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Fall 2020 Colloquium, Wednesdays

Beginning immediately, until at least December 31, all seminars will take place virtually, through Zoom. 

The 2020-2021 Colloquium will take place every Wednesday from 9:00 to 10:00am ET virtually, using zoom. All CMSA postdocs/members are required to attend the weekly CMSA Members’ Seminars, as well as the weekly CMSA Colloquium series. Please email the seminar organizers to obtain a link. This year’s colloquium will be organized by Wei Gu and Sergiy Verstyuk. The schedule below will be updated as speakers are confirmed.

To learn how to attend, please fill out this form.

Information on previous colloquia can be found here.

9/23/2020David Kazhdan (Hebrew University)Title: On Applications of Algebraic Combinatorics to Algebraic Geometry 

Abstract: I present a derivation of a number of  results on morphisms of a high Schmidt’s rank from a result in Algebraic Combinatorics. In particular will explain the flatness of such morphisms and show their fibers have rational singularities.

Mariangela Lisanti (Princeton University)

Title: Mapping the Milky Way’s Dark Matter Halo with Gaia 

Abstract: The Gaia mission is in the process of mapping nearly 1% of the Milky Way’s stars—-nearly a billion in total.  This data set is unprecedented and provides a unique view into the formation history of our Galaxy and its associated dark matter halo.  I will review results based on the most recent Gaia data release, demonstrating how the evolution of the Galaxy can be deciphered from the stellar remnants of massive satellite galaxies that merged with the Milky Way early on.  This analysis is an inherently “big data” problem, and I will discuss how we are leveraging machine learning techniques to advance our understanding of the Galaxy’s evolution.  Our results indicate that the local dark matter is not in equilibrium, as typically assumed, and instead exhibits distinctive dynamics tied to the disruption of satellite galaxies.  The updated dark matter map built from the Gaia data has ramifications for direct detection experiments, which search for the interactions of these particles in terrestrial targets.
10/14/2020Gil Kalai (Hebrew University and IDC Herzliya)

Title: Statistical, mathematical, and computational aspects of noisy intermediate-scale quantum computers 

Abstract: Noisy intermediate-scale quantum (NISQ) Computers hold the key for important theoretical and experimental questions regarding quantum computers. In the lecture I will describe some questions about mathematics, statistics and computational complexity which arose in my study of NISQ systems and are related to
a) My general argument “against” quantum computers,
b) My analysis (with Yosi Rinott and Tomer Shoham) of the Google 2019 “quantum supremacy” experiment.
Relevant papers:
Yosef Rinott, Tomer Shoham and Gil Kalai, Statistical aspects of the quantum supremacy demonstration, https://gilkalai.files.

Gil Kalai, The Argument against Quantum Computers, the Quantum Laws of Nature, and Google’s Supremacy Claims, https://gilkalai.files.

Gil Kalai, Three puzzles on mathematics, computations, and games, https://gilkalai.files.

10/21/2020Marta Lewicka (University of Pittsburgh)

Title: Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models

Abstract: We propose results that relate the following two contexts:
(i) Given a Riemann metric G on a thin plate, we study the question of what is its closest isometric immersion, with respect to the distance measured by energies E^h which are modifications of the classical nonlinear three-dimensional elasticity.
(ii) We perform the full scaling analysis of E^h, in the context of dimension reduction as the plate’s thickness h goes to 0, and derive the Gamma-limits of h^{-2n}E^h for all n. We show the energy quantization, in the sense that the even powers 2n of h are the only possible ones (all of them are also attained).
For each n, we identify conditions for the validity of the corresponding scaling, in terms of the vanishing of Riemann curvatures of G up to appropriate orders, and in terms of the matched isometry expansions. Problems that we discuss arise from the description of elastic materials displaying heterogeneous incompatibilities of strains that may be associated with growth, swelling, shrinkage, plasticity, etc. Our results display the interaction of calculus of variations,
geometry and mechanics of materials in the prediction of patterns and shape formation.
10/28/2020Jonathan Heckman (University of Pennsylvania)

