Abstract: p-adic numbers are cousins of real numbers, defined via a completion of rational numbers. The field of p-adic numbers has both analytic and algebraic structures, given by the p-adic norm and the Galois group. As a consequence, geometry over p-adic numbers provides a tidy connection between various areas of mathematics, such as number theory, representation theory, and algebraic geometry. We give an overview of this connection, with emphasis on the context of the Langlands program, and discuss some recent developments. 

We introduce intrinsic Sobolev-Slobodeckij spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak Hörmander condition.  We prove continuous embeddings into Lorentz and intrinsic Hölder spaces. We also prove approximation and interpolation inequalities by means of an intrinsic Taylor expansion, extending analogous results for Hölder spaces.  The embedding at first order is proved by adapting a method by Luc Tartar which only exploits scaling properties of the intrinsic quasi-norm, while for higher orders we use uniform kernel estimates.

p-adic representations are important objects in number theory, and stable lattices serve as a connection between the study of ordinary and modular representations. These stable lattices can be understood as stable vertices in Bruhat-Tits buildings. From this viewpoint, the study of fixed point sets in these buildings can aid research on p-adic representations. The simplicial balls, in particular, hold an important role as they possess the most symmetry and fastest growth, and are closely related to the Moy-Prasad filtrations. In this talk, I'll explain those new findings, provide a characterization of such simplicial balls, and compute their simplicial volume under certain conditions.

We present a novel approach to the numerical resolution of McKean-Vlasov stochastic differential equations via stochastic gradient descent. The algorithm is presented for a class of mean-field diffusions with separable coefficients. After introducing the algorithm, we present a convergence result and some numerical tests to confirm the theoretical findings. We finally discuss further ongoing investigations about the performance of the algorithm and possible generalizations. This is a joint work with Ankush Agarwal and Gonçalo dos Reis.

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
Zoom   https://arizona.zoom.us/j/85889389967 
Password:  applied

Abstract:

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. 

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.

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.

February 20, 2023, 2:30pm, PAS 522, In-person

Title: A Distributed Approach for Learning Spatial Heterogeneity 

Abstract:

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.

Bio:

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. 

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.