The spectacular proof of Fermat’s Last Theorem not only answered a famous open question in mathematics, it also introduced a powerful new technique ("The Taylor-Wiles method”) for studying problems in arithmetic. The aim of this talk is to give an overview of developments in the field since then, but also to address the larger context of what (my part) of the Langlands program is really all about and what problems it hopes to solve. The audience for this talk is decidedly supposed to be first year graduate students and mathematicians with no background in either number theory or algebraic geometry.

A spanner of a given graph is a sparse subgraph which approximately preserves distances of the original graph, up to some error. Spanners have application in many areas including graph compression, computing approximate all-pairs shortest distances, and motion planning. In this talk, we survey some theoretical results involving the size or lightness of spanners with a given error, including those on multiplicative and additive spanners, as well as recent results on additive spanners in weighted graphs. We conclude with open problems in this area.

We study the problem of decision-making in the setting of a scarcity of shared resources when the preferences of agents are unknown a priori and must be learned from data. Taking the two-sided matching market as a running example, we focus on the decentralized setting, where agents do not share their learned preferences with a central authority. Our approach is based on the representation of preferences in a reproducing kernel Hilbert space, and a learning algorithm for preferences that accounts for uncertainty due to the competition among the agents in the market. Under regularity conditions, we show that our estimator of preferences converges at a minimax optimal rate. Given this result, we derive optimal strategies that maximize agents' expected payoffs and we calibrate the uncertain state by taking opportunity costs into account. We also derive an incentive-compatibility property and show that the outcome from the learned strategies has a stability property. Finally, we prove a fairness property that asserts that there exists no justified envy according to the learned strategies. This is a joint work with Michael I. Jordan.

How the engineering question of thermal engine efficiency spawned two hundred years of ever-accelerating progress in fundamental science, communications and computations. Thermodynamic and statistical entropy. The puzzle of the second law of thermodynamics. How irreversible entropy growth appears from reversible flows in phase space via dynamical chaos. Entropy decrease and non-equilibrium fractal measures.

For more information see: https://appliedmath.arizona.edu/events/zakharov-lectureship-2022

Let $p>2$ be a prime and let $X$ be a proper smooth formal scheme over $\mathcal{O}_K$ where $K/\mathbb{Q}_p$ is a local number field. In this talk, we describe a series of conditions $(\mathrm{Cr}_s)$ that provide control on the Galois action on the Breuil--Kisin cohomology $\mathrm{R}\Gamma_{\Delta}(X/\mathfrak{S})$ inside the $A_{\inf}$--cohomology $\mathrm{R}\Gamma_{\Delta}(X_{\mathbb{C}_K}/A_{\inf})$. When $s=0$, the resulting condition is essentially the crystallinity criterion of Gee and Liu for Breuil--Kisin--Fargues $G_K$--modules, and it leads to an alternative proof of crystallinity of the $p$--adic \'{e}tale cohomology $H^i_{\mathrm{et}}(X_{\mathbb{C}_K}, \mathbb{Q}_p)$. Adapting a strategy of Caruso and Liu, the conditions $(\mathrm{Cr}_s)$ for higher $s$ then lead to upper bounds on ramification of the mod $p$ \'{e}tale cohomology $H^i_{\mathrm{et}}(X_{\mathbb{C}_K}, \mathbb{Z}/p\mathbb{Z})$, expressed in terms of $i, p$ and $e=e(K/\mathbb{Q}_p)$ that work without any restrictions on the size of $i$ and $e$.

Co-Chair: Josh Levine, Computer Science

Co-Chair: Andrew Gillette, Lawrence Livermore Laboratory

Consider a Boolean function f:{0,1,…,N-1}?{0,1}, where x?{0,1,…,N-1} is called a good element if f(x)=1 and a bad element otherwise. Estimating the proportion of good elements is one of the most fundamental problems in statistics and computer science. The error bound of the confidence interval based on the classical sample proportion scales as O(1/vn) according to the central limit theorem, where n is the number of times the function f is queried. Quantum algorithms can break this fundamental limit.  We propose an adaptive quantum algorithm for proportion estimation and prove that the error bound achieves O(1/n) up to a double-logarithmic factor. The algorithm is based on Grover’s algorithm, without the use of quantum phase estimation. We show with an empirical study that our algorithm uses a similar number of quantum queries to achieve the same level of precision compared to other algorithms, but the non-quantum part of our algorithm is significantly more efficient.

