This talk is intended as an semi-informal introduction to my recent work on Arithmetic Teichmuller Spaces and its relationship to Mochizuki's Inter Universal Teichmuller Theory which is at the center of his work on the abc-conjecture. My work establishes a p-adic analog of classical Teichmuller Theory of Riemann surfaces. Notably in this theory, the etale fundamental group remains fixed (just as in the classical theory of Riemann surfaces) while the holomorphic structure varies. Given the nature of this work and its close relationship to Mochizuki's work, I have intentionally divided this talk into two talks  on (10/3/2023) and (10/10/2023) both to provide a gentle introduction to my ideas and also to allow for plenty of questions. [There will be plenty of technical material presented as well but the emphasis will be on providing an introduction to my ideas.] I will also include a discussion of  Mochizuki-Scholze-Stix issues in the context of Mochizuki's Theory (and answer any questions in this context).

Abstract:  This talk will explore the equivalence between neural networks and kernel methods (and reproducing kernel spaces more generally) by deriving a representation of any finite-size parametric classification model trained with gradient descent as a kernel machine. We will compare this exact representation to the well-known Neural Tangent Kernel (NTK) and discuss approximation accuracy. We will apply this kernel in several experimental contexts to demonstrate that this theoretical foundation informs cutting edge algorithms on several machine-learning tasks. Last, we will discuss several natural applications of this theory which we are only just beginning to explore.

We present a physics-informed, market-aware machine learning-driven algorithm to solve the DC optimal power flow problem. Namely, we employ active set learning to construct a system of linear equations whose solution is equivalent to the primal and dual solution of DC-OPF. The scheme, which we refer to as physics-informed, market-aware, active-set OPF (PIMA-AS-OPF), is validated on the New York ISO 1814-bus system, with results shown for different unit commitment, levels of wind penetration, and levels of noise. In addition, we demonstrate that the LMPs and dispatches adhere to established principles of market design, including revenue adequacy and cost recovery. Finally, we discuss how similar ML-driven schemes may be developed for unit commitment, which is mixed-integer and thus inherently more challenging in nature than DC-OPF.  

I will be rehearsing a conference talk, so feedback on the clarity of the presentation and ways to make the content more concise/accessible would be appreciated. I am not necessarily expecting detailed questions about the topic, though I welcome them as well.

Fix a finite extension of fields L/K of degree n. We present a Galois-cohomological approach to finding genus g curves X/K having a closed point with residue field L. There is a coarse moduli space M_{g,L/K} for this problem, which is a twisted form of the classical moduli space M_{g,n} which parametrizes n-pointed genus g curves. Our strategy basically consists in twisting a given G-equivariant morphism V --> M_{g,n} (with G a subgroup of S_n and V/K a variety having many rational points)  to get a similar morphism of twisted forms V' --> M_{g,L/K}. We discuss some limitations with this idea, and also survey some of the known results about the spaces M_{g,L/K}.

Abstract:         The budgets of turbulent kinetic energy (TKE) and turbulent potential energy (TPE) at different scales in sheared, stably stratified turbulence (SSST) are analyzed using a filtering approach together with data from direct numerical simulations. We consider the competing effects in the flow along with the physical mechanisms governing the energy fluxes between scales. The theoretical work of our energy budget analysis is used to analyze data from direct numerical simulation (DNS). Various quantities in the energy budget equations are evaluated based on DNS data of SSST, with detailed discussions on both the mean-field behavior of the flow, as well as fluctuations about this mean-field state. Importantly, it is shown that the TKE and TPE fluxes between scales are both downscale on average and their instantaneous values are positively correlated, but not strongly. The relative weak correlation occurs mainly due to the different physical mechanisms that govern the TKE and TPE fluxes.

 I consider the problem of estimating a n x n covariance matrix from a set of m sampled vectors, each of dimension n. The caveat is that m is much less than n, i.e., the dimension is huge and the number of samples is small. I will explain why this seemingly mundane problem is so important across all of Earth science and then report how this problem has been partially solved in numerical weather prediction (NWP) via a procedure called "covariance localization." We will then construct a theory of optimal localization. A numerical and analytical comparison of optimal and practical approaches reveals that (i) the practical approach, used for decades in NWP, very much mimics an optimal approach, implying that there is mathematical support for the practical, ad hoc solution; and (ii) the details of how covariance matrices are localized have only a second order effect, implying that "even bad localization is good localization." The latter point is quite subtle, but has practical implications which I will discuss. This talk summarizes work done in collaboration with Dr. Daniel Hodyss of the Naval Research laboratory and is supported by an ONR grant (N00014-21-1-2309).

