|William J. Keith|
Michigan Tech University Math Department
My courses for the Fall of 2022 are:
|My regular office hours are Monday and Wednesday 11am - 12pm. My office is Fisher 314. Should students have class during this hour I warmly encourage making an appointment for a more convenient time to meet. I also have a Zoom office if students prefer, https://michigantech.zoom.us/j/7738766078, where I can take appointments.|
|Outside of my office hours, students should get in touch with me using my MTU email, wjkeith [at] mtu.edu (no spaces). I am generally available at other times if a student emails me well in advance. This and other class information is available on the syllabus for each class, which is also on the course webpages linked above.|
I currently serve on the Undergraduate Committee and am the advisor for math majors with the Discrete Mathematics concentration. I am the faculty Senator for the Math Department and coordinate assessment activities for the Department.
Here is the Algebra and Combinatorics Seminar schedule for previous semesters, with schedules back to the Spring of 2013. The Seminar is no longer running.
Here is the Partitions Seminar schedule and archival page for the current and previous semesters, with schedules, and videos and slides, of talks back to its start in Spring 2021.
|My research is in combinatorics, specializing in partition theory and related q-series and identities.|
For the standard outline of my research, please help yourself to a copy of my CV which includes a full publication list. For more detail, I list below a few of my papers (and my thesis). Preprints of all my work are available on the arXiv.
Selected Publications and Preprints
In 2016-2017 I supervised the master's thesis of J. T. Davies in permutation statistics. Mr. Davies is currently in doctoral study at the University of Waterloo in Canada. Our motivating question: the major index is symmetric over some sets of pattern-avoiding permutations in Sn with fixed descent number, and (maj, des) form a Mahonian pair. Are there conditions analogous to pattern avoidance (and hopefully equally interesting) for other pairs such as (den, exc) which are known to be Mahonian but are not distributed symmetrically over pattern-avoidance classes?
In 2020-2021 I supervised the master's thesis of Emily Anible in the distribution of major index over tableaux with fixed descent number. Ms. Anible is currently in doctoral study at Penn State. Our motivating question: if we fix the descent number and ask for the distribution of the major index over all standard Young tableaux of a given shape, it seems that in many cases some beautiful and combinatorially interesting formulas arise. A full restriction giving such a formula would refine Stanley's q-Frame-Robinson-Thrall formula.
I am currently overseeing the doctoral work of Hunter Waldron, who has just submitted his first paper on Schmidt-type theorems.
If graduate students are interested in research with me they should let me know. I am presently open to becoming a graduate advisor for another student.
These are a few of the ongoing research questions which interest me. I am always happy to receive comments from interested colleagues, and would be pleased to collaborate with someone who has useful ideas in these directions. Graduate students considering combinatorics who find some of these questions interesting are encouraged to contact me as well.1.) Schmidt-type theorems, such as those discussed by Mr. Waldron and in my recent submission with George Andrews.
2.) Extending the refinement of Stanley's formula listed given in the first listed paper. I think it would be an exciting result if this could be generalized to standard Young tableaux of any shape.
3.) I am interested in m-regular partitions, especially their low-modulus congruences. Related to this, I would like to show properties of singular overpartitions related to known theorems such as the Pak-Postnikov (m,c) theorem.
4.) Dousse and Kim's conjectures on unimodality for the overpartition analogue of the Gaussian coefficients seems quite interesting to me.
5.) Bergeron's ad - bc conjecture.