This quarter

Spring 2019

Robert Adler, "Topology and Probability, in the service of the most humungous data set of all"
Thursday May 23, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. There are data sets that are normal sized, data sets that are large, and there are those that qualify as “Big Data”. And then there are the humungous ones, the humungousest of which is The Universe; i.e. cosmological data. Cosmological data sets are so overwhelming that, if you look hard enough, you can find almost anything that you want (or do not want) to find in them, and this calls for the development of powerful techniques that, on the one hand, summarize the data in a meaningful way and, on the other hand, allow for distinguishing between important physical phenomena and those which `you just happened to see by chance’. Abstract mathematical concepts from the rather esoteric area of Algebraic Topology provide a set of techniques, for the first of these problems, while Probability Theory provides the second set. I will explain how all all three of these topics - Cosmology, Algebraic Topology, and Probability Theory - fit together, although I will not assume any prior knowledge of any of them. (i.e. Lots of pictures, hardly any formulae, and definitely not even an attempt at a proof (despite the fact that a lot of the fun is in the proofs).

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Elliot Lipnowski, "Incomplete Information and Coordination"
Thursday May 16, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. We'll talk about how a little bit of uncertainty can have extreme consequences for strategic interactions. Then, we'll show how an organization can leverage this idea to resolve coordination issues in teams.

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James Evans, "Embeddings for the Science of Science and Society"
Thursday April 25, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. Here I explore the use of Euclidean, hyperbolic and mixed auto-encoder and parametric embeddings for the purpose of understanding human culture, language, scientific discovery and social networks. I begin with the case of human culture, and how dimensions induced by word differences (e.g., man – woman, rich – poor, black – white, liberal – conservative) in these vector spaces closely correspond to dimensions of cultural meaning, and the projection of words onto these dimensions reflects widely shared cultural connotations when compared to surveyed responses and labeled historical data. I show how nonparametric subsample and bootstrap approaches can reveal the stability of these associations, and then demonstrate these methods in a longitudinal analysis of the coevolution of class and gender associations in the United States and Great Britain over the 20th century. Then I use embeddings to explore similarities and differences across the world's languages, which reveal that while languages tend to have similar semantic clusters, with more concrete concepts tending to be clustered the most consistently, those clusters are networked in radically different ways around the world, mapping out different organizations of meaning. Then I exemplify the use of hyperbolic embeddings for the purpose of recovering not social and semantic dimensions, but hierarchies in data on 21st Century physics. Finally, I explore the concepts of geometric curvature applied to social networks, and the meaning and potential for embedding networks with mixed positive, negative and neutral curvature for mapping out the social and cultural universes in ways resonant with our modern understanding of the physical universe.

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Daniil Rudenko, "Elliptic Curves and Applications"
Thursday April 18, 2019 — 6 PM @ Ryerson 255
Abstract. I will talk about elliptic curves, one of the most simple and rich mathematical object. You encounter it, when you play “Snake 2” on Nokia, snack a donut or try to prove the Fermat’s Last Theorem. Also it might help to encrypt your proof afterwards.

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Alexander Bogatskiy, "Random Matrices and Their Applications"
Thursday April 11, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. We will discuss how the classic Central Limit Theorem for uncorrelated events is not the only fundamental universal statistical law appearing in nature. Random Matrix Theory (a.k.a. "noncommutative probability theory") lies on the intersection of the theory of integrable systems, probability theory, asymptotic analysis, complex analysis, orthogonal polynomials... It sheds light on many extremely relevant statistical phenomena, from nuclear physics to zeta functions. Using the Gaussian Unitary Ensemble as the simplest example, we will clarify the connections between Random Matrices and orthogonal polynomials, and sketch the derivations of the basic statistical properties of the eigenvalues.

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Kyle Kawagoe, "The Toric Code: A Mathematical Introduction to Topological Physics"
Thursday April 4, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. Exactly solvable spin models serve as crucial examples for studying topological physics. In addition to being physically relevant for applications such as quantum computing, these models elegantly blend discrete and continuous mathematics. In this lecture, we will explore the simplest non-trivial example: the Toric Code.

