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Thursday, February 2, 2023
Humans have been discussing the benefits and drawbacks of democratic vs. authoritarian governance for millennia, with perhaps the earliest and most famous discussion favoring enlightened authoritarian rule contained in Plato's 'Republic'. The commonly cited benefits of an enlightened dictatorial regime are the efficiency of governance and long-term horizon in planning due to the independence of frequent election cycles. To analyze the claims of potential superiority of an authoritarian rule, we develop a simple mathematical theory of a dictatorship in the ideal case of a dictator wanting the best outcome for the country, with the additional external noise describing external and internal challenges in the country's path.
We assume the linear proportional feedback control based on the information provided by the output from the advisors. The resulting stochastic differential equations (SDEs) describe the evolution of both the trajectory of a country's well-being and the accuracy of advisors' information. We show the system's inherent instability due to the corruption of the advisor's information provided to the dictator. While the system without noise does possess a large amount of phase space with stable solutions, the noise pushes all solutions to the unstable regime. We show that there is a typical unstable time scale, and describe the long-term evolution of the system using asymptotic solutions, some results from the theory of SDEs and phase space analysis. We also discuss the application of the theory to historical data on grain harvest in the Soviet Union.
No previous knowledge of Ito's calculus or theory of SDEs is assumed; I will provide all necessary background from that theory during the lecture.
Math, 402 and
When a solid surface is bombarded with a broad ion beam, self-assembled patterns, such as hexagonal arrays of nanodots, form. Ion bombardment could become a widely employed method of fabricating large-scale nanostructures with wavelengths as short as 10 nm if these patterns were not typically ridden with defects. We build a model for nanoscale pattern formation in binary materials. Motivated by understanding and controlling defects in these as well as other nanoscale patterns, we develop quantitative measures of order for imperfect Bravais lattices in the plane that help compare the model with experimental data. A tool from topological data analysis called persistent homology, combined with a metric on point distributions, are the key components for defining these measures. We also develop a method, called persistent images, that employs persistent homology and machine learning to help determine parameter values from experimental data. Of particular interest is understanding the role of the local geometry compared to the global topology of the data in the method of persistent images.
Tuesday, February 7, 2023
The final frontier of extraterrestrial planetary exploration is the exploration of subsurface environments, such as caves and oceans. In particular, the existence of subsurface oceans on celestial bodies – e.g., Europa and Enceladus – known as ocean worlds has been backed by varying levels of evidence since the 1980s, but there has been no direct confirmation as of yet. Such environments are largely shielded from radiation, and in combination with the hypothesized presence of water, are prime candidate environments for finding extant or extinct life. However, the in-situ exploration of these subsurface oceans at hypothesized depths ranging anywhere from 1km to 100km (including terrestrial oceans up to 11km) necessitates disruptive advances in the design of robotic subsurface explorers capable of operating in such extreme aqueous environments, i.e., at such depths/pressures and temperatures. The field of underwater exploration systems is currently dominated by rigidly framed robots whose designs convey a philosophy of having wide arenas in which to move about without interruption. However, such a design ideology is less suitable for confined environments, which might limit a rigid explorer’s ability to navigate, and for extreme environments characterized by high pressures and low temperatures, i.e., extreme aqueous environments, such as Titan’s hydrocarbon lakes. Limited efforts have gone towards designing underwater exploration systems with a soft robotics philosophy, which overcomes the limitations of stiff robotic systems and permits more flexibility in underwater exploration. It is in this soft robotics underwater exploration context that the research of this Ph.D.-Thesis takes root: The design of a multisensory soft robotic explorer with a biologically inspired propulsion system for extreme aqueous environments using silicone rubber, thereby reframing its flexibility in a space which physically hinders the use of commonly employed pneumatic or hydraulic drivers. Taking inspiration from various biological sources including jellyfish, squid, octopus, and even the chambers of the human heart, this work presents the prototype for an autonomous underwater soft robotic exploration system featuring a novel propulsion system for navigation that is devoid of any (electric) motors or actuators and thus well-suited for extreme aqueous environments. The soft robotic explorer features onboard sensors for depth/pressure and temperature, and an onboard computer in charge of data recording, navigation, and propulsion control. Silicone rubber, widely used in other soft robotics applications due to its flexibility, forms the overall shell of the soft robot and its thrusters, and encases the onboard electronics. In addition, its electric inertness to permit direct electronics enclosure, resistance to saltwater degradation, maintained flexibility in low temperatures, and suitability for an additive manufacturing process were reconfirmed. The resulting soft robotic explorer system is fully sealed for all electrical components and fully open in its propulsion design.
Thursday, February 9, 2023
Enumerative geometry is concerned with counts of geometric objects satisfying given constraints. In algebraic geometry, these can be related to geometric spaces, such as curves, and objects of more algebraic flavor, like sheaves, which encode equations and are a generalization of vector bundles. Under certain assumptions, one hopes to consider a moduli space that parametrizes the objects in question and then obtain numerical invariants by integrating cohomology classes capturing the constraints against an appropriate fundamental class in its homology. In this talk, we will review sheaf counting and then discuss how to extend this approach to cases of interest where the usual assumptions do not hold. An important application motivated by physics is the construction of new generalized Donaldson-Thomas invariants enumerating sheaves on Calabi-Yau threefolds.
