The Planar Rook Algebra

April 30, 2020   /  

Henry Potts-Rubin

Student: Henry Potts-Rubin
Major: Mathematics
Advisor: Robert Kelvey

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Henry will be online to field comments on May 8:
10am-noon EDT (Asia: late evening, PST 6am-8am, Africa/Europe: late afternoon)

45 thoughts on “The Planar Rook Algebra”

  1. Very interesting. Not sure what it means but it looks cool. Good luck at Syracuse.

  2. Thank you for sharing your IS? What excites you most about Algebra and this project in particular?

    1. We’re used to thinking about mathematical systems and operations as particulars, like the integers and addition. Algebra is exciting because it takes particulars and views them as more general structures. Rather than the integers and addition, we can simply talk about groups. This allows us to relate many (at-first-seemingly-unrelated) things. This project in deals greatly with the concept of actions, which allows us to take these abstract structures and make them interact with each other, which I find exciting, as through interaction, new information can be gleaned.

  3. I have been semi-listening to your explain your thesis all year, and I can promise you that I still have ABSOLUTELY NO IDEA what any of this says or means⁠—but you do, and that’s pretty cool! Congratulations on finishing, and I cannot wait to visit you at Syracuse!

  4. It seems to me that the presentation of the Monoid (and perhaps the whole Algebra) seems related to the presentation of the Symmetric group. Any word on their relationship? or is it strictly coincidental?

    Fantastic to finally get to see everything together!

    1. Yes! Good eye! The diagrams look very similar to permutation diagrams, don’t they? You can view the rook monoid as a monoid of “partial permutations” and the planar rook monoid as a monoid of “planar partial permutations.” The “partial” part is telling us that it is only a monoid, and not a group, because not all partial permutations are bijections.

      Thanks, Isaac!

  5. Congratulations, Henry … we look forward to hearing about your experience in the grad program at Syracuse. Perhaps you (and Ralph Xu) will open doors there for future Wooster students who have similar interests in theoretical math.

    1. Thank you! I’ll be sure to keep in touch. It’s crazy that Ralph and I will both be there, isn’t it?

  6. Hi Henry. Congratulations on all your hard work and focus! Just wondering if this modelling could be applied effectively to solving the classic “lights on the chain bridge” conundrum?

    1. Hi there! A very interesting thought you’ve had there. I feel that’s more of a Number Theory or Combinatorics problem, but, that being said, counting strategies do arise in Algebra, as well. Thank you for your comment!

  7. Well done, Henry! Congratulations on all of your successes at Wooster. Wish you the best at Syracuse!

  8. Congratulations Henry! Best wishes for your future in mathematics at Syracuse! -J. Bowen

  9. Excellent work, Henry! It was a pleasure second-reading your IS. Best of luck at Syracuse, and I’ll be not too far away in Buffalo.

  10. Even though I don’t know anything about the field your IS is in, let alone the specific topic, you do an excellent job of making me want to know more.

    What was one of the most surprising challenges you faced while wiring on this project?

    1. Thanks, Brendan! A good amount of this project was trying to explain higher-level math at an undergraduate level. That’s certainly a task by itself, but one of the main papers I worked through just simply didn’t explain well what was going on. It was (and is) surprising to see how some papers, while solid mathematical work, just aren’t accessible. I know that what I’ve written isn’t accessible to a lot of people, but trust me when I say it’s far more accessible than some of the papers I read during this process.

  11. I think I understood some of that…? Just kidding, awesome job Henry! Can’t wait to hear what you end up studying at Syracuse!

  12. Very cool, Henry! I wish I were able to understand more, but it really is amazing to see how all of your hard work turned out.

  13. Henry, this stuff is awesome! I’m so glad that you continued to pursue abstract algebra (and I’m sad I wasn’t able to take AA2 with you because it would have been so fun!). You are destined for greatness at Syracuse, I am certain!

  14. Congratulations on a year’s work well-done! I love how passionate you are about math and the effort you put into this project, now for all to see/attempt to understand 🙂 What was your favorite part about conducting/writing your IS and/or excites you most about moving forward in the math world?

    1. Thank you, Gracie! I can’t wait for it to be your turn to share your IS (hopefully in person).

      I really enjoyed this project, and maybe the best part was getting to work on it with my advisor. His guidance was key, and his support was genuine. He’s a good guy to bounce ideas off of, for sure. In terms of moving forward, I’m excited that there’s so much more to learn. I’m really only scratching the surface of one small aspect of one area of math with this IS.

  15. Awesome job, Henry – you have worked hard, and we are very proud of you!

  16. I so much wish that I could talk to you about this in person! I know that you would explain it beautifully, as you typically do. It looks very interesting, and it appears that you did a really nice job here. Heartfelt congratulations! It has been a pleasure having you as a math major at Wooster, and I wish you all the best at Syracuse. Please keep in touch!

  17. I can’t understand a single word of this and that is a good thing! So proud of you 🙂

  18. Hey man, congratulations!! This is obviously so smart and advanced! I am really excited to see what amazing math you will do and go through grad school at the same time as you do. It will be fun 🙂

  19. Excellent poster, Henry! It’s clear you’ve put a lot of good work into this project, and I’m excited to see what your future holds!

  20. Congratulations Henry, now I see the results of those long days of work at the table outside Dr. Kelvey’s office.

  21. Congratulations on your IS, Henry!! Wishing you all the best for the future. I hope you will have the opportunity to apply the skills you learned doing this work, wherever you go after graduation.

    All my best,
    Pres. Bolton

    1. Thank you so much! And thank you for everything you’ve done during this wild time.

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