Started in Fall 2020, the Algebra & Logic Lab at Sacramento State explores problems and facilitates learning around various mathematical topics, focusing on those lying in the intersection of group theory and model theory. It aims to drive research, build excitement and capacity for advanced mathematics, and support students in pursuing advanced degrees and other professional goals.

The Lab warmly acknowledges the support of the National Science Foundation. Much of its work has been supported in part by the National Science Foundation under grant number DMS-1954127. Any opinions, findings, and conclusions or recommendations expressed on this website are those of Joshua Wiscons and do not necessarily reflect the views of the National Science Foundation.

Recent Research

Actions of $\operatorname{Alt}(n)$ on groups with a dimension

Luis-Jaime Corredor, Adrien Deloro, and Joshua Wiscons developed a new context of “modules with an additive dimension”, and in that context, they classified the faithful modules of minimal dimension for $\operatorname{Sym}(n)$ and $\operatorname{Alt}(n)$ provided $n$ is large enough. Preprint on the arXiv.

Two masters students, Barry Chin and Andy Yu, together with Adrien Deloro and Joshua Wiscons continued working in the abstract context of modules with an additive dimension to investigate the low values of $n$ left open in the previous project. Their main results determined the minimal dimension of faithful $\operatorname{Alt}(n)$-modules when $n\le 6$ and also classified several of the modules of minimal dimension. Preprint on the arXiv.

Master’s student Lauren McEnerney and Joshua Wiscons worked to develop an abstract recognition theorem for the icosahedral modules for $\operatorname{Alt}(5)$. The initial motivation was to classify the modules of minimal dimension for $\operatorname{Alt}(5)$ in characteristics not 2 or 5 (again in the context of modules with an additive dimension), which was left open in the previous project. This was achieved, but the recognition theorem developed holds under more general conditions where there need not be a notion of dimension present. Preprint on the arXiv.

Tuna Altınel and Joshua Wiscons worked in the model-theoretic setting of finite Morley rank to understand the minimal dimension (i.e. minimal Morley rank) of a not necessarily abelian group that carries a faithful action of $\operatorname{Alt}(n)$. The focus of this project was the case when the acted upon group has no elements of order 2. Preprint on the arXiv.

Tuna Altınel and Joshua Wiscons continue to investigate the minimal Morley rank of a not necessarily abelian group that carries a faithful action of $\operatorname{Alt}(n)$. This was determined in the case when the acted upon group is algebraic as a part of a different project, but the problem remains open in general. The main motivation for this project is in its application to actions with a high degree of generic transitivity.

Permutation groups with a high degree of generic transitivity

Tuna Altınel and Joshua Wiscons are working to show that a generically sharply t-transitive permutation group of finite Morley rank must satisfy $t \le n +2$ where $n$ is the rank of the set being acted upon. The case when the stabilizer of a generic (t−2)-tuple is a so-called L-group has been solved (which covers when $n=3,4,5$ and when this stabilizer is solvable), but the general case remains open. Preprint on the arXiv.

Tuna Altınel and Joshua Wiscons are working to classify the generically t-transitive permutation groups of finite Morley rank for which $t \ge n +2$ where $n$ is the rank of the set being acted upon. The case when $n=2$ was previously solved, and the current focus is on when the stabilizer of a generic (t−2)-tuple is an L-group.

Relational complexity of finite permutation groups

Gregory Cherlin and Joshua Wiscons are working to understand the relational complexity of the symmetric and alternating groups $\operatorname{Sym}(nk)$ and $\operatorname{Alt}(nk)$ when acting on partions of $\{1,\dots,nk\}$ into $n$ blocks, each of size $k$. The problem has been solved for $k=2$, and an article on this is being finalized.

Master’s student Meagan Pham together with Gregory Cherlin and Joshua Wiscons worked towards determining the precise relational complexity of the group $\operatorname{A\Gamma{}L}(n,F)$ in its natural action on $F$. The main focus is on the case of $n=1$ where it is known that the complexity is either 3 or 4. The problem has been solved when the dimension of F over the base field is divisible by 4 or 5 (as well as for other small, specific dimensions via computer calculations), but it remains open in general.

Additional projects

The Lab has various other ongoing projects (both theoretical and computational) mostly around the relational complexity of finite permutation groups as well as geometries associated to certain groups of finite Morley rank. Please reach out if you’re interested [joshua.wiscons@csus.edu]).

Change Maker Series

The Lab collaborates with the the Department of Mathematics and Statistics, Women in STEM and Math Club to organize the Change Maker Series at Sacramento State. This is a yearly series—launched in Spring of 2021—that aims to connect the Sacramento State community with leaders in the mathematical sciences who are transforming the discipline by advancing knowledge and improving access for all. More information can be found on the Change Maker Series webpage.

• Spring 2021. Inaugural event! The series featured Dr. Candice Price. It took place virtually and was attended by over 65 members of our campus community.
• Spring 2022. The series featured Dr. Pamela E. Harris. The event again took place virtually and was attended by nearly 40 members of our campus community.
• Spring 2023. The series featured Dr. Marissa Kawehi Loving. This time it was a hybrid event with nearly 40 members of our campus community attending in-person and virtually.
• Spring 2024. The series featured Dr. Omayra Y. Ortega. This was the first time the speaker came in person; over 40 members of our campus community attended in-person and virtually.

ANTC Seminar Series

Spring 2021: Representation theory of the symmetric group

In Spring 2021, we ran a seminar series on the representation theory of the symmetric group. It was a joint offering between the Lab and the Algebra, Number Theory, and Combinatorics (ANTC) seminar at Sacramento State. The target audience was undergraduate (and beyond) math-interested folk with some exposure to linear algebra, and the main goals were for folks to have fun, learn something interesting they wouldn’t see in a regular class, and hopefully get excited to study more math. It is hoped that the seminar might also be a starting point for an independent study or research project. The series drew about 10 people at each meeting (including undergraduates, masters students, and faculty).

Topics included: the symmetric group: basic structure and properties; an introduction to representations and modules for groups; Young tableaux and the classification of the irreducible $\operatorname{Sym(n)}$-modules. More information is on the seminar series webpage.

Spring 2022: Topics in Permutation Groups

In Spring 2022, we ran a seminar series around the theory of finite permutation groups. As with the 2021 series, the target audience was math-interested folk with exposure to linear algebra. About 15 people attended the series.

Topics included: an introduction to permutation groups: notation and terminology; examples: symmetric and alternating groups, affine and projective groups, and some Mathieu groups (via actions on a Dogic); the classification of permutation groups with a high degree of transitivity; relational complexity. More information is on the seminar series webpage.