turing, investigation of decidability has become a mainstream area of mathematical logic. For a decidable theory, the next important aspect is its computational complexity: how many resources (time or memory) can decision procedures take? This area has been intensively developing in recent decades, but many fundamental problems remain open. The most famous of them is the equality of the complexity classes P and NP, one of the Millennium Prize Problems.

Modal and intuitionistic logics have numerous simple and elegant algorithmic problems related to finite structures (checking the modal satisfiability problem in a finite graph, calculating the Heyting algebra of a finite poset, verifying the existence of morphisms between finite structures). Such problems can be addressed and algorithmically implemented at the undergraduate level. Algorithmic problems in logic are closely related to semantic and algebraic properties like completeness and the finite model property;

these topics can be addressed at the graduate level. There are many open problems in the area which have different levels of difficulty. Some of them can be addressed by graduate students. At the advanced level, deeper problems can be considered that are currently investigated by our faculty.

**Thematic Problem: Logic and Geometry in Quantum Computing**

The notion of projection operators plays a key role in quantum computing from the most basic interpretation as a measurement process, through deep applications in operator algebras lying at the heart of topological quantum computing. Projections are the bridge between operator algebras and the logical and order-theoretic focus of our research group.

Vertical integration in this setting could have undergraduates reconstruct their quantum computing course from the perspective of measurement-based quantum computing. They could work with graduate students to develop the connection between operator algebras and lattices in the setting of matrix algebras and rings of functions. For the graduate students, projections and rings of continuous functions will lead from the Stone duality they learn in logic to Gelfand duality. In considering projections of von Neumann algebras, graduate students will see connections to modular lattices. There are many problems related to work of our faculty in these areas that a group of graduate students, under guidance, could solve in year for a small publication or conference presentation. At the advanced level, work on subfactors of von Neumann algebras and on abstract treatments of rings of continuous functions are current topics for our faculty.

**Thematic problem: Rings of Functions**

Characterizing the ring C(X) of continuous functions on a compact Hausdorff space was done over the complex numbers by Gelfand and Naimark, and independently over the reals by Stone. In the former case these are commutative C*-algebras. The latter case involves a combination of commutative ring theory, lattice theory, and topology. The classic book by Gillman and Jerison collected everything known about the subject by 1960. More categorical approaches to the study of rings of functions has been done in recent years, including considerable work by project personnel.

Vertical integration in this setting could have undergraduates work through ring and lattice-theoretic properties of $C(X)$. This would require them to develop familiarity with several branches of mathematics. Graduate students can then work on understanding Gelfand-Naimark-Stone duality. Finally, the postdoc can address some open problems in this direction as these relate to current work of our faculty.

To implement these thematic problems, we will have weekly research seminars, followed by four-week long summer research projects which will build on and consolidate the acquired knowledge. The seminars and projects will be designed and overseen by the project personnel. They will be led by the postdoc with the help of two supported graduate students. We anticipate that our students will present the findings of these research projects at regional and possibly national levels.

**Support **

Each year, the program will support 5 undergraduate students. These undergraduates should typically be past our Introduction to Higher Mathematics course, but do not need to have prior exposure to foundations courses. Undergraduates receive a stipend of $5,000 for the academic year for participating in a weekly seminar and weekly group meetings. In the summer, undergraduates that are not placed into an internship receive a stipend of $5,200 for participating in a 4-week group research project and writing their results. Undergraduate students may be supported by the project for more than one year, or move on in the project to be a graduate student participant.

Each year, the project will provide generous support for 2 graduate students. These students will conduct the weekly seminar and group meetings with the oversight of faculty and the postdoc and provide the primary interface with the undergraduate students. They will serve as mentors for the undergraduates. The graduate students will participate in the summer group project and possibly participate in travel to conferences related to group themes. The graduate student funding is expected to be limited to one year to provide opportunity for more graduate students and to allow the students to obtain practical training in teaching required for academic positions. Unfunded graduate students in the Foundations group are welcome to participate in group seminars and projects.

All students funded by the project must be US citizens or permanent residents

The postdoctoral position for the group will be advertised in Fall 2023.

**Contact**

Please contact John Harding jharding@nmsu.edu or Ilya Shapirovskiy ilshapir@nmsu.edu for further information.