Chip-firing processes are discrete dynamical systems. A commodity (chips, sand, dollars) is exchanged between sites of a network according to simple local rules. Although governed by local rules, the long-term global behavior of the system reveals unexpected properties, including intricate fractal-like patterns. Early results related chip-firing to classic combinatorial objects such as spanning trees, parking functions, and matroids. In recent years, chip-firing has seen much activity in new directions. Connections have been made, for example, between chip-firing and Coxeter groups, binomial ideals, and Riemann surfaces. In this talk, I will give a broad survey of the theory of chip-firing and its many ties to algebraic combinatorics.

Classifying lattice walks in restricted lattices is an important problem in enumerative combinatorics. Recently, computer algebra has been used to explore and to solve a number of difficult questions related to lattice walks. We give an overview of recent results on structural properties and explicit formulas for generating functions of walks in the quarter plane, with an emphasis on the algorithmic methodology.

Everybody knows that identical regular triangles or squares can tile the whole plane. Many people know that identical regular hexagons can tile the plane properly as well. In fact, even the bees know and use this fact! Is there any other convex domain which can tile the Euclidean plane? Of course, there is a long list of them! To find the list and to show the completeness of the list is a unique drama in mathematics, which has lasted for more than one century and the completeness of the list has been mistakenly announced not only once! Up to now, the list consists of triangles, quadrilaterals, fifteen types of pentagons, and three types of hexagons. In 2017, Michael Rao announced a computer proof for the completeness of the list. Meanwhile, Qi Yang and Chuanming Zong made a series of unexpected discoveries in multiple tilings in the Euclidean plane. For examples, besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form any two-, three- or four-fold translative tiling in the plane. However, there are two types of octagons and one type of decagons which can form nontrivial five-fold translative tilings. Furthermore, a convex domain can form a five-fold translative tiling of the plane if and only if it can form a five-fold lattice tiling. In this talk we will report these progresses.

We are faced with a flood of molecular and clinical data. We are measuring interactions between various bio-molecules in a cell that form large, complex systems. Patient omics datasets are also increasingly becoming available. These systems-level data provide heterogeneous, but complementary information about cells, tissues and diseases. The challenge is how to mine them collectively to answer fundamental biological and medical questions. This is nontrivial, because of computational intractability of many underlying problems, necessitating the development of heuristic methods for finding approximate solutions.
We develop methods for extracting new biomedical knowledge from the wiring patterns of systems-level, heterogeneous, networked biomedical data. Our methods uncover the patterns in molecular networks and in the multi-scale network organization indicative of biological function, translating the information hidden in the topology into domain-specific knowledge. We introduce a versatile data fusion (integration) framework to address key challenges in precision medicine: better stratification of patients, prediction of driver genes in cancer, and re-purposing of approved drugs to particular patients and patient groups. Our new methods stem from novel network science approaches coupled with graph-regularized non-negative matrix tri-factorization, a machine learning technique for dimensionality reduction and co-clustering of heterogeneous datasets. We utilize our new framework to develop methodologies for performing other related tasks, including disease re-classification from modern, heterogeneous molecular level data, inferring new Gene Ontology relationships, aligning multiple molecular networks, and uncovering new cancer mechanisms.

The classical Torelli theorem says that a Riemann surface can be recovered from its Jacobian, which is a principally polarized Abelian variety. There is an analogous theorem for graphs, due to Artamkin and Caporaso–Viviani that the 2-isomorphism class of a graph can be recovered from its cycle space, equipped with its cycle pairing. We ask what happens when one considers mildly non-abelian data as in the unipotent Torelli theorem for Riemann surfaces due to Hain and Pulte. This leads us to introducing the analogue of iterated integrals on graphs and encoding them in a particular structure. This structure turns out to recover pointed bridgeless graphs up to isomorphism. We discuss some of the application of this result and connections to Hodge theory, tropical geometry, and number theory. This is joint work with Raymond Cheng.

Homotopy type theory is a foundation of mathematics that both expands and refines the classical foundations built on first-order logic and set theory. It accounts properly and formally for various aspects of informal mathematics that are otherwise difficult to formalize, for instance the idea that equivalent structures may be identified. It is also able to express the distinction between concrete and abstract existence arguments. It thus offers an answer, however useful it might be, to the question: what precisely is an explicit bijection?

This talk will focus on the distributions of various parameters associated with random trees, and on the limit distributions of such parameters. Different families of trees will be considered, such as simply generated (Galton–Watson) trees, Pólya trees and increasing trees. The notion of an additive tree functional will play a major role, as it provides us with a rather general approach to study seemingly unrelated parameters of trees, such as the number of leaves, the multiplicity of eigenvalues, and the cardinality of the automorphism group. General asymptotic schemes allow us to prove that many different tree parameters follow a normal or log-normal limit law.

The discriminant of a polynomial with prescribed monomials is an irreducible polynomial in the coefficients vanishing when the corresponding polynomial has multiple roots. It is an important tool in numerical studies of polynomials and describes a rich geometry of tangency of an associated algebraic variety. I will recall this interplay and introduce the notion of discriminant for a system of polynomials (the mixed discriminant). I will then show how a generalised Schäfli method leading to the notion of iterated discriminant can simplify the analysis of mixed discriminants. The talk will mostly be introductory to the subject. The latest results are based on joint work with Alicia Dickenstein and Ralph Morrison.

Much modern combinatorics involves finite reflection groups, both real and complex. Mysteriously, many results work particularly well for the so-called "coincidental" reflection groups. These are the groups generated by n reflections acting in n-dimensional space whose exponents form an arithmetic sequence – they are the real reflection groups of types A, B/C, H₃, dihedral groups, and all non-real Shephard groups (the symmetries of regular complex polytopes).
This talk will discuss recent work featuring the coincidental groups. This includes recent work with Shepler and Sommers uncovering their extra elegant invariant theory, leading to product formulas for the face numbers and h-vectors of their associated cluster complexes, and a q-analogue of the transformation taking the h-vector to f-vector. We also hope to discuss theorems and conjectures of various others, such as Alex Miller, Barnard–Reading, Hamaker–Patrias–Pechenik–Williams, and Sam Hopkins.