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Summer Graduate School

Topological Methods for the Discrete Mathematician July 25, 2022 - August 05, 2022
Parent Program: --
Location: St. Mary's College, Moraga, California
Organizers Pavle Blagojevic (Freie Universität Berlin), Florian Frick (Carnegie Mellon University), Shira Zerbib (Iowa State University)
Lecturer(s)

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Description
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Recently, progress in the field of topological methods in discrete mathematics has been rapid and has generated a lot of activity with the resolution of major open problems, the emergence of new lines of inquiry, and the development of new tools. These exciting new developments have not been digested into a textbook treatment. The two main goals of this school are to:

  1. Provide graduate students with a thorough introduction to novel topological techniques and to a handful of their applications in the fields of combinatorics and discrete geometry with short glimpses into mathematical mechanics and algorithm complexity.
  2. Expose students to current research, and guide them in research on open problems in discrete mathematics using modern topological tools.

The summer school will lead participants from appealing, simple-to-state problems at confluence of combinatorics, geometry, and topology to sophisticated topological methods that are required for their resolution. In recent years topological methods have found numerous novel applications in mathematics and beyond, such as in data science, machine learning, economics, the social sciences, and biology.  The problems we will discuss are particularly well-suited to rapidly put students in a position to approach related research questions.

School Structure
There will be up to three lectures per day. Lectures will be split into parallel sessions whenever appropriate and depending on student feedback. For example, we will use small parallel sessions to give brief introductions to a variety of further applications of topological methods that students can attend based on their particular interests:

  • Shira Zerbib will lead sessions with a combinatorial focus that delve deeper into applications in graph theory, packing and covering problems, and problems of fair division;
  • Florian Frick will discuss topological methods in analysis and geometry, specifically in functional and harmonic analysis, in data analysis, and for the theory of convex neural codes;
  • Pavle Blagojević’s sessions will develop additional topological tools and discuss the topology of configuration spaces, hyperplane mass partitions, and billiard trajectories.

The afternoons will usually be devoted to discussing (solved and unsolved) problems in small groups, under the guidance of lecturers and TAs. The three lecturers will offer close mentorship while students explore topological methods in discrete mathematics in a problem-based fashion. Students will work in small groups on specific problems; each group will be mentored by one of the lecturers. Our aim is that the summer school will lead to collaboration among the participants past the two-week summer period.

Suggested Prerequisites
First degree or equivalent in mathematics. Introductory courses in analysis, algebra, and combinatorics. Familiarity with point-set topology recommended. A certain level of mathematical maturity will be assumed.
Possible introductory and background reading materials for participants who would like to acquire basic knowledge before the summer school include popular readings:

  • Ziegler, G. (2011). 3N Colored Points in a Plane. Notices of AMS, 58(4), 550-557.
  • Ziegler, G. (2015). Cannons at Sparrows, Newsletter of EMS, 95, 25-32.
  • Blagojevic, P., Barany, I., and Ziegler, G. (2016). Tverberg’s Theorem at 50: Extensions and Counterexamples. Notices of AMS, 63(7), 732-739.
  • Soberon, P. (2017). Gerrymandering, Sandwiches, and Topology. Notices of AMS, 64(9), 1010-1013.

Survey Materials:

  • Matousek, J. (2003). Using the Borsuk–Ulam theorem: Lectures on topological methods in combinatorics and geometry. Springer Science & Business Media.
  • De Loera, J., Goaoc, X., Meunier, F., and Mustafa, N. (2019). The discrete yet ubiquitous theorems of Caratheodory, Helly, Sperner, Tucker, and Tverberg. Bulletin of the American Mathematical Society, 56(3), 415-511.
  • Barany, I. and Soberon, P. (2018). Tverberg’s theorem is 50 years old: a survey. Bulletin of the American Mathematical Society, 55(4), 459-492.
  • Blagojevic, P. and Ziegler, G. (2017). Beyond the Borsuk-Ulam theorem: The topological Tverberg story. in “Journey Through Discrete Mathematics. A Tribute to Jirı Matousek”, 273-341. Springer International publishing.

and classical textbooks and monographs:

  • Bredon, G. (1993). Topology and Geometry. Springer Science & Business Media.
  • Tom Dieck, T. (1987). Transformation Groups. De Gruyter.
  • Adem, A. and Milgram, J. (2004). Cohomology of Finite Groups. Springer Science & Business Media.

For eligibility and how to apply, see the Summer Graduate Schools homepage.

Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • combinatorics

  • discrete math

  • discrete geometry

  • combinatorial topology

  • topological methods

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
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