General Information

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Research project log

Week 0 (05/31-06/03)

• Arrived at JFK (NYC) after a long flight and we were greeted with pizza on the first night! Thanks Larry and Lazaros.
• Prepared the initial student presentation introducing the problem I will be working on.
• An HTML tutorial was organized for designing user homepages, like this one.
• Visited Washington DC for the weekend, had quite fun!

Week 1 (06/06-06/10)

• Presented the problem and plan of research to REU participants.
• James provided more historical background for the problem and sent some papers for a more in-depth literature review.
• Mikhail Khovanov presented the talk "Regular languages and cobordisms of decorated manifolds".
• Lazaros conducted the research ethics workshop.

Week 2 (06/13-06/17)

• We explored a few more related problems surrounding the topic at hand, like reconstruction of partially balanced tableau and diameter of the graphs of persistent graphs.
• Met Bhargav Narayanan to discuss some problem in extremal graph theory (Ramsey-Turán type).
• Robert Bosch delivered the lecture "Connecting the Dots: Using Combinatorial Optimization to Design Visual Artwork".
• Biked along the D&R Canal to Princeton.

Week 3 (06/20-06/24)

• James suggested me to work on a simpler class of persistent graphs called single blocking vertex graph for which we have a way of realizing them as polygons.
• Furthered my discussion with Bhargav on the problem and I will be presenting the notes I am reading in a couple of weeks.
• A fun social event Culture/Student day was organized this week that I enjoyed.
• Went to NYC (finally!).

Week 4 (06/27-07/01)

• For a while James and I thought to have obtained a class of persistent graphs not realizable as polygons but then we observed the mistake we made. Anyway, so we continued formulating and analyzing the different cases in single-vertex blocking graphs. The next aim is to generalize it.
• The seminar "A Systematization of Content Moderation in End-to-End Encryption" by Sarah Scheffler was organized.
• Another workshop by Lazaros, this one on scientific writing. It was good to hear on this from an actual scientist.
• Had been looking forward to the 4th of July celebrations, which was quite something.

Week 5 (07/05-07/08)

• This week we worked out the case of two concave regions having a convex vertex connecting them. Seems that piecing this altogether we might have a way to prove the general case of persistent graphs.
• Attended a seminar on the graph cities project ("Visual Exploration of Billion Edge Graphs") by James Abello.
• Nicole Shanahan presented "Perspectives on IP from Machine Learning and AI".

Week 6 (06/11-07/15)

• Went on an amazing trip to the Nokia Bell Labs and saw some cool stuff like the anechoic chamber.
• Worked more on the problem in the case when we have more than two concave regions with some convex vertices connecting them.
• Attended the guided tour of Zimmerli Art Museum.

Week 7 (06/18-07/22)

• Had the final presentation and didn't manage to solve the problem but I am happy with the experience.
• Very excited to go back to Prague.
• Also met with students from other REU programs working in Combinatorics sometime during the program.

References

• [1]  J. Abello, Ö. Egecioglu, K. Kumar, Visibility Graphs of Staircase Polygons and the Weak Bruhat Order, I: from Visibility Graphs to Maximal Chains*. Discrete & Computational Geometry. 1995. 331-358.
• [2]  V. Froese, M. Renken. Persistent graphs and cyclic polytope triangulations. Combinatorica 41 (3) (2021), 407-423.
• [3]  P. Edelman, C. Greene. Balanced Tableaux. Advances in Mathematics Vol 63 (1) (1987), 42-99.

Acknowledgements

This work was carried out while the author was a participant in the 2022 DIMACS REU program, supported by CoSP, a project funded by European Union’s Horizon 2020 research and innovation programme, grant agreement No. 823748. The author is grateful to the organizers at Rutgers and Charles for providing this opportunity.