Organizing Committee
Abstract

The Summer@ICERM faculty advisers will present a variety of research projects on the combinatorics of parking functions. This overarching theme will allow participants to study and analyze parking functions by leveraging computational techniques and theory. Faculty will also guide the development of open-source computational tools for analyzing parking functions and their statistics, with time devoted to creating a database of parking functions and their generalizations.

Throughout the eight-week program, 18-22 students will work on their projects in groups of two to four, supervised by faculty advisors and aided by teaching assistants. Students will meet daily, give regular talks about their findings, attend mini-courses, guest talks, and professional development seminars, and will acquire skills in free software development. Students will learn how to collaborate mathematically, working closely in their teams to write up their research into a paper.

Image for "Summer@ICERM 2022: Computational Combinatorics"

Program Details

2022 Proposed Research Topics
  • Invariant and Prime Parking Sequences
  • Statistics on parking functions and their generalizations
  • Connection to posets and graph theory, algebraic structures, and discrete geometry
Detailed Project Descriptions (PDF)
Important Notes
  • Applicants must be matriculating undergraduate students at the time of the start of Summer@ICERM 2022.
  • Funding is available for 2-3 students who are not US citizens or permanent residents.
Funding Includes*
  • $3,800 stipend
  • Providence, RI room, board, and travel funding provided for in-person programming, pandemic permitting. (Brown students who choose not to live in the dorm will waive ICERM provided housing and meal funding.)
To Apply

Applications accepted via MathPrograms.org. Applications submitted by February 14, 2022 will receive full consideration.

Confirmed Speakers & Participants

Talks will be presented virtually or in-person as indicated in the schedule below.

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee
  • Cara Bennett
    Georgia Institute of Technology
  • Steven Bradt
    Arizona State University
  • Josephine Brooks
    University of Toronto
  • Lucas Chaves Meyles
    University of California, Los Angeles
  • Douglas Chen
    Johns Hopkins University
  • Michael Dougherty
    Lafayette College
  • Jennifer Elder
    Rockhurst University
  • Susanna Fishel
    Arizona State University
  • Pamela E. Harris
    Williams College
  • Max Hlavacek
    UC berkeley
  • Richter Jordaan
    MIT
  • Zoe Markman
    Swarthmore College
  • Lucy Martinez
    Rutgers University
  • Juan Carlos Martinez Mori
    Cornell University
  • Ava Mock
    Wellesley College
  • Alejandro Morales
    University of Massachusetts, Amherst
  • Eric Pabón-Cancel
    University of Puerto Rico, Mayagüez Campus
  • Eva Reutercrona
    Pacific Lutheran University
  • Gordon Rojas Kirby
    Arizona State University
  • Sophie Rubenfeld
    Hamilton College
  • Gabriel Sargent
    University of Notre Dame
  • Sam Sehayek
    University of California, Santa Barbara
  • Ethan Spingarn
    Amherst College
  • Izah Tahir
    Georgia Institute of Technology
  • Bianca Teves
    Haverford College
  • Steve Trettel
    Stanford University
  • Robin Truax
    Stanford University
  • Amanda Verga
    Trinity College
  • Andres Vindas Melendez
    University of California, Berkeley
  • Susan Wang
    Mount Holyoke College
  • Juliet Whidden
    Vassar College

