Wednesday, February 1 |
07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
09:00 - 10:00 |
Emil Saucan: The Versatile Forman-Ricci curvature and its Networks Applications ↓ We present the adaptations of Forman's discretization of Ricci curvature to the setting of networks and their higher dimensional generalizations and we explore their applications to a variety of real-life applications, such as: brain networks, chemical reactions, financial market crushes, stem cells and cancer research, autism understanding, intelligence of communication and social networks, deep learning and semantics.
We also show how it naturally allows for the understanding of the long-time evolution of networks, their sampling as well as their study through persistent homology. (Online) |
10:00 - 10:30 |
Coffee Break (TCPL Foyer) |
10:30 - 11:30 |
Hans Riess: Lattice Theory in Social Choice and Mutli-Agent Systems ↓ Combinatorial thinking has entered the field of economics in social choice theory, pioneered by Arrow (1951). We argue order lattices are a befitting object for modeling preferences, choice, and information on the level of the individual, as well as a theoretical tool for modeling aggregation. Recently introduced (Ghrist & Riess, 2022), an operator, called the Tarski Laplacian, acting on the (product) lattice of 0-cochains of a lattice-valued sheaf, induces “heat flow” dynamics for lattice data supported on an (e.g. social) network. A Hodge-style theorem characterizes the time-invariant solutions of the heat equation as global sections of a lattice-valued sheaf. This talk aims to draw connections between both lines of work and encourage future conversation and collaboration. (TCPL 201) |
11:30 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
13:00 - 14:00 |
Discussion (Online and In Person) |
14:00 - 15:00 |
Dane Taylor: Homological analysis of network dynamics ↓ Social, biological, and physical systems are widely studied through the modeling of dynamical processes over networks, and one commonly investigates the interplay between structure and dynamics. I will discuss how cyclic patterns in networks can influence models for collective and diffusive processes, including generalized models in which dynamics are defined over simplicial complexes and multiplex networks. Our approach relies on homology theory, which is the subfield of mathematics that formally studies cycles (and more generally, k-dimensional holes). We will make use of techniques including persistent homology and Hodge theory to examine the role of cycles in helping organize dynamics onto low-dimensional manifolds. (TCPL 201) |
15:00 - 15:30 |
Kang-Ju Lee: Simplicial electrical networks and applications ↓ We extend the notion of effective resistance from the classical circuit theory to simplicial electrical networks. Our approach, based on combinatorial Hodge theory, is to assign a unique harmonic class to a current generator, an extra top-dimensional simplex to be attached to the simplicial resistor network. The harmonic class gives rise to the current (cycle) and the voltage (cocyle) satisfying Thomson's energy-minimizing principle and Ohm's law for simplicial networks. We introduce a simplicial analogue of Kirchhoff index, the sum of all pairwise effective resistances, proposing the quantity as a measure of robustness of simplicial complexes. Also, we present a method for counting spanning trees in simplicial complexes by using a combinatorial interpretation of effective resistance. This talk is based on joint works with Kook and Duval-Kook-Martin. (Online) |
15:30 - 16:20 |
Alexander Strang: Applications of Hodge Theory to Nonequilibrium Steady States ↓ Combinatorial Hodge theory is a powerful framework for analyzing skew-symmetric functions on graph edges. We explore applications of the discrete Helmholtz-Hodge Decomposition to the steady state of reversible nonequilibrium stochastic processes. (TCPL 201) |
16:20 - 16:30 |
Discussion (Online) |
16:30 - 17:00 |
Christopher Cebra: Similarity Promotes Transitivity in Generic Competitive Systems ↓ Real-world competitive systems are often more transitive (hierarchical) systems than intransitive (cyclic), although the space of intransitive systems is larger. Why? We propose a domain-independent mechanism that promotes transitivity in these systems, namely if selection promotes similarity among the competitors then similarity leads to transitivity. We analyze the rate of convergence to transitivity based on the characteristics of the performance function relating the competitors. We also test these findings numerically for game theory and random examples, using the Helmholtz-Hodge Decomposition to measure the size of the transitive and intransitive components. (TCPL 201) |
17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |