MCAIM Colloquiums meet Wednesdays at 4pm

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**FALL SEMESTER 2021
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**September 15, 2021
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**Gérard Ben Arous, The Courant Institute , New York University**

**Title:**Topological Complexity and Optimization of High Dimensional Random Functions

**Abstract:**Smooth random functions of very many variables can be topologically very complex, and thus it can be terribly hard to find their minimum. One does not need to look very far for such an example: pick at random a homogeneous polynomial of degree p (with p larger than 3) of a large number of variables and restrict it to the (high-dimensional) unit sphere. Important examples of such functions include many Hamiltonians of statistical mechanics in disordered media (as Spin Glasses or Random Interfaces in high disorder). They can also include the loss functions of high dimensional inference problems, and naturally the landscapes defined by Machine Learning.

We will cover some of the recent progress in our understanding of both questions: the statics or geometric question about the topological complexity and the transition to simple landscapes (the so-called topological trivialization), as well as the dynamics and optimization questions.

**WINTER SEMESTER 2021**

**April 14, 2021**

**Sheperd S. Doeleman, Founding Director of the Event Horizon Telescope, Harvard University, Center for Astrophysics, Harvard and Smithsonian, Black Hole Initiative**

**Title:** Black Hole Imaging: First Results and Future Vision

**Abstract: **In April 2017, the Event Horizon Telescope (EHT) carried out a global Very Long Baseline Interferometry (VLBI) observing campaign at a wavelength of 1mm that led to the first resolved image of a supermassive black hole. For the 6.5 billion solar mass black hole in the giant elliptical galaxy M87, the EHT estimated the spin orientation and constrained models of accretion on Schwarzschild radius scales. This work relied on two decades of technical advances in ultra-high resolution interferometry and theoretical General Relativistic Magnetohydrodynamic (GRMHD) simulations. This talk will review these advances and recent new EHT results. We will also look to the next decade when a next-generation EHT (ngEHT) that doubles the number of participating radio dishes in the VLBI network will enable time-lapse movies of M87 that link the black hole to the relativistic jet it powers. For SgrA*, the Galactic Center black hole that evolves on time scales 1000 times faster, ngEHT will produce real-time video.

**April 7, 2021**

**Mete Soner, Princeton University, Department of Operations Research and Financial Engineering**

**Title:** Deep Neural Networks for High-dimensional Uncertain Decision Problems

**Abstract: ** Stochastic optimal control has been an effective tool for many decision problems. Although, they provide the much needed quantitative modeling for such problems, until recently they have been numerically intractable in high-dimensional settings. However, several recent studies that use deep neural networks report impressive numerical results in high dimensions when the structure of the uncertainty is assumed to be known. The main tool is a Monte-Carlo type algorithm combined with deep neural networks proposed by Han, E and Jentzen. In this talk, I will outline this approach and discuss its properties; in particular, the difficulties that data-driven problems face as opposed to model-driven ones. Numerical results, while validating the power of the method in high dimensions, they also show the dependence on the dimension and the size of the training data. This is joint work with Max Reppen of Boston University.

**March 31, 2021**

**Bärbel Finkenstädt Rand, University of Warwick, Department of Statistics**

**Title:** Inference for Circadian Pacemaking

**Abstract:** Organisms have evolved an internal biological clock which allows them to temporally regulate and organize their physiological and behavioral responses to cope in an optimal way with the fundamentally periodic nature of the environment. It is now well established that the molecular genetics of such rhythms within the cell consist of interwoven transcriptional-translational feedback loops involving about 15 clock genes, which generate circa 24-h oscillations in many cellular functions at cell population or whole organism levels. We will present statistical methods and modelling approaches that address newly emerging large circadian data sets, namely spatio-temporal gene expression in SCN neurons and rest-activity actigraph data obtained from non-invasive e-monitoring, both of which provide unique opportunities for furthering progress in understanding the synchronicity of circadian pacemaking and address implications for monitoring patients in chronotherapeutic healthcare.

**March 10, 2021**

**Corinna Ulcigrai, University of Zurich, Institute for Mathematics**

**Title:** Slowly Chaotic Behavior

**Abstract: **How can we understand chaotic behavior mathematically? A well popularized feature of chaotic systems is the butterfly effect: a small variation of initial conditions may lead to a drastically different future evolution, a mechanism at the base of the so-called ‘deterministic chaos’. We will introduce and focus on ‘slowly chaotic’ dynamical systems’, for which the butterfly effect happens “slowly” (e.g. at polynomial speed). These include many fundamental examples coming from physics, such as the Ehrenfest billiard and the Novikov model of electrons in a metal. In the talk we will survey some of the recent advances in our understanding of their typical chaotic features as well as common mechanisms for chaos.

**FALL SEMESTER 2020**

**Josselin Garnier, Ecole Polytechnique, France
Title: **Passive Imaging and Communication

**Abstract:**In this talk we consider the propagation of waves transmitted by ambient noise sources.

We discuss a generalized Helmholtz-Kirchhoff identity that derives from Green’s identity and Sommerfeld radiation condition. The inspection of this identity makes it possible to design passive imaging methods, i.e., imaging methods using only passive receiver arrays and ambient noise illumination. More surprisingly, it is also possible to design an original passive communication scheme between two passive arrays that uses only ambient noise illumination. The passive transmitter array does not transmit anything but it is a tunable metamaterial surface that can modulate its scattering properties and encode a message in the modulation.

**November 18, 2020**

**Gigliola Staffilani, Massachusetts Institute of Technology (MIT)
Title:** The Many Faces of Dispersive Equations.

**Abstract:**In recent years great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a variety of techniques from Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of problems connected with dispersive and wave equations, such as the derivation of a certain nonlinear Schrodinger equation from a quantum many-particles system, periodic Strichartz estimates, the concept of energy transfer, the invariance of a Gibbs measure associated to an infinite dimension Hamiltonian system and non-squeezing theorems for such systems when they also enjoy a symplectic structure.

**October 28, 2020**

**Martin Lesourd, Harvard University**

**Title:** On the Formation of Black Holes in General Relativity.

**Abstract:** We describe what is known – including some recent progress – and the major outstanding conjectures concerning the formation of black holes in general relativity. The recent progress part of the talk will be about recent joint work with Nikos Athanasiou https://arxiv.org/

**October 21, 2020 **

**Monica Valluri, University of Michigan Dept. Astronomy
Title: **The Dynamical Inference of the Properties of Dark Matter Halos

**Abstract:**Dark Matter is thought to constitute about 85% of the matter in the Universe. The inference of its properties is largely based on astronomical observations of normal matter (stars and gas) in the outskirts of galaxies and on comparisons of observations with cosmological simulations. I will give a brief overview of the astrophysical evidence for dark matter on various scales. I will then describe what cosmological simulations predict regarding the properties of the dark matter halos that galaxies are embedded in. Finally, I will describe the dynamical modeling and simulation methods that are being used to model the 3-dimensional motions of stars in order to characterize the properties of the Milky Way’s dark matter halo.