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Lecture archive

Winter 2019

Christopher Henderson, "Population dynamics: traveling waves and invasion speeds"
Thursday Feb. 28th, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. Most people are familiar with a number of examples of invasive species, that is, when a new species is introduced to some habitat. In Chicago, for example, there are the Asian carp in the Chicago river and zebra mussels in Lake Michigan. In this talk, I will go over the derivation of some basic PDE models for these invasions. The focus will be on developing mathematical tools to understand the influence of populations to various factors (e.g., the drift of a river current, genetic variation, geographic features that "slow down" or "speed up" the species).

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Guillaume Bal, "From coefficients to solutions and back"
Thursday Feb. 21st, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. We will talk about the forward map, which to known coefficients and sources associates solutions of equations. We will then talk about the inverse map, which roughly corresponds to the reverse direction.

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Aaron Dinner, "Accelerating stochastic simulations with stratified sampling"
Thursday Feb. 13th, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. A problem in computational modeling in all fields is that averages can involve contributions from relatively rare states and thus be slow to converge. Stratified sampling is a long-standing approach to addressing this problem: essentially, one samples from rare and common populations independently and then combines the information from the different populations with appropriate weights. I will talk about a recent mathematical framework that extends the power of this approach significantly and show applications in bio-molecular simulations.

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David Bindel, "The Many Applications of Eigenvalues"
Thursday Feb. 7th, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. Eigenvalue analysis is one of the power tools in applied mathematics. Apart from the central role in the theory of linear dynamical systems, eigenvalue problems are among the few families of nonlinear equations and non-convex optimization problems that we know how to solve quickly and reliably on a computer. In this talk, we describe three different applications where eigenvalue analysis plays a central role: the analysis of resonant micro-electro-mechanical systems (MEMS), game-theoretic models of opinion formation, and the spectral analysis of networks. In the process, we explore perturbations, approximations, nonlinearities, and the many other ways in which eigenvalue analysis touches on a wide range of mathematical topics.

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Savdeep S. Sethi, "String Theory and Physical Mathematics"
Thursday Jan. 24th, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. String theory and mathematics have enjoyed a wonderful productive interplay over the past few decades. I will roughly sketch how physical mathematics uses physical reasoning to arrive at often unexpected mathematical conjectures. I’ll also try to discuss a couple of mathematics questions of current interest to the string landscape.

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Rebecca Willett, "Nonlinear Models for Matrix Completion"
Thursday Jan. 17th, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. The past decade of research on matrix completion has shown it is possible to leverage linear dependencies to impute missing values in a low-rank matrix. However, the corresponding assumption that the data lies in or near a low-dimensional linear subspace is not always met in practice. Extending matrix completion theory and algorithms to exploit low-dimensional nonlinear structure in data will allow missing data imputation in a far richer class of problems. In this talk, I will describe how models of low-dimensional nonlinear structure can be used for matrix completion. In particular, we will explore matrix completion in the context of unions of subspaces, in which data points lie in or near one of several subspaces, and nonlinear algebraic varieties, a polynomial generalization of classical linear subspaces. Low Algebraic-Dimension Matrix Completion (LADMC) is a novel and efficient method for imputing missing values and admits new bounds on the amount of missing data that can be accurately imputed. Theproposed algorithms are able to recover synthetically generated data up to predicted sample complexity bounds and outperform standard low-rank matrix completion in experiments with real motion capture data. This is joint work with Daniel Pimentel-Alarcon, Gregory Ongie, Laura Balzano, and Robert Nowak.