Summer Schedule

Monday, June 13, 2022
  • 8:00 - 8:45 am EDT
    Breakfast -
    Breakfast - 10th Floor Collaborative Space
    Abstract
    For Undergraduate Researchers - come to the 11th floor of 121 South Main Street to check in prior to breakfast on the 10th floor.
  • 8:45 - 9:15 am EDT
    Check In
    11th Floor Collaborative Space
  • 9:15 - 9:30 am EDT
    Welcome
    10th Floor Collaborative Space (Office 1015 Side)
  • 10:00 am - 12:00 pm EDT
    Scavenger Hunt
    Scavenger Hunt - Brown University, Providence, RI 02912, USA
  • 12:00 - 1:30 pm EDT
    Welcome Lunch
    Working Lunch - 10th Floor Collaborative Space (Office 1015 Side)
  • 2:00 - 3:00 pm EDT
    Goals of Summer+Community agreement
    10th Floor Collaborative Space
    • Susanna Fishel, Arizona State University
    • Gordon Rojas Kirby, Arizona State University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:00 - 4:30 pm EDT
    Python and Sage intro
    Tutorial - 10th Floor Collaborative Space
    • Juan Carlos Martinez Mori, Cornell University
  • 4:30 - 6:00 pm EDT
    Reception
    11th Floor Collaborative Space
Tuesday, June 14, 2022
  • 10:00 - 11:00 am EDT
    Lecture- posets,perms
    10th Floor Collaborative Space
    • Jennifer Elder, Rockhurst University
  • 11:00 am - 12:00 pm EDT
    Problem Session
    10th Floor Collaborative Space
  • 12:00 - 2:00 pm EDT
    Lunch/Free Time
  • 2:00 - 3:00 pm EDT
    Lecture- posets,perms
    10th Floor Collaborative Space
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:00 - 4:30 pm EDT
    Problem Session
    10th Floor Collaborative Space
Wednesday, June 15, 2022
  • 10:00 - 11:00 am EDT
    Lecture-Catalan, parking function
    10th Floor Collaborative Space
    • Steven Bradt, Arizona State University
  • 11:00 am - 12:00 pm EDT
    Problem Session
    10th Floor Collaborative Space
  • 12:00 - 2:00 pm EDT
    Lunch/Free Time
  • 2:00 - 3:00 pm EDT
    Lecture-Catalan, parking function
    10th Floor Collaborative Space
    • Steven Bradt, Arizona State University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:00 - 4:30 pm EDT
    Problem Session
    10th Floor Collaborative Space
Thursday, June 16, 2022
  • 10:00 - 11:00 am EDT
    Intro to graph theory
    10th Floor Collaborative Space
    • Max Hlavacek, UC berkeley
  • 11:00 am - 12:00 pm EDT
    Problem Session
    10th Floor Collaborative Space
  • 12:00 - 2:00 pm EDT
    Lunch/Free Time
  • 2:00 - 3:00 pm EDT
    Intro to chip firing
    10th Floor Collaborative Space
    • Lucy Martinez, Rutgers University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:00 - 4:30 pm EDT
    Problem Session
    10th Floor Collaborative Space
Friday, June 17, 2022
  • 10:00 - 11:00 am EDT
    Hyperplanes
    10th Floor Collaborative Space
    • Sam Sehayek, University of California, Santa Barbara
  • 11:00 am - 12:00 pm EDT
    Problem Session
    10th Floor Collaborative Space
  • 12:00 - 2:00 pm EDT
    Lunch/Free Time
  • 2:00 - 3:00 pm EDT
    Hyperplanes
    10th Floor Collaborative Space
    • Sam Sehayek, University of California, Santa Barbara
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:00 - 4:30 pm EDT
    Problem Session
    10th Floor Collaborative Space
Monday, June 20, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, June 21, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, June 22, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, June 23, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 6:00 - 8:00 pm EDT
    Game Night
    - 11th Floor Collaborative Space
Friday, June 24, 2022
  • 2:00 - 3:00 pm EDT
    Colloquium - Geometric combinatorics: an intro to Ehrhart theory
    11th Floor Lecture Hall
    • Andres Vindas Melendez, University of California, Berkeley
    Abstract
    Geometric combinatorics is an area of mathematics concerned with counting properties of geometric objects described by a finite set of building blocks. Polytopes are geometric objects that can be formed by taking the convex hull of finitely many points. When extracting combinatorial information from geometric objects, a natural approach is to subdivide the combinatorial object into smaller, more accessible pieces. In the case of polytopes a natural approach is to triangulate the polytope into simplices or parallelepipeds. The Ehrhart polynomial of a lattice polytope P encodes the number of integer lattice points in dilates of P, and the h*-polynomial of P is the numerator of the accompanying generating function. In this talk I will present background on polytopes, lattice-point enumeration, and share my favorite family of polytopes. I will conclude the talk with some open problems.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, June 27, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, June 28, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, June 29, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, June 30, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 6:00 - 8:00 pm EDT
    Game Night
    - 10th Floor Collaborative Space
Friday, July 1, 2022
  • 2:00 - 2:45 pm EDT