Modern data science applications aim to efficiently and accurately extract important features and make predictions from high dimensional and large data sets. Naturally occurring structure in the data underpins the success of many contemporary approaches, but large gaps between theory and practice remain. In this talk, I will present two different methods for nonparametric regression that can be viewed as a projection of a lifted formulation of the problem with a simple stochastic or convex geometric description, allowing the projection to encapsulate the data structure. In particular, I will first describe how the theory of stationary random tessellations in stochastic geometry addresses the computational and theoretical challenges of random decision forests with non-axis-aligned splits. Second, I will present a new approach to non-polyhedral convex regression that returns convex estimators compatible with semidefinite programming. These works open new questions at the intersection of stochastic and convex geometry, machine learning, and optimization.

Asymptotic equipartition as a universal mathematical tool of statistical physics and information theory. Information as a choice. Entropy as the rate of information transfer. Two greatest code inventions: alphabet and positional numeral system. Entropy of the language and the genetic code.

For more information see: https://appliedmath.arizona.edu/events/zakharov-lectureship-2022

Dendrochronology - the study of the rings in trees - allows us to reconstruct climate variability over the past ca. 2,000 years and to put current anthropogenic climate change in a long-term context. We can use tree rings to study past mean climate, but also climate extremes - such as drought, hurricanes, and wildfires - and climate dynamical patterns, such as the jet stream. In addition to this, dendrochronology sits at the nexus of climatology, ecology, and archeology and helps us to link climate history to forest history and human history.

 In my talk, I will present tree-ring based studies aimed at providing long-term records of (1) climate variability and (2) California wildfires. I will show how our century-long proxy records have improved our understanding of the interactions between the climate system, human systems, and ecosystems.

Bio:  Valerie Trouet is a Professor in the Laboratory of Tree-Ring Research at the University of Arizona. She received her PhD in Bioscience Engineering at the KULeuven in Belgium in 2004 and has worked at PennState University and at the Swiss Federal Research Institute WSL before moving to Tucson in 2011.  She is a dendrochronologist whose research focuses on past climate variability and how it has influenced human systems and ecosystems. She has published more than 80 scientific papers and is the author of Tree Story, a broad audience book about dendrochronology published by Johns Hopkins University Press in 2020 that is translated in 7 languages. She leads the ‘Spatiotemporal Interactions between Climate and Ecosystems’ research group (trouetlab.arizona.edu), is a University of Arizona Distinguished Scholar, the laureate of the 2019 Willi Dansgaard award of the American Geophysical Union, and a Kavli Fellow of the National Academy of Sciences.

We introduce the first DG methods with time-operators in their numerical traces. The time-operators are devised in such a way that the resulting semi discrete methods conserve the energy. Taking advantage of the Hamiltonian structure of Maxwell’s equations, a time-marching symplectic method is applied which renders the resulting fully discrete method energy-conserving, or with an energy which does not drift. 

This is joint work with Shukai Du and Manuel Sanchez.

Place: Hybrid: Math, 501 and
Zoom https://arizona.zoom.us/j/86997964863    
Password:   Locute

Mutual Information as a universal measure of correlation and a channel capacity. Hypothesis testing and relative entropy. Second of law of thermodynamics made trivial. Distribution from information. Exorcising Maxwell demon, Landauer limit, fluctuating particle path in a potential.

For more information see: https://appliedmath.arizona.edu/events/zakharov-lectureship-2022

“What lies at the heart of every living thing is not a spark of life. It is information.” Flies and spies. Rate distortion theory and information bottleneck. Information is money and life: proportional gambling and phenotype switching. Learning by forgetting – renormalization group.

For more information see: https://appliedmath.arizona.edu/events/zakharov-lectureship-2022

Random walks and Brownian motion. Stochastic Web surfing and Google’s PageRank. Fluctuation theorems and modern forms of the second law of thermodynamics.

For more information see: https://appliedmath.arizona.edu/events/zakharov-lectureship-2022