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Mary Silber, "How my research in pattern formation led to some virtual travel via Google Maps"
Thursday Jan. 10th, 2019 — 6 PM @ The Barn (Ryerson 352)
Abstract. Studying how spatial patterns form, spontaneously, in nonlinear dynamical systems has a long tradition in physics and applied math, and takes a lot of its inspiration from nature. I will describe an example of this from my own research, where we investigated large-scale patterns of vegetation in dry-land ecosystems that self-organize, in response to aridity stress, at a community scale. We investigated the dynamics, or time-evolution, of the vegetation on timescales of decades by aligning modern satellite images with old aerial survey photographs from the 1950s. Our armchair explorations were so much fun for me. I’ll show a lot of pictures from our virtual travels, and describe some of the mathematics that led us to take these journeys.

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Autumn 2018

David McAllester, "Universality in Deep Learning and Models of Computation"
Nov. 29th, 2018 — 6 PM @ The Barn (Ryerson 352)
Abstract. Deep learning trains circuits to perform sophisticated functions such as speech recognition, machine translation, or playing chess or go. In spite of its success, deep learning has been described as alchemy. Deep learning methods have largely evolved through empirical experimentation and folklore rather than analytical design. This talk will argue that theoretical design is likely impossible. Instead we should be trying to understand principles behind effective general purpose (Turing complete) programming language features (deep architectures). A variety of specific theorems and architectures will be discussed.

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Sidney Nagel, "Fluids Meets Applied Math"
Nov. 15th, 2018 — 6 PM @ The Barn (Ryerson 352)
Abstract. Many complex phenomena are so familiar that we hardly realize that they defy our normal intuition. Examples include the anomalous flow of granular material, the long messy tendrils left by honey spooned from one dish to another, the pesky rings deposited by spilled coffee on a table after the liquid evaporates or the common splash of a drop of liquid onto a countertop. Aside from being uncommonly beautiful to see, many of these phenomena involve non-linear behavior where the system is far from equilibrium. Although most of the world we know is beyond description by equilibrium theories, we are still only at the threshold of learning how to deal with such deep and complex behavior.

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Shmuel Weinberger, "Topological Robotics"
Nov. 8th, 2018 — 6 PM @ The Barn (Ryerson 352)
Abstract. One of the goals of robotics is creating of autonomous robots. Such robots should be able to accept high-level descriptions of tasks and execute them without further human intervention. Using concepts from Topology, we can understand and address a few of the problems that arise in the endeavor to create such robots.

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Gregory Lawler, "The Self-Avoiding Random Walk"
Nov. 1st, 2018 — 6 PM @ The Barn (Ryerson 352)
Abstract. A Self-Avoiding walk (SAW) is a path on the integer lattice that never returns to any point. SAWs arose as a model for polymers. Many of the original problems are still open but there has been a lot learned by mathematicians and physicists in trying to understand them. I will describe the model and some of the mathematics that arises from it such as: why are SAWs easy to understand in high dimensions but not in two or three dimensions?

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Panagiotis Souganidis, "From Applications, to Mathematics, to Applications"
Oct. 25th, 2018 — 6 PM @ The Barn (Ryerson 352)
Abstract. I plan to describe, using simple examples, that the paradigm of applied mathematics is the close interaction and feedback that connects applications, modeling and mathematics.

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Mihai Anitescu, "Mathematical and Computational Challenges in Energy Systems"
Oct. 18th, 2018 — 6 PM @ The Barn (Ryerson 352)
Abstract. The electrical power grid (the electricity transmission and distribution system) is one of the most complex engineering achievements of the 20th century. It is also at the center of massive changes in the way we create and consume energy. A distinguishing feature of power grid applications is that optimization is ubiquitous and that it must accommodate simultaneously multiple complexity drivers. These include not only discrete variables, non-convexity, or stochasticity but also ordinary and, with the increased usage of natural gas, partial differential equations. We outline a number of existing and emerging fundamental research challenges and discuss some recent promising avenues in the area. We will discuss in some detail the issue of long-horizon dynamic optimization problems that appear in planning problems and stochastic optimization problems that accommodate the uncertainty stemming from massive penetration of renewable resources.

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