    Pattern avoidance in Parking Functions
    Virtual
    • Virtual Speaker
    • Ayo Adeniran, Colby College
    Abstract
    Parking functions have been well-studied in combinatorics. We extend the classical definition of patterns in permutations to parking functions. In particular, we study parking functions that avoid permutations of length 3. A number of well-known combinatorial sequences arise in our analysis, and this talk will highlight several enumeration results and conjectures. This project is joint work with Lara Pudwell(Valparaiso U.)
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, July 4, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, July 5, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, July 6, 2022
  • 4:00 - 6:00 pm EDT
    Ice Cream Social
    Coffee Break - 11th Floor Collaborative Space
Thursday, July 7, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 6:00 - 8:00 pm EDT
    Game Night
    - 11th Floor Collaborative Space
Friday, July 8, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:00 - 3:45 pm EDT
    Colloquium - Geometry and Poincare Conjecture
    11th Floor Lecture Hall
    • Steve Trettel, Stanford University
    Abstract
    Manifolds are ubiquitous in modern mathematics, from the familiar low dimensional examples of curves and surfaces in calculus to higher dimensional abstract examples in geometry, physics, data science, and beyond. They come in a bewildering variety - with the basic question of "what kinds of manifolds are possible" providing a powerful guiding light in topology. While the 19th century witnessed a near-complete understanding of the 1 and 2 dimensional cases, much progress during the 20th century in 3 dimensions was guided by a conjecture of Poincare, first formulated in 1904. Poincare's conjecture - essentially that simple 3 dimensional spaces can be probed effectively using 1-dimensional loops - proved much more difficult than originally hoped, remaining unsolved for nearly 100 years. Following a century of work, its eventual resolution by Perelman in 2002 provided a new and powerful tool - called Geometrization - to the study of all 3-dimensional spaces. And while the arguments involved get quite technical, the big-picture story is a beautiful interplay of shape, symmetry and geometry which deserves to be more widely known. My goal in this talk is to give an overview of this exciting story lying at the heart of modern topology, from what was asked to what the mathematical community has learned.
Monday, July 11, 2022
  • 11:00 - 11:45 am EDT
    Noncrossing Partitions and Complex Polynomials
    10th Floor Collaborative Space
    • Michael Dougherty, Lafayette College
    Abstract
    The lattice of noncrossing partitions began as a combinatorial curiosity in the early 1970s, but is now connected to several modern areas of mathematics, including parking functions and the combinatorics of reflection groups. In this talk I will describe some work of mine with Jon McCammond in which we provide a new connection between noncrossing partitions and the topology of complex polynomials.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, July 12, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, July 13, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, July 14, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 6:00 - 8:00 pm EDT
    Game Night
    - 11th Floor Collaborative Space
Friday, July 15, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, July 18, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, July 19, 2022
  • 2:30 - 3:30 pm EDT
    Visit to American Mathematical Society
    External Event
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, July 20, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 5:30 - 7:00 pm EDT
    Math Craft Night
    Working Lunch - 11th Floor Collaborative Space
Thursday, July 21, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 6:00 - 8:00 pm EDT
    Game Night
    - 11th Floor Collaborative Space
Friday, July 22, 2022
  • 2:00 - 3:00 pm EDT
    Colloquium
    11th Floor Lecture Hall
    • Virtual Speaker
    • Alejandro Morales, University of Massachusetts, Amherst
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, July 25, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, July 26, 2022
  • 9:00 - 11:00 am EDT
    Ethics I & II
    Professional Development - 11th Floor Lecture Hall
  • 11:45 am - 12:00 pm EDT
    Group Photo
    At stairs between 10th & 11th Floor
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, July 27, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, July 28, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 6:00 - 8:00 pm EDT
    Game Night
    - 11th Floor Collaborative Space
Friday, July 29, 2022
  • 2:15 - 3:15 pm EDT
    Colloquia
    11th Floor Lecture Hall
    • Virtual Speaker
    • Caroline Klivans, Brown University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, August 1, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, August 2, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, August 3, 2022
