Titles and abstracts

• Mihai Ciucu (Indiana University): A variant of the factorization theorem for perfect matchings and applications (slides, video)
  The factorization theorem for perfect matchings of symmetric graphs opens up the possibility of simple proofs for results stating that the number of perfect matchings of a given family of symmetric planar graphs is expressed by an explicit product formula. In this talk I will present a variant of the factorization theorem that allows dealing also with some regions that are not symmetric. Applications include the enumeration of tilings of several new families of regions. This is joint work with Seok Hyun Byun and Yi-Lin Lee.
  
  
• David desJardins (Institute for Defense Analysis): Counting Tilings by Enlightened Brute Force (slides, video)
  Much of Propp’s recent work on enumeration of tilings involves tiling problems that aren’t amenable to algebraic shortcuts like the permanent-determinant trick, yet he presents empirical results in which the number of tilings is enormous, with twenty digits or more. He obtains these numbers using a program I wrote for him that combines the power of modern processors with some familiar tricks (divide-and-conquer) and some less-familiar ones. In this talk I’ll explain how the program works and give some examples.
  
  
• Emily Gunawan (University of Massachusetts, Lowell): Jim's contribution to cluster algebras and friezes (slides, videos)
  I will talk about Jim's pioneering 2005 article "The combinatorics of frieze patterns and Markoff numbers" on snake graphs, Conway—Coxeter frieze patterns, and other combinatorial interpretation of type A cluster algebras. It has since inspired hundreds of research papers, and I will discuss some of them, including my own work on binomial coefficients of words, friezes, and superunitary regions of cluster algebras.
  
  
• Sam Hopkins (Howard University): Order polynomial product formulas and poset dynamics (slides, video)
  Dynamical algebraic combinatorics (DAC) is an emerging subfield of combinatorics that has been spearheaded by Jim Propp over the last fifteen years. DAC studies the actions of natural dynamical operators on objects from algebraic combinatorics. While these operators are defined on general families of objects, it is only for special families of objects that they exhibit really good behavior. We put forward a powerful heuristic which allows us to uncover these special families. Our heuristic roughly says that the families which exhibit good dynamical behavior are the same as those which exhibit good enumerative behavior.
  
  
• Rick Kenyon (Yale University): The multinomial dimer model (slides, video)
  We study a variant of the dimer model/domino tiling model, called the multinomial dimer model, which is tractable for general graphs. We find formulas for the partition function, limit shapes and fluctuations in some natural settings, including a three-dimensional version of the Aztec Diamond. This is joint work with Catherine Wolfram (MIT).
  
  
• Greg Kuperberg (University of California, Davis): What is a bijective proof? (slides, video)
  In combinatorics, we celebrate when we can prove an enumeration; we celebrate more when we can find a bijective proof. But what is a bijective proof? Intuitively, bijective proofs make perfect sense and are clearly valuable; yet rigorously, the concept has always been controversial. In this talk, I will sketch a formal version of bijective proofs borrowed from circuit complexity in computer science. It is probably not the last word on the question, but it is rigorous and it is a start. I will explain how several enumerative methods are bijective by this standard, and I will discuss evidence (also borrowed from computer science) that there are enumerations that don't have a bijective proof.
  
  
• Tri Lai (University of Nebraska, Lincoln): Jim's list of tiling problems (slides, video)
  Jim's list of 32 open problems has significantly advanced the field of tiling enumeration. We will discuss my favorite Jim's problems, their solutions, and developments. We will also explore several new tiling problems if time allows.
  
  
• Lionel Levine (Cornell University): Math for AI Safety (slides, video)
  AI holds great promise and, many believe, great peril. What can mathematicians contribute to ensuring that promise is fulfilled, and peril avoided? I’ll highlight some math research directions relevant to AI safety. I’ll also touch on the question of how to empower students to succeed in a world pervaded by AI, and to shape that world for the better: After all, many of today’s math majors will be tomorrow’s AI engineers. This talk will be interactive, and I’ll mostly supply questions rather than answers, so please come prepared to debate!
  
  
• Olya Mandelshtam (University of Waterloo): Combinatorics of Macdonald polynomials through the ASEP and TAZRP (slides, video)
  The ASEP and TAZRP are related one-dimensional interacting particle processes from statistical mechanics. Due to Cantini, de Gier, and Wheeler, it was known that the stationary distribution of the ASEP can be obtained through specializations of Macdonald polynomials. I will describe the combinatorial objects that make this connection explicit (joint work with Corteel and Williams). I will then explain how we can similarly obtain the modified Macdonald polynomials through the TAZRP (joint work with Ayyer and Martin), deriving new tableaux formulas for these polynomials along the way.
  
  
• Robin Pemantle (University of Pennsylvania): Chasing the Three (slides, video)
  Jim Propp, via his research on Aztec Diamond tilings, was largely responsible for my starting a new and fruitful line of research into Analytic Combinatorics in several variables (ACSV). This talk gives a brief outline of this history, and of the development of ACSV, focusing on its ability to explain the placement probabilities for random tilings via harmonic analysis. By the end, at least one intriguing open problem will be presented, which goes by the name of "Chasing the 3". Expect pictures. Expect digressions.
  
  
• Kyle Petersen (De Paul University): Napkin Problems (slides, video)
  Suppose a number of mathematicians sit down at a circular banquet table that has napkins evenly spaced between each place setting. When a particular diner sits, they might encounter two napkins (in which case they choose their preferred napkin), they might encounter one napkin because a neighbor already took one (in which case they take the other napkin), or they might encounter zero napkins because both their napkins were already taken by neighbors. If people sit down in a random order and grab napkins from the left or right side of their place at random, what is the expected proportion of napkinless diners? What is the worst order in which people might sit?
  In this talk I will tell you the answers to both these questions, as well as some related open questions. Along the way, I will tell you the human story of my engagement with these questions in two different projects, separated by almost 20 years.
  