Matthias Morzfeld Website:  https://igppweb.ucsd.edu/~mmorzfeld/

Place: Math Building, Room 501  https://map.arizona.edu/89

This talk is intended as an semi-informal introduction to my recent work on Arithmetic Teichmuller Spaces and its relationship to Mochizuki's Inter Universal Teichmuller Theory which is at the center of his work on the abc-conjecture. My work establishes a p-adic analog of classical Teichmuller Theory of Riemann surfaces. Notably in this theory, the etale fundamental group remains fixed (just as in the classical theory of Riemann surfaces) while the holomorphic structure varies. Given the nature of this work and its close relationship to Mochizuki's work, I have intentionally divided this talk into two talks  on (10/3/2023) and (10/10/2023) both to provide a gentle introduction to my ideas and also to allow for plenty of questions. [There will be plenty of technical material presented as well but the emphasis will be on providing an introduction to my ideas.] I will also include a discussion of  Mochizuki-Scholze-Stix issues in the context of Mochizuki's Theory (and answer any questions in this context).

Title: Nonparametric inference on network effects for relational data with dependent edges

Abstract: In recent years, the relationship network has received significant attention for its ability to provide unique insights into agent interactions across various fields. Most existing studies have primarily focused on modeling the association between the relationship network and other covariates using arguably restrictive parametric models, while largely overlooking the inference of network effects, such as the reciprocity or sender-receiver effect. Testing network effects within a relationship network is particularly challenging due to edge dependence, which renders permutation-based methods inapplicable. Our testing statistics utilize the reduced U-statistics and admit analytically tractable limiting distributions, overcoming the nontrivial sampling distributions of network moment statistics on relationship network effects caused by degeneracy and indeterminacy of degeneracy order. We establish the theoretical guarantee of our testing framework by investigating the Berry-Esseen bounds for our testing statistics. To showcase the practicality of our methods, we apply them to two real-world relationship networks, one in international trade and the other in faculty hiring networks.

Abstract:         In this talk, we first present a new rapid estimation method (REM) to perform stochastic impact analysis of grid-edge technologies (GETs) to the power distribution networks. The evolution of network states' probability density functions (PDFs) in terms of GET penetration levels are characterized by the Fokker-Planck equation. The approach is illustrated on a large-scale, realistic distribution network using a modified version of the IEEE 8500 feeder, where electric vehicles (EVs) or photovoltaic systems are installed at various penetration rates. Second, we design an incentive-based mitigation strategy for equipment overloading leveraging the REM. The objective is to shift the EV charging during the peak hours when the equipment is overloaded to the off-peak hours and maintain equipment service lifetime. The incentive level and corresponding contributions from customers to alter their EV charging habits are determined by a search algorithm with provable convergence guarantee. The strategy is illustrated on the same IEEE 8500 feeder with a high EV penetration to mitigate overloads on the substation transformer.

Bio: Antoine Lesage-Landry is an Assistant Professor in the Department of Electrical Engineering at Polytechnique Montréal, QC, Canada, a member of the Group for Research in Decisions Analysis (GERAD), and an associate academic member of Mila — Quebec’s AI Institute. He received the B.Eng. degree in Engineering Physics from Polytechnique Montréal, QC, Canada, in 2015, and the Ph.D. degree in Electrical Engineering from the University of Toronto, ON, Canada, in 2019. From 2019 to 2020, he was a Postdoctoral Scholar in the Energy & Resources Group at the University of California, Berkeley, CA, USA. His research interests include optimization, online learning, machine learning, and their application to power systems with renewable generation.

This talk is an Applied Math Guest Lecture and Special Data and Dynamics Group meeting. 