  • 9:00 - 9:45 am EDT
    On Flattened Parking Functions
    11th Floor Lecture Hall
    • Zoe Markman, Swarthmore College
    • Izah Tahir, Georgia Institute of Technology
    • Amanda Verga, Trinity College
    Abstract
    We recall that a permutation of length n is said to be a flattened partition if the leading terms of maximal chains of ascents (called runs) are in increasing order. In this talk, we define an analogous set of objects which we call flattened parking functions: a subset of parking functions for which the leading terms of maximal chains of weak ascents (also called runs) are in weakly increasing order. We say that a parking function is flattened if its runs satisfy this condition. We show that the number of flattened parking functions with one run are enumerated by the Catalan numbers, as they are in bijection with nondecreasing parking functions. For n ≤8, where there are at most four runs, we give data for the number of flattened parking functions and it remains an open problem to give formulas for their enumeration in general. We then specialize to a subset of flattened parking functions that we call flattened S-insertion parking functions. These can be obtained by inserting all numbers of a multiset Swhose elements are in [n] := {1, 2, . . . , n}, into a permutation of [n] and checking that the result is flattened. We provide bijections between flattened S-insertion parking functions and flattened S′-insertion parking functions, where Sand S′ have certain relations. We then specialize to the case S:= r, the multiset with r ones, and we establish a bijection between flattened r-insertion parking functions and set partitions of [n +r] with the first r integers in different subsets. As a corollary, we establish that flattened r-insertion parking functions are enumerated by the r-Bell numbers. We also construct exponential generating functions of these enumerations and give the closed differential form. These results generalize the work of Nabawanda, Rakotondrajao, Bamunoba and Beyene, Mantaci on flattened partitions.
  • 10:00 - 10:45 am EDT
    Out of the Parking Lot and into the Forest: Parking Functions, Bond Lattices, and Unimodal Forests
    11th Floor Lecture Hall
    • Josephine Brooks, University of Toronto
    • Sophie Rubenfeld, Hamilton College
    • Bianca Teves, Haverford College
    Abstract
    Parking functions are a well-studied combinatorial object with numerous connections to other areas of math. They can be seen in the maximal chains of the noncrossing partition lattice N Cn, an important partially ordered set (poset) that provides an ordering relation to all noncrossing set partitions on the integers 1 through n. Just as there are (n + 1)(n−1) parking functions of length n, there are (n + 1)(n−1) maximal chains in N Cn+1. Stanley defined a bijection between the two sets in 1996. In 1964, Rota introduced the bond lattice, a subposet derived from a given graph. When the graph is of a certain form, the bond lattice is a subposet of N Cn+1, encoding a subset of the parking functions of length n. Our work builds on the work of a previous REU group, who found that the bond lattices of certain triangulation graphs have the same number of maximal chains as there are ordered cycle decompositions on n integers. Anders and Archer further showed that each ordered cycle decomposition corresponds to a rooted unimodal forest. Although there exist many bijections between parking functions and rooted forests, our work develops a recursive bijection between parking functions of a triangulation graph and unimodal unordered rooted labelled forests.
  • 11:00 - 11:45 am EDT
    Parking Functions with Fixed Ascent and Descent Sets
    11th Floor Lecture Hall
    • Eva Reutercrona, Pacific Lutheran University
    • Susan Wang, Mount Holyoke College
    • Juliet Whidden, Vassar College
    Abstract
    Given a parking function α= (a1, . . . , an) define the ascent set of αto be the set of i such that ai < ai+1 and the descent set as the set of i such that ai > ai+1. Our work focuses on enumerating parking functions with certain ascent or descent sets in order to refine previous counts of ascents and descents in parking functions. In fact, these descent and ascent sets exhibit a remarkable symmetry, which we prove. Additionally, we investigate a specialization of ordinary parking functions called unit interval parking functions (UPFs), in which each car parks at most one spot from its preferred parking spot. The enumeration of UPFs with fixed ascent or descent sets is particularly nice. For example, we give a new, independent proof that UPFs are enumerated by the Fubini numbers via a bijection with Fubini rankings. Then, we extend this connection between Fubini rankings and unit interval parking functions to r-Fubini rankings through an investigation of UPFs based on the indices of their ascents, descents, and ties.
  • 12:00 - 1:00 pm EDT
    Lunch/Free Time
  • 1:00 - 1:45 pm EDT
    Repetitions of Pak-Stanley Labels in the G-Shi Arrangement: Playing Games on Paths, Trees, and More
    11th Floor Lecture Hall