  
• Dan Romik (University of California, Davis): Exact solutions and area bounds in the moving sofa problem (slides, video)
  The moving sofa problem is a well-known open problem in geometry that asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width. In addition to the intrinsic interest of the question, its challenging nature has led to the development of fruitful ideas in geometry, experimental mathematics, computer-assisted proofs, and geometric optimization. In this talk I will survey the known results on the problem, focusing on two areas: 1. analytically described shapes conjectured to be the correct solutions. 2. rigorous bounds for the maximal area shape, including intriguing conditional bounds obtained in new work by Jineon Baek.
  The talk is based on works by Joseph Gerver, Jineon Baek, myself, and myself jointly with Yoav Kallus.
  
  
• Marianna Russkikh (Notre Dame University): Perfect t-embeddings of Aztec diamond (slides, video)
  A new type of graph embedding called a t-embedding, was recently introduced and used to prove the convergence of dimer model height fluctuations to a Gaussian Free Field (GFF) in a naturally associated metric, under certain technical assumptions. We study the properties of t-embeddings of uniform Aztec diamond graphs, and in particular utilize the integrability of the “shuffling algorithm” on these graphs to provide a precise asymptotic analysis of t-embeddings and verify the validity of the technical assumptions required for convergence. As a consequence, we complete a new proof of GFF fluctuations for the dimer model height function on the uniformly weighted Aztec diamond.
  
  
• Scott Sheffield (MIT): 3D dimers and gauge theory (video)
  25 years ago (plus or minus a few) Propp and co-authors wrote a highly influential series of papers about dimers and related topics, including large deviations, local statistics, tree bijections and sampling. I'll discuss recent 3D extensions. This is joint work with Nishant Chandgotia and Catherine Wolfram.
  In 2D we all know about the distinct phases for interior height function slopes:
  1. Localized, smooth, gaseous, exponential-correlation-decay, surface tension cusp.
  2. Delocalized, rough, liquid, polynomial-correlation-decay, no surface tension cusp.
  These phases can also be explored via "defects" — i.e. what happens to a partition function when you remove a black square and a far away white square? There is a continuum off-Gaussian hybrid model (the Sine-Gordon model) that somehow interpolates.
  In gauge theory applications (recent joint work with Sky Cao, Minjae Park and Joshua Pfeffer) one replaces "random height functions" (with a surface tension for each possible constant gradient) with "random one-forms" (with a surface tension for each possible constant curvature). In this setting, a "defect" is a loop rather than a pair of points. One again expects two phases:
  1. Exponential decay, area law, surface tension cusp, quark confinement, mass gap.
  2. Polynomial decay, perimeter law, no surface tension cusp, no confinement or mass gap.
  But constructing the continuum hybrid remains a major open problem. Might our dimer intuition give us some clues?
  
  
• Jessica Striker (North Dakota State University): Alternating sign matrices and the many faces of dynamical algebraic combinatorics (slides, video)
  As I began to prepare for this talk, I realized nearly all my work relates in some way to Jim’s. In this talk, I’ll discuss what I’ve learned from my favorites of Jim’s early papers and how it relates to more recent dynamical algebraic adventures.
  
  
• Bridget Tenner (De Paul University): Propperties and Proppositions (slides, video)
  Jim has had an immeasurable impact on the combinatorics community and how that community functions and grows. I will talk about Jim's lasting influence on my own career path, particularly in the direction of permutation patterns -- which somehow makes for an extra-compelling illustration, given that I never studied them with Jim.
  
  
• Julianna Tymoczko (Smith College): Webs and quantum representations (slides, video)
  A web is a directed, labeled plane graph satisfying certain conditions coming from representation theory. Each web corresponds to a specific invariant vector in a tensor product of fundamental representations of a quantum group. In this talk, we introduce a process called stranding, which encodes the monomial terms in a web’s associated vector as a collection of paths in the web graph.  We also describe how the strandings connect seemingly-unrelated ideas in combinatorics (e.g. noncrossing matchings) and geometry (e.g. certain algebraic varieties called Springer fibers).  Throughout, we'll describe how undergraduate and postbaccalaureate work was central to the evolution of these ideas.
  
  
• Peter Winkler (Dartmouth College): Changing Horses among Markov Chains (slides, video)
  You are at some state in each of several Markov chains, and trying to reach as quickly as possible a target state on one of them. You can choose any chain to move in, and can change your mind between moves. What's your strategy?
  Joint work with Ioana Dumitriu and Prasad Tetali, based on great stuff from John Gittins and Richard Weber. Included: the Whirling Tour, whatever that is . . .
  
  
• Benjamin Young (University of Oregon): The Squish map, the SL₂(C) Double dimer model, and stone-bone-snake tilings (slides, video)
  We relate the (relatively easy) 2-periodically weighted single dimer model on the hexagon lattice to an instance of the (relatively hard) SL_2(C) double-dimer model of Kenyon, via the squish map. In a special case, the loops which appear are conjecturally tileable by stones and bones (the same tile set used in Propp's benzel tilings), as well as a third tile, the "snake". Joint with Leigh Foster.
  
  
• Open Problem session (notes)
  2^{(3 choose 2)} mathematicians — Aaron Abrams, Colin Defant, Sam Hopkins, Lionel Levine, Jim Propp, Mikhail Skopenkov, David Speyer and Peter Winkler — presented open problems. Thanks to Sam Hopkins for taking (notes) on the discussion.