Research: Number Theory
Summer School 2019: Computational Aspects of Buildings
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From June 24 to 28, 2019, the University of North Carolina Greensboro will host the UNCG Summer School in Computational Number Theory and Algebra: Computational Aspects of Buildings.
Buildings provide a rich class of simplicial complexes with very nice local structure. They play key roles in several areas of mathematics, including number theory, geometry, algebra, and combinatorics. In this workshop, we will begin by studying the basic combinatorial and geometric structure of buildings. We will then explore applications and open questions related to their large scale geometry, structure of their automorphism groups, existence of buildings with cocompact lattices and prescribed vertex link, and their role in algebraic combinatorics and the study of symmetric functions.
On a typical day, external and local experts give talks in the morning, and in the afternoon students solve problems related to this material. The talks early in the week introduce the students to the subject. Talks later in the week cover related areas of current research and unsolved problems. The problems given to the students might be of a theoretical nature but could also involve programming problems and computer experiments. All problems are aimed at increasing the students’ understanding of the material by working with it.
Speakers
 Thibaut Dumont (University of Jyväskylä, Finland)
 Yusra Naqvi (University of Sydney, Australia)
 Xiangdong Xie (Bowling Green State University)
Schedule
Talks are held in Petty Building 213.
Time  Monday 6/24  Tuesday 6/25  Wednesday 6/26  Thursday 6/27  Friday 6/28 
9:00  Welcome  Coffee and Lightning Talks  
9:30  Yusra Naqvi: Coxeter Groups, Chamber Systems, and Buildings I  Thibaut Dumont: Low dimensional Euclidean buildings [slides (pdf)]  Thibaut Dumont: Regularity and Parameter Systems  Yusra Naqvi: Hecke Algebra II  Yusra Naqvi: Folded Galleries [papera2 (pdf), paperb2 (pdf), paperg2 (pdf)] 
10:45  Coffee  
11:15  Yusra Naqvi: Coxeter Groups, Chamber Systems, and Buildings II  Xiangdong Xie: Retraction, curvature aspects of buildings [slides (pdf), paper (pdf)]  Yusra Naqvi: Hecke Algebras I  Xiangdong Xie: Some basics on Coxeter groups and cross ratio on the boundary [slides (pdf), paper (pdf)]/Thibaut Dumont: Low dimensional Euclidean buildings II [slides (pdf)]  Xiangdong Xie: Quasiisometric rigidity of Fuchsian buildings [slides (pdf)]/Thibaut Dumont: BruhatTits Buildings of \(\mathrm{SL}_{n + 1}(K)\) 
12:30  Lunch  
14:00  Xiangdong Xie: Geometric realizations of Coxeter groups and buildings [slides (pdf), paper (pdf)]  Excursion: Greensboro Science Center and SKYWILD  Thibaut Dumont: Low dimensional Euclidean buildings III [slides (pdf)]  
15:30  Paper and pen lab: [Yusra exercises (pdf), Xiandong exercises (pdf)]  Yusra Naqvi: BNpairs 
Thibaut Dumont: Computer Lab [Further projects (pdf)]


Preliminaries  
Shared on open problems (or to conclude a minicourse)


Lab  
Core of the minicourse  Lightning talks (9:059:20am): Jenny Beck, Kalani Thalagoda, Kristen Schekelhoff [slides (pdf)] 
Lightning talks (9:059:20am): Amy Aswani, Fikreab Solomon [slides (pdf)]

Lightning talks (9:059:20am): Shivam Arora, Daniel Belin, Brian Grove [slides (pdf)]

Lightning talks (9:059:20am): Christopher Ratigan, Richard Mandel, Peter Huxford [slides (pdf)]

Participants
 Shivam Arora (Memorial University of Newfoundland)
 Amy Aswani (University at Buffalo)
 Jenny Beck (UNCG)
 Daniel Belin (University of Wisconsin – Madison)
 Sahana Balasubramanya (UNCG)
 Thibaut Dumont (University of Jyväskylä)
 Talia Fernos (UNCG)
 Brian Grove (The University of Toledo)
 Peter Huxford (The University of Auckland)
 Richard Mandel (Stevens Institute of Technology)
 Yusra Naqvi (University of Sydney)
 Sebastian Pauli (UNCG)
 Christopher Ratigan (Tufts University)
 Sandi Rudzinski (UNCG)
 Filip Saidak (UNCG)
 Bakul Sathaye (Wake Forest)
 Cliff Smyth (UNCG)
 Fikreab Solomon (The Graduate Center, CUNY)
 Brett Tangedal (UNCG)
 Kalani Thalagoda (UNCG)
 Xiangdong Xie (Bowling Green State University)
 Dan Yasaki (UNCG)
Additional Materials
See schedule above for course materials.
 Brent Everitt: A (very short) introduction to buildings
 James Parkinson: Buildings and Hecke Algebras
 Nicolas Radu: A lattice in a residually nonDesarguesian Ã 2building
 Anne Thomas: Geometric and topological aspects of Coxeter groups and buildings
Organizers
Talia Fernos, Sebastian Pauli, Filip Saidak, Brett Tangedal, Dan Yasaki
Acknowledgements
The summer school in computational number theory is supported by UNCG and the NSF (DMS1602025, DMS1802448).