||Finding Unique Rectification Targets for d-Complete Posets
|| Oliver Pechenik
Hello world! My name is Mik Zlatin and this summer I will be investigating the existence of Unique Rectification Targets (URTs) in d-Complete posets. The existence of URTs in a poset allows for the computation of the structure constants for the Cohomological ring of the geometric space associated with the partial order.
We are looking for analagous results in the K-theory rings of spaces associated with d-complete posets.
- Week 1:
- The first week was intense. I met all my fellow researchers for the summer, including Rahul Ilango, who will be my research partner for this project. We met Professor Pechenik and he gave us more background on the research topic. He also assigned us to investigate the existance of URT's in the special case when the posets can be represented by a tree. Rahul and I believe we have a result in this case and will be presenting to the other DIMACS participants on Monday.
- Week 2:
- This week started off great! On Monday, Rahul and I presented our summer project to our fellow researchers and heard all about their projects as well.
Then we had a long meeting with our advisor about our next tasks. After a lot of slogging through complicated papers, we decided our next objective would be to understand and characterize the d-Complete posets algebraically.
Every d-Complete poset can be constructed by combining a set of 15 different irreducible posets.
Tuesday and Wednesday, I attended the DIMACS Workshop on Algorithms for Data Center Networks, a series of workshops and lectures on the hosted by DIMACS.
Some of the talks were extremely interesting and it makes me want to learn more about the field. It also showed me just how useful the algorithms and techniques I am learning are in practice and in industry.
We have made progress in our understanding of d-Complete posets, and will now be working on determining the existence of URT's for the 15 irreducible components that make up all d-Complete posets. Then we will try to see how combining them affects this unique rectification property.
- Week 3:
- How is it already the end of the third week? It feels like we just got started. This week I attended several seminars. On Tuesday, Sorelle Friedler gave a talk about auditing black box machine learning algorithms, an issue that is becoming increasingly important in today's world. On Wednesday, I dragged myself out of bed at 9:00 AM to go to a talk on Combinatorial Game Theory, where I learned about how to find winning strategies in Combinatorial Games using the Sprague-Grundy number.
On Thursday I attended a talk on Scientific Writing (how to write an abstract, publish papers etc.), which was quite boring but I suppose it's necessary. Finally, Friday was culture day, where I heard about the culture and history of my fellow researchers. I chose to share the origins and meaning behind my name.
In terms of the research, this week was spent developing (and mainly debugging) a program to compute all the rectifications of all skew partially ordered sets generated from a generating partially ordered set that is inputted by the user. We also have begun typing up our results formally in a Latex document.
- Week 4:
- We have begun focusing on one of the 15 irreducible components called the "near-bat" family. The "bat" has already been shown to have the unique rectification property, so we thought that it would be productive to see what we could learn about the near-bat (which is closely related to the bat poset).
However, our program takes days to run on even the smallest posets in the near-bat family. We are somewhat stuck until we can improve the runtime of the program, because we want to have good evidence that our conjectures are actually true before trying to prove them.
In the mean time, I have been reading this paper by Robert A. Proctor to try and understand how some of the other irreducibles are defined and the connection to geometry of Grassmanians.
- Week 5:
- This week we proved and wrote up some interesting properties about unique rectification in increasing-tableaux that might be useful in proving things about the d-Complete posets. We also managed to glean some ideas about rectification to minimal fillings in the near-bat structure using the results of our program.
We had a field trip on Friday to IBM headquarters in New York, which was incredibly interesting. We saw the famous jeopardy-playing Watson as well as IBM's experimental set-up that maintains a logical group of qubits for quantum computing. IBM actually allows users to use some of their qubits through the Quantum Experience. I've been playing around and experimenting with quantum circuits and programs; it's really fun.
- Week 6:
- It turns out we drew some incorrect conclusions from the results of the program. Now we are back to trying to prove things about each of the irreducibles, but now with a different strategy. We draw on the idea of Knuth-equivalence of the reading words associated with each tableau.
If we can figure out a way to define a generalized reading word for d-Complete posets, we can use the same methods Buch and Samuel use to prove unique rectification to standard fillings. In general, it would make things a lot easier if we just had to deal with reading words instead of the specific structure of each poset.
- Week 7:
- In our struggle to define slide invariant reading words for the d-Complete posets, we have been focusing our attention on the irreducible component called the “inset”, because it is similar to the rectangle poset (which is minuscule and therefore has the Jeu de Taquin property). We have enumerated a set of moves on reading words which we think characterize the equivalence of words under slides. My next task is to write a computer program that takes two reading words and checks if they are in the same equivalence class, i.e. it checks whether it is possible to transform one of the words into the other using only these moves.
We had final presentations this week! So we presented our findings to our fellow DIMACS researchers and learned about what they have been doing all summer.
- Week 8:
- This week was spent developing the program to test equivalence of reading words. There were several objectives that I wanted to achieve with the project. Most importantly, it must correctly determine if two reading words are equivalent under a given set of moves. But I also wanted to make the program general enough so that new moves could be added and removed easily (because we will almost certainly need this functionality in the future).
Another challenge is the case where the two words are not equivalent. In this case, we want the program to terminate instead of running indefinitely.
Next week is the final week of the program, and it looks like it will be a busy one. We have a lot more things we want to try in terms of our research, but we also have to write up our final report for the DIMACS program. Luckily, we've been writing up our results along the way so hopefully it won't be too bad!
- Week 9:
- I was right about this final week being a busy one. Rahul and I spent long hours writing up our final report to submit to DIMACS. This mainly involved polishing up our proofs and notation and writing an abstract and an introduction to our paper.
Writing out the introduction was tough because I went into some detail about the geometry that motivates our work, which I didn't understannd very well and rarely impacted our solutions.
However, it was definitely helpful to gain an understanding of where our research fits into the big picture of the mathematical history and literature about this topic, and I felt like we were able to contribute something (albeit a small piece) to this decades-long effort.