Title: Top Down Approach to Quantum Fields

Abstract: Quantum Field theory (QFT) is the common language of particle physicists, cosmologists, and condensed matter physicists. Even so, many fundamental aspects of QFT remain poorly understood. I discuss some of the recent progress made in understanding QFT using the geometry of extra dimensions predicted by string theory, highlighting in particular the special role of seemingly “exotic”  higher-dimensional supersymmetric QFTs with no length scales known as six-dimensional superconformal field theories (6D SCFTs). We have recently classified all examples of such 6D SCFTs, and are now using this to extra observables from strongly correlated systems in theories with more than four spacetime dimensions, as well as in spacetimes with four or fewer spacetime dimensions. Along the way, I will also highlight the remarkable interplay between physical and mathematical structures in the study of such systems
9:00pm ET
Surya Ganguli (Stanford)

Title: Weaving together machine learning, theoretical physics, and neuroscience through mathematics

Abstract: An exciting area of intellectual activity in this century may well revolve around a synthesis of machine learning, theoretical physics, and neuroscience.  The unification of these fields will likely enable us to exploit the power of complex systems analysis, developed in theoretical physics and applied mathematics, to elucidate the design principles governing neural systems, both biological and artificial, and deploy these principles to develop better algorithms in machine learning.  We will give several vignettes in this direction, including:  (1) determining the best optimization problem to solve in order to perform regression in high dimensions;  (2) finding exact solutions to the dynamics of generalization error in deep linear networks; (3) developing interpretable machine learning to derive and understand state of the art models of the retina; (4) analyzing and explaining the origins of hexagonal firing patterns in recurrent neural networks trained to path-integrate; (5) delineating fundamental theoretical limits on the energy, speed and accuracy with which non-equilibrium sensors can detect signals
Selected References:
M. Advani and S. Ganguli, Statistical mechanics of optimal convex inference in high dimensions, Physical Review X, 6, 031034, 2016.
M. Advani and S. Ganguli, An equivalence between high dimensional Bayes optimal inference and M-estimation, NeurIPS, 2016.
A.K. Lampinen and S. Ganguli, An analytic theory of generalization dynamics and transfer learning in deep linear networks, International Conference on Learning Representations (ICLR), 2019.
H. Tanaka, A. Nayebi, N. Maheswaranathan, L.M. McIntosh, S. Baccus, S. Ganguli, From deep learning to mechanistic understanding in neuroscience: the structure of retinal prediction, NeurIPS 2019.
S. Deny, J. Lindsey, S. Ganguli, S. Ocko, The emergence of multiple retinal cell types through efficient coding of natural movies, Neural Information Processing Systems (NeurIPS) 2018.
B. Sorscher, G. Mel, S. Ganguli, S. Ocko, A unified theory for the origin of grid cells through the lens of pattern formation, NeurIPS 2019.
Y. Bahri, J. Kadmon, J. Pennington, S. Schoenholz, J. Sohl-Dickstein, and S. Ganguli, Statistical mechanics of deep learning, Annual Reviews of Condensed Matter Physics, 2020.  
S.E. Harvey, S. Lahiri, and S. Ganguli, A universal energy accuracy tradeoff in nonequilibrium cellular sensing, https://arxiv.org/abs/2002.10567
11/11/2020Kevin Buzzard?(Imperial College London)

Title: Teaching proofs to computers

Abstract: A mathematical proof is a sequence of logical statements in a precise language, obeying some well-defined rules. In that sense it is very much like a computer program. Various computer tools have appeared over the last 50 years which take advantage of this analogy by turning the mathematical puzzle of constructing a proof of a theorem into a computer game. The newest tools are now capable of understanding some parts of modern research mathematics. In spite of this, these tools are not used in mathematics departments, perhaps because they are not yet capable of telling mathematicians *something new*.
I will give an overview of the Lean theorem prover, showing what it can currently do. I will also talk about one of our goals: using Lean to make practical tools which will be helpful for future researchers in pure mathematics.
11/18/2020Jose A. Scheinkman?(Columbia)

