About the Lab

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 also serves as a space to build community around a shared love of mathematics. Please email Joshua Wiscons [joshua.wiscons@csus.edu] if you have any questions or want to drop by.

Launched in late Fall 2020, the Lab warmly acknowledges the support of the National Science Foundation. Much of this material is based upon work supported 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.

Research

Modules with an additive 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 the symmetric and alternating groups, $\operatorname{Sym}(n)$ and $\operatorname{Alt}(n)$, provided $n$ is large enough. Preprint on the arXiv.
A small team of Masters students, Barry Chin and Andy Yu, together with Adrien Deloro and Joshua Wiscons investigated the low values of $n$ left open in the work mentioned above. Their main result was determining the minimal dimension of faithful $\operatorname{Alt}(n)$-modules when $n\le 6$; classifying those modules of minimal dimension remains open. A preprint will be released soon.
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 $\operatorname{Alt}(n)$. Work is complete in the case where the acted upon group has no elements of order 2. Preprint on the arXiv.
More info:
- slides from a talk on the Representations of $\operatorname{Sym}(n)$ of minimal dimension given at the Sacramento State ANTC seminar
- slides from a talk on the The minimal faithful $\operatorname{Sym}(n)$- and $\operatorname{Alt}(n)$- modules given at the joint Logic Seminar of Imperial College and Queen Mary University
- slides and recording from a talk on the Minimal representations of Sym(n) and Alt(n) given at the conference Groupes rangés : le retour at Institut Camille Jordan in celebration of celebrate the return to France of Tuna Altınel

Relational complexity of finite permutation groups

A Masters student Meagan Pham together with Gregory Cherlin and Joshua Wiscons are working to determine the precise relational complexity of the (full) affine semilinear group in its natural action on affine space.
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 case when $k=2$ is currently being finalized.
More info:
- slides from a talk on An introduction to relational complexity given at the Workshop on the Model Theory of Finite and Pseudofinite Structures at the University of Leeds

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 have any questions.
More info: email 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.

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.
More info: Change Maker Series webpage

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). We are looking forward to the next series in Spring 2022.

Topics included:
- The symmetric group: basic structure and properties
- Introduction to representations and modules for groups
- Young tableaux and the classification of the irreducible $\operatorname{Sym(n)}$-modules
More info: webpage for seminar series

Spring 2022: Topics in Permutation Groups

In Spring 2022, we developed 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. Familiarity with permutations and modular arithmetic (as one would see in Math 110A) will also be useful, but not necessary.

Topics included:
- Introduction to permutation groups: notation and terminology.
- Examples:
  - symmetric and alternating groups
  - affine and projective groups
  - some Mathieu groups (via actions on a Dogic)
- Classification of permutation groups with a high degree of transitivity
- Relational complexity
More info: webpage for seminar series