We present a framework to compute and study two-dimensional water waves that are quasi-periodic in space and/or time. This means they can be represented as periodic functions on a higher-dimensional torus by evaluating along irrational directions. In the spatially quasi-periodic case, we consider both traveling waves and the general initial value problem. In both cases, the nonlocal Dirichlet-Neumann operator is computed using conformal mapping methods and a quasi-periodic variant of the Hilbert transform.  We obtain traveling waves either as a generalization of the Wilton ripple problem or through bifurcation from large-amplitude periodic waves. In the temporally quasi-periodic case, we devise a shooting method to compute standing waves with 3 quasi-periods as well as hybrid traveling-standing waves that return to a spatial translation of their initial condition at a later time. Many examples will be given to illustrate the types of behavior that can occur.

We explore three directions extending the traditional theory of complex quadratic iterations to: (1) complex quadratic networks; (2) template iterations and (3) mutated systems. In all three cases, we define Julia and Mandelbrot sets, and describe how their properties are different than those of the traditional sets for single map iterations. In the case of networks, the dynamics of each node are driven by the same complex quadratic map f(z) = z2 +c, with coupling specified by the adjacency matrix Aij and weights gij , so that the system takes the form of an iteration in Cn. We discuss how the topology of the network Mandelbrot changes under perturbation of the network architecture, and how this can be used to create and analyze a fractal profile for a person’s brain network. In the case of templates, we study non-autonomous iterations of a pair of complex quadratic functions, fc0 and fc1 , applied according to a general binary symbolic sequence s. Here, the Mandelbrot set is a subset of the product space C2 ×{0, 1}∞. We discuss its slices in either subspace, present a Hausdorff convergence theorem and illustrate how the framework can be used in genetics, to understand the how cells differentiate and specialize. In the case of mutations, we insert an “erroneous” function fc0 to replace the intact replication function fc1 within a disk around a mutation epicenter. We analyze the perturbations to the Julia set induced by mutations with different parameter c0, different radius and different location. We discuss how this can be used to understand the role of mutations in cancer modeling research.

Title and Abstract: TBA

Vasali Perebenios Website: 

https://engineering.buffalo.edu/ee/faculty/faculty_directory/vasili-perebeinos.html

Join a panel of students and faculty of the Statistics & Data Science Graduate Interdisciplinary Program for a short presentation and a Q&A period.  This is an opportunity for undergraduate students to learn more about the Grad Program in Statistics & Data Science.

Pizza and cookies will be served.

Title and Abstact: TBA

Pavel Bochev: https://www.sandia.gov/ccr/staff/pavel-b-bochev/

High Dimensional Gaussian Graphical Regression

Models with Covariates

Though Gaussian graphical models have been widely used in many scientific fields, relatively limited progress has been made to link graph structures to external covariates. We propose a Gaussian graphical regression model, which regresses both the mean and the precision matrix of a Gaussian graphical model on covariates. In the context of co-expression quantitative trait locus (QTL) studies, our method can determine how genetic variants and clinical conditions modulate the subject-level network structures, and recover both the population-level and subject-level gene networks. Our framework encourages sparsity of covariate effects on both the mean and the precision matrix. In particular for the precision matrix, we stipulate simultaneous sparsity, i.e., group sparsity and element-wise sparsity, on effective covariates and their effects on network edges, respectively. We establish variable selection consistency first under the case with known mean parameters and then a more challenging case with unknown means depending on external covariates, and establish in both cases the  convergence rates and the selection consistency of the estimated precision parameters. The utility and efficacy of our proposed method is demonstrated through simulation studies and an application to a co-expression QTL study with brain cancer patients.

Title and Abstract: TBA

Paul Atzberger Research Group:  https://web.math.ucsb.edu/~atzberg/pmwiki_intranet/index.php?n=AtzbergerHomePage.Homepage?setskin=atzbergerHomepage4

Abstract: TBA

Simon Tavener Website: https://www.math.colostate.edu/~tavener/

The Mathematics Educator Appreciation Day Conference (MEAD) is Arizona's largest mathematics education event. In 2024, MEAD will be celebrating its 20th anniversary, so come join us for this unique opportunity to grow professionally and show appreciation for everything teachers do.

In 2023, MEAD hosted almost 1000 participants who attended more than 120 sessions. We hope to make the 20th anniversary bigger and better to honor the decades of service provided by the UArizona's Mathematics Department's Center for Recruitment and Retention of Mathematics Teachers (CRR).