Dr Zoe Copperband is a member of the Penn Engineering GRASP Laboratory. Her recent preprint, Towards Homological Methods in Graphic Statics, can be found here.
Andrea Jo-Wei Liu is the Hepburn Professor of Physics at the University of Pennsylvania, where she holds a joint appointment in the Department of Chemistry. She is a theoretical physicist studying condensed matter physics and biophysics.
In order for artificial neural networks to learn a task, one must solve an inverse design problem. What network will produce the desired output? We have harnessed AI approaches to design physical systems to perform functions inspired by biology, such as protein allostery. But artificial neural networks require a computer in order to learn in top-down fashion by the global process of gradient descent on a cost function. By contrast, the brain learns by local rules on its own, with each neuron adjusting itself and its synapses without knowing what all the other neurons are doing, and without the aid of an external computer. But the brain is not the only biological system that learns by local rules; I will argue that the actin cortex and the amnioserosa during the dorsal closure stage of Drosophila development can also be viewed this way.
Title: “Mathematical models of curiosity”
Prof. Bassett is the J. Peter Skirkanich Professor at the University of Pennsylvania, with appointments in the Departments of Bioengineering, Electrical & Systems Engineering, Physics & Astronomy, Neurology, and Psychiatry. They are also an external professor of the Santa Fe Institute. Bassett is most well-known for blending neural and systems engineering to identify fundamental mechanisms of cognition and disease in human brain networks.
What is curiosity? Is it an emotion? A behavior? A cognitive process? Curiosity seems to be an abstract concept—like love, perhaps, or justice—far from the realm of those bits of nature that mathematics can possibly address. However, contrary to intuition, it turns out that the leading theories of curiosity are surprisingly amenable to formalization in the mathematics of network science. In this talk, I will unpack some of those theories, and show how they can be formalized in the mathematics of networks. Then, I will describe relevant data from human behavior and linguistic corpora, and ask which theories that data supports. Throughout, I will make a case for the position that individual and collective curiosity are both network building processes, providing a connective counterpoint to the common acquisitional account of curiosity in humans.
Random walks are a common model for the exploration and discovery of complex networks. While numerous algorithms have been proposed to map out an unknown network, a complementary question arises: in a known network, which nodes and edges are most likely to be discovered by a random walker in finite time? In this talk we introduce exposure theory, a statistical mechanics framework that predicts the learning of nodes and edges across several types of networks, including weighted and temporal, and show that edge learning follows a universal trajectory. While the learning of individual nodes and edges is noisy, exposure theory produces a highly accurate prediction of aggregate exploration statistics. As a specific application, we extend exposure theory to better understand human learning with its typical mental errors, and thus account for distortions of learned networks.
This talk is based on https://arxiv.org/abs/2202.11262
Pablo G. Cámara is an Assistant Professor of Genetics at the University of Pennsylvania and a faculty member of the Penn Institute for Biomedical Informatics. He received a Ph.D. in Theoretical Physics in 2006 from Universidad Autónoma de Madrid. He performed research in string theory for several years, with postdoctoral appointments at Ecole Polytechnique, the European Organization for Nuclear Research (CERN), and University of Barcelona. Fascinated by the extremely interesting and fundamental open questions in biology, in 2014 he shifted his research focus into problems in quantitative biology, and joined the groups of Dr. Rabadan, at Columbia University, and Dr. Levine, at the Institute for Advanced Study (Princeton). Building upon techniques from applied topology and statistics, he has devised novel approaches to the inference of ancestral recombination, human recombination mapping, the study of cancer heterogeneity, and the analysis of single-cell RNA-sequencing data from dynamic and heterogeneous cellular populations.
One of the prevailing paradigms in data analysis involves comparing groups of samples to statistically infer features that discriminate them. However, many modern applications do not fit well into this paradigm because samples cannot be naturally arranged into discrete groups. In such instances, graph techniques can be used to rank features according to their degree of consistency with an underlying metric structure without the need to cluster the samples. Here, we extend graph methods for feature selection to abstract simplicial complexes and present a general framework for clustering-independent analysis. Combinatorial Laplacian scores take into account the topology spanned by the data and reduce to the ordinary Laplacian score when restricted to graphs. We show the utility of this framework with several applications to the analysis of gene expression and multi-modal cancer data. Our results provide a unifying perspective on topological data analysis and manifold learning approaches to the analysis of point clouds.
The class of sheaf Laplacians can be characterized as the convex closure of a certain set of sparse semidefinite matrices. From this viewpoint, the study of sheaf Laplacians becomes a question of linear algebra on sparse matrices. I will discuss the applications of this perspective to the problems of approximating, sparsifying, and learning sheaves.
There is a conjecture by Colliot-Thelene (about 2005) that under specific hypotheses, a morphism of Q-varieties f : X --> Y has the property that for almost all prime numbers p, the corresponding map X(Q_p) --> Y(Q_p) is surjective. A sharpening of the conjecture was solved by Denef (2016), and later, "if and only if" conditions on f were given by Skorobogatov et al. The plan for the talk is to explain in detail the conjecture and the results mentioned above, and to report on work in progress on a different method to attack the conjecture under quite relaxed hypotheses.
In contrast to the Artin-Schreier Theorem, its $p$-adic analog(s) involve infinite Galois theory, e.g., the absolute Galois group of $p$-adic fields. We plan to give a characterization of $p$-adic $p$-Henselian valuations in an essentially finite way. This relates to the $Z/p$ metabelian form of the birational $p$-adic Grothendieck section conjecture.