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ESE Ph.D. Thesis Defense: “Lattice Theory in Multi-Agent Systems”

September 26, 2022 at 2:00 PM - 3:30 PM

Ordered sets model signals such as binary relations, concepts, partitions, rankings, matchings, events, as well as other taxa of information, temporal, hierarchical, relational, or, in general, logical in nature. We argue that (order-) lattice-based (networked) multi-agent systems constitute a broad class of systems in which data fusion, consensus, synchronization, and other collaborative tasks are described with lattices and Galois connections (maps between lattices that preserve structure). Mathematically speaking, these systems are network sheaves. Motivated by analogous vector diffusion and consensus algorithms, we initiate a discrete Hodge theory with the Tarski Laplacian, a diffusion operator—analogous to the graph Laplacian and the graph connection Laplacian—acting on assignments of lattice-valued data to the nodes of a network. The Hodge-Tarski theorem (Main Theorem) relates the fixed point theory of the Tarski Laplacian to the global sections (consistent signals, equilibria, if you will) of the sheaf. We present novel applications in signal processing and multi-modal semantics where we design a consensus algorithm on statements such as “I know that she knows that he doesn’t know that I’m defending my thesis.”

Hans Reiss

ESE Ph.D. Candidate

Hans Riess was born and raised in Bloomington, Indiana. He attended Duke University and graduated with a Bachelor’s in Mathematics in 2016. In 2017, he joined the Department of Electrical Systems Engineering at the University of Pennsylvania as a doctoral candidate under the supervision of Robert Ghrist. At Penn, he was a recipient of the Ganster Fellowship as well as the Legget Family Endowed Fellowship. At various conferences and workshops, Hans has presented on a diversity of topics from control theory and machine learning to algebraic topology and lattice theory. Hans has also presented to panels of military and civilian personnel as a participant in grants funded by the Undersecretary for Research and Engineering and the Defense Advanced Research Projects Agency (DARPA). Hans’ specific research interests include topological data analysis, geometric deep learning, and formal methods in robotics. He has refereed articles for SIAM and IEEE journals, and, since 2017, has served on the Duke Alumni Admissions Advisory Committee. Hans will be beginning a postdoc at Duke University in November.


September 26, 2022
2:00 PM - 3:30 PM
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Electrical and Systems Engineering
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Moore 317
200 S 33rd Street
Philadelphia, PA 19104 United States
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