Title: Re-pricing avalanches

Abstract: Monthly aggregate price changes exhibit chronic fluctuations but the aggregate shocks that drive these fluctuations are often elusive.  Macroeconomic models often add stochastic macro-level shocks such as technology shocks or monetary policy shocks to produce these aggregate fluctuations. In this paper, we show that a state-dependent  pricing model with a large but finite number of firms is capable of generating large fluctuations in the number of firms that adjust prices in response to an idiosyncratic shock to a firm’s cost of price adjustment.  These fluctuations, in turn, cause fluctuations  in aggregate price changes even in the absence of aggregate shocks. (Joint work with Makoto Nirei.)

Eric J. Heller?(Harvard)

Title: Branched Flow

Abstract: In classical and quantum  phase space flow, there exists a regime of great physical relevance that is belatedly but rapidly generating a new field. In  evolution under smooth, random, weakly deflecting  but persistent perturbations, a remarkable regime develops, called branched flow. Lying between the first cusp catastrophes at the outset, leading to fully chaotic  statistical flow much later, lies the visually beautiful regime of branched flow.  It applies to tsunami wave propagation, freak wave formation, light propagation, cosmic microwaves arriving from pulsars, electron flow in metals and devices, sound propagation in the atmosphere and oceans, the large scale structure of the universe, and much more. The mathematical structure of this flow is only partially understood, involving exponential instability coexisting with “accidental” stability. The flow is qualitatively universal, but this has not been quantified.  Many questions arise, including the scale(s) of the random medium,  and the time evolution of manifolds and “fuzzy” manifolds in phase space.  The classical-quantum (ray-wave)  correspondence in this flow is only partially understood.  This talk will be an introduction to the phenomenon, both visual and mathematical, emphasizing unanswered questions
12/2/2020Douglas Arnold?(U of Minnesota)

Title: Preserving geometry in numerical discretization

Abstract: An important design principle for numerical methods for differential equations is that the discretizations preserve key geometric, topological, and algebraic structures of the original differential system.  For ordinary differential equations, such geometric integrators were developed at the end of the last century, enabling stunning computations in celestial mechanics and other applications that would have been impossible without them.  Since then, structure-preserving discretizations have been developed for partial differential equations.  One of the prime examples has been the finite element exterior calculus or FEEC, in which the structures to preserve are related to Hilbert complexes underlying the PDEs, the de Rham complex being a canonical example.  FEEC has led to highly successful new numerical methods for problems in fluid mechanics, electromagnetism, and other applications which relate to the de Rham complex.  More recently, new tools have been developed which extend the applications of FEEC far beyond the de Rham complex, leading to progress in discretizations of problems from solid mechanics, materials science, and general relativity.
12/9/2020Manuel Blum and Lenore Blum (Carnegie Mellon)

Title: What can Theoretical Computer Science Contribute to the Discussion of Consciousness?

Abstract: The quest to understand consciousness, once the purview of philosophers and theologians, is now actively pursued by scientists of many stripes. We study consciousness from the perspective of theoretical computer science. This is done by formalizing the Global Workspace Theory (GWT) originated by cognitive neuroscientist Bernard Baars and further developed by him, Stanislas Dehaene, and others. We give a precise formal definition of a Conscious Turing Machine (CTM), also called Conscious AI, in the spirit of Alan Turing’s simple yet powerful definition of a computer. We are not looking for a complex model of the brain nor of cognition but for a simple model of (the admittedly complex concept of) consciousness.
After formally defining CTM, we give a formal definition of consciousness in CTM. We then suggest why the CTM has the feeling of consciousness. The reasonableness of the definitions and explanations can be judged by how well they agree with commonly accepted intuitive concepts of human consciousness, the range of related concepts that the model explains easily and naturally, and the extent of the theory’s agreement with scientific evidence

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