Title: Hilbert modular forms from orthogonal modular forms on binary lattices

Abstract: We show the explicit connection between Hilbert modular forms and orthogonal modular forms arising from positive definite binary lattices over the ring of integers of a totally real number field. Our work uses the even Clifford algebra to generalize Gauss composition in a categorical way over a Dedekind domain. This allows us to classify the class sets of different types of genera in terms of the class groups of the associated quadratic algebras. This is joint work with John Voight.

Title: Maximally Even Sets

Abstract:``Maximally Even Sets" make appearances in a diverse slice of the modern world: as the arrangement of the black keys among the white keys on a piano; as the arrangement of magnets in spallation neutron source accelerators in nuclear physics; in the drawing of straight lines on pixelated screens; in the structure of traditional rhythms in cultures worldwide. At the heart of each of these problems is a natural problem in extremal graph theory.

We discuss recent work on the mathematical formalization of such maximally even sets based on a recent characterization for cycles and paths; we also extend these ideas to a multicolor context. Along the way we establish a connection with the majorization order on tuples of integers, originally defined by Muirhead and studied by Lorenz in his mathematical treatment of wealth inequality.

This is based joint work with Viktoriya Bardenova, Brent Cody, Paul Fay, Luke Freeman, Chris Leffler, Maya Tennant, and Toby Whitaker.

Title: Cluster Synchronization via Graph Laplacian Eigenvectors

Abstract: Almost equitable partitions (AEPs) have been linked to cluster synchronization in oscillatory systems, providing a mathematical framework for understanding collective behavior. Using the spectral properties of AEPs, we can describe this synchronization behavior in terms of the graph Laplacian eigenvectors. Our results also shed light on transient hierarchical clustering as well as multi-frequency clustering, and the conditions under which they can occur. Through our analysis we are able to relate dynamical clustering phenomena directly to network symmetry and community detection, joining the structural and functional notions of clustering in complex networks. This bridges a crucial gap between static network topology and emergent dynamic behavior. Additionally, this description of the problem allows us to define a relaxation of an AEP we call a quasi-equitable partition (QEP). Perfect AEPs are rare in real-world networks since most have some degree of irregularity or noise. While relaxing these strict conditions, QEPs are able to maintain many of the useful properties allowing for qualitatively similar clustering behavior as with AEPs. Our findings have important implications for understanding synchronization patterns in real-world networks, from neural circuits to power grids.

Title: How norms shape the evolution of prosocial behavior: A C.U.R.E for social dilemmas

Abstract: How cooperation evolves and particularly maintains at a large scale remains an open problem for improving humanity across domains ranging from climate change to pandemic response. To shed light on how behavioral norms can resolve the social dilemma of cooperation, here we present a formal mathematical model of individuals’ decision making under general social norms, encompassing a variety of concerns and motivations an individual may have beyond simply maximizing their own payoffs. Using the canonical game of the Prisoner’s Dilemma, we compare four different norms: compassion, universalizability, reciprocity, and equity, to determine which social forces can facilitate the evolution of cooperation, if any. We analyze our model through a variety of limiting cases, including weak selection, low mutation, and large population sizes. This is complemented by computer simulations of population dynamics via a Fisher process, which confirm our theoretical results. We find that the first two norms lead to the emergence of cooperation in a wide range of games, but the latter two do not on their own. Due to its generality, our framework can be used to investigate many more norms, as well as how norms themselves emerge and evolve. Our work complements recent work on fair-minded learning dynamics and provides a useful bottom-up perspective into understanding the impact of top-down social norms on collective cooperative intelligence.

Title: Generalized Ramsey-Turan Numbers

Abstract: We study the generalized Ramsey-Tur'an function RT(n,K_q,K_p,o(n)), the maximum number of copies of K_q in an n-vertex K_p-free graph with independence number sublinear in n. The case q=2 was resolved by Erdos, Sos, Bollobas, Hajnal, and Szemeredi in the 1980s. More recently, Balogh, Liu, and Sharifzadeh resolved the q=3 and p=q+2 cases. We extend these results to the cases of q=4 and q=5, as well the general cases p\geq 5q and q+3\leq p\leq q+5, along the way disproving a conjecture of Balogh, Liu, and Sharifzadeh. Joint work with Jozsef Balogh and Cory Palmer.

Title: On secondary invariants of spin manifolds

Abstract: We discuss secondary invariants and their topological formulas.

Title: Nonlinear Bayesian Update

Abstract: Bayesian updating is a fundamental tool for improving the statistical accuracy of uncertain systems using measurement data, with applications spanning numerical weather prediction, robotics, and neural network training. In this talk, I will review sample-based Bayesian update methods and examine their behavior under various assumptions on the prior distribution and measurement model. Building on this analysis, I will introduce a stable Bayesian update scheme designed to handle non-Gaussian statistics. The effectiveness of the proposed approach will be demonstrated through numerical examples, including nonlinear measurements in chaotic systems, robot localization, and PDE-constrained optimization.