    • Cara Bennett, Georgia Institute of Technology
    • Ava Mock, Wellesley College
    • Robin Truax, Stanford University
    Abstract
    Given a simple graph G, one can define a hyperplane arrangement called the G-Shi arrangement. The Pak-Stanley algorithm labels the regions of this arrangement with G•-parking functions in a way that some G•-parking functions may appear more than once. These repetitions of Pak-Stanley labels are a topic of interest in the study of G-Shi arrangements and G•-parking functions, as well as the many combinatorial objects they are connected to. The key insight of our work is the introduction of a combinatorial model called the “Three Rows Game”. Analyzing the histories of this game and the ways in which they can induce the same outcomes allows us to completely characterize the repetitions of the Pak-Stanley labels for path, cycle, and star graphs, and make substantial progress towards understanding the repetitions of the Pak-Stanley labels for trees and general graphs.
  • 2:00 - 2:45 pm EDT
    Parking Functions with Fixed Displacement
    11th Floor Lecture Hall
    • Lucas Chaves Meyles, University of California, Los Angeles
    • Richter Jordaan, MIT
    • Ethan Spingarn, Amherst College
    Abstract
    Displacement in the context of classical parking functions measures the total number of spots passed over when all the cars have parked. We obtain enumerative results for the number of classical and prime parking functions where each car has at most one displacement. Bijections are obtained between these parking functions and types of surjective functions, as well as with the k-faces of the permutohedron. Beyond this flavor of parking functions, we enumerate parking functions where up to three cars are displaced and suggest a direction to generalize these formulae for any number of cars. Moreover, an efficient algorithm is obtained that computes the number of parking functions exhibiting specific displacements
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EDT
    On Permutation Invariant Parking Sequences
    11th Floor Lecture Hall
    • Douglas Chen, Johns Hopkins University
    • Eric Pabón-Cancel, University of Puerto Rico, Mayagüez Campus
    • Gabriel Sargent, University of Notre Dame
    Abstract
    Parking sequences are a generalization of parking functions in which cars can have different lengths and there are as many parking spots as the sum of the car lengths. A list of parking spot preferences is said to be a parking sequence for the given car lengths if all cars are able to park. A parking sequence is said to be permutation invariant if all of its rearrangements are parking sequences. While all parking functions (i.e., parking sequences given cars of unit length) are invariant, this is not the case for parking sequences. The overarching goal of this work is to provide necessary and sufficient conditions for a parking sequence to be invariant. While obtaining a full characterization remains elusive, we do so for a number of foundational cases. Our main result is a concise characterization of minimally invariant car lengths, wherein the only invariant parking sequence is the all ones preference list. We moreover provide a full characterization of invariant parking sequences given two and three cars. Lastly, we give alternate proofs of known results for constant and strictly increasing car lengths and derive certain closure properties of invariant parking sequences. We conclude with a conjecture regarding a Boolean formula for minimally invariant parking sequences with four cars, as well as open questions concerning invariance under subgroups of the permutation group and the computational complexity of characterizing invariant parking sequences in their full generality.
Thursday, August 4, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 6:00 - 8:00 pm EDT
    Game Night
    - 11th Floor Collaborative Space
Friday, August 5, 2022
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space

All event times are listed in ICERM local time in Providence, RI (Eastern Standard Time / UTC-5).

All event times are listed in .

Other Summer@ICERM Events

Summer@ICERM 2012: Geometry and Dynamics
Image for "Summer@ICERM 2012: Geometry and Dynamics"
Summer@ICERM 2013: Geometry and Dynamics
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Summer@ICERM 2014: Polygons and Polynomials
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Summer@ICERM 2015: Computational Dynamics and Topology
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Summer@ICERM 2016: Dynamics and Stochastics
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Summer@ICERM 2017: Topological Data Analysis
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Summer@ICERM 2018: Low Dimensional Topology and Geometry
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Summer@ICERM 2019: Computational Arithmetic Dynamics
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Summer@ICERM 2021: Computational Polygonal Billiards
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