Friday, February 10, 2023
Action potential propagation along the axons and across the dendrites is the foundation of the electrical activity observed in the brain and the rest of the central nervous system. In this work, a novel Poisson-Nernst-Planck based treatment of the underlying electro-diffusive activity in the neurons is presented. This model is shown to produce results similar to the established cable-theory based electrical models, but in addition, the rich spatio-temporal evolution of the underlying ionic transport is captured. Specifically, saltatory conduction due to the presence of myelin sheath and the peri-axonal space is investigated. Further, we apply this model to numerically estimate conduction velocity in a rat and a squid axon. Time permitting, extensions of this framework to model mechano-chemo-electrostatic processes underlying traumatic brain injury (TBI) will be discussed.
Bio: Shiva Rudraraju is an Assistant Professor in the Department of Mechanical Engineering at the University of Wisconsin-Madison. He heads the Computational Mechanics and Multiphysics Group at UW-Madison, and his research interests are broadly in computational modeling of mechanics and multiphysics driven morphological processes in materials (structural, functional and biological). His current research projects are supported by ONR, ARO and NSF. He received his PhD in Mechanical Engineering and Scientific Computing from the University of Michigan Ann Arbor, and his undergraduate degree from India.
Math, 501 and
Monday, February 13, 2023
[CANCELLED] Special Colloquium
Unipotent flows have some striking rigidity properties: for instance, every orbit is recurrent, and every orbit closure as well as every invariant measure is homogeneous. In particular, the isomorphism rigidity theorem tells us that a measurable isomorphism between two flows in the class of unipotent flows on quotients of semisimple groups implies an algebraic isomorphism between their corresponding groups and lattices. Moreover, such systems always have zero entropy and thus they cannot be distinguished by classical entropy invariants.
We extend the isomorphism rigidity for unipotent flows to its time changes. More precisely, one parameter unipotent flows on quotients of semisimple groups fall into two categories: 1. unipotent flows that are time changes of linear irrational flows on T^2 and hence are all time changes of each other; 2. The measurable isomorphism between their time changes implies the much stronger (algebraic) equivalence as above. We also show that the complexity of time changes of unipotent flows can be described explicitly in terms of the corresponding adjoint action, and associated Jordan block-like structures. Moreover, this is also true for the complexity of high rank abelian unipotent actions.
This is based on joint works with Adam Kanigowski, Philipp Kunde, Elon Lindenstrauss and Kurt Vinhage.
Monday, February 20, 2023
February 20, 2023, 2:30pm, PAS 522, In-person
Title: A Distributed Approach for Learning Spatial Heterogeneity
Spatial regression is widely used for modeling the relationship between a spatial dependent variable and explanatory covariates. In many applications there are spatial heterogeneity in such relationships, i.e., the regression coefficients may vary across space. It is a fundamental and challenging problem to detect the systematic variation in the model and determine which locations share common regression coefficients and where the boundary is. In this talk, we introduce a Spatial Heterogeneity Automatic Detection and Estimation (SHADE) procedure for automatically and simultaneously subgrouping and estimating covariate effects for spatial regression models, and present a distributed spanning-tree-based fused-lasso regression (DTFLR) approach to learn spatial heterogeneity in the distributed network systems, where the data are locally collected and held by nodes. To solve the problem parallelly, we design a distributed generalized alternating direction method of multiplier algorithm, which has a simple node-based implementation scheme and enjoys a linear convergence rate. Theoretical and numerical results as well as real-world data analysis will be presented to show that our approach outperforms existing works in terms of estimation accuracy, computation speed, and communication costs.
Dr. Zhengyuan Zhu is the College of Liberal Arts and Sciences Dean's Professor, Director of the Center for Survey Statistics Methodology, and Professor of Statistics in the Department of Statistics at Iowa State University. He received his B.S. in Mathematics from Fudan University and Ph.D. in Statistics from the University of Chicago. His research interests include spatial statistics, survey statistics, machine learning, statistical data integration, and applications in environmental science, agriculture, remote sensing, and official statistics. He is a fellow of the American Statistics Association, and an elected member of the International Statistical Institute.
How can observational data be used to improve the design and analysis of randomized controlled trials (RCTs)? We first consider how to develop estimators to merge causal effect estimates obtained from observational and experimental datasets, when the two data sources measure the same treatment. To do so, we extend results from the Stein shrinkage literature. We propose a generic "recipe" for deriving shrinkage estimators, making use of a generalized unbiased risk estimate. Using this procedure, we develop two new estimators and prove finite sample conditions under which they have lower risk than an estimator using only experimental data. Next, we consider how these estimators might contribute to more efficient designs for prospective randomized trials. We show that the risk of a shrinkage estimator can be computed efficiently via numerical integration. We then propose algorithms for determining the experimental design -- that is, the best allocation of units to strata -- by optimizing over this computable shrinker risk.
Monday, March 13, 2023
Nash equilibrium (NE) is one of the most important concepts in game theory that captures a wide range of phenomena in engineering, economics, and finance. An NE is characterized by observing that in a stable game, no player can lower their cost by changing their action within their designated strategy. An equilibrium in the Nash game can be found by solving a variational inequality (VI) problem. Solving VI and stochastic VI (SVI) problems become more difficult if we consider that the players interact also at the level of the feasible sets. This situation arises naturally if the players share some common resource. This problem is modeled as Generalized NE (GNE) which can be formulated as Quasi VI (QVI) or Stochastic QVI (SQVI). In this talk, we present efficient iterative schemes to solve SVI and SQVI problems and present their convergence guarantees. To validate our theoretical findings, we provide some preliminary simulation results on different problems such as stochastic Nash Cournot competition over a network.