The MCAIM Colloquium is a distinguished lecture series, featuring international experts presenting their latest research results to a broad mathematical and scientific audience at the University of Michigan. Scholars are invited to speak on campus during the Fall and Winter semesters. Follow the hyperlinks below for more information.

*WINTER SEMESTER 2024*

**May 24, 2024****Jae Kyoung Kim, Korea Advanced Institute of Science & Technology (KAIST)****Title:** Inference of Dynamic Networks in Biological Systems**Abstract:** Biological systems are complex dynamic networks. In this talk, I will introduce GOBI (General Model-based Inference), a simple and scalable method for inferring regulatory networks from time-series data. GOBI can infer gene regulatory networks and ecological networks that cannot be obtained with previous causation detection methods(e.g., Granger, CCM, PCM). I will then introduce Density-PINN (Physics-Informed NeuralNetwork), a method for inferring the shape of the time-delay distribution of interactions in a network. The inferred shape of time-delay distribution can be used to identify the number of pathways that induce a signaling response against antibiotics, which solves the long-standing mystery, the major source of cell-to-cell heterogeneity in response to stress. Finally, I will talk how to infer the dynamic information from just network structure information, which can be used to identify the targets (nodes) perturbing the homeostasis of the systems.

[1] Jo H, Hong H, Hwang HJ, Chang W, Kim JK, Density Physics-Informed Neural Network identifies sources of cell heterogeneity in signal transduction under antibiotic stress, Cell Patterns (2024)

[2] Park SH, Ha S, Kim JK, A general model-based causal inference overcomes the curse of synchrony and indirect effect, Nature Communications (2023)

[3] Hirono Y, Moon SH, Hong H, Kim JK, Robust Perfect Adaptation of Reaction Fluxes Ensured by Network Topology, Arxiv (2023)

**April 12, 2024****Boyce Griffith, University** **of North Carolina, Chapel Hill****Title:** Immersed methods for fluid-structure interaction**Abstract:** The immersed boundary (IB) method is a framework for modeling systems in which an elastic structure interacts with a viscous incompressible fluid. The fundamental feature of the IB approach to such fluid-structure interaction (FSI) problems is its combination of an Eulerian formulation of the momentum equation and incompressibility constraint with a Lagrangian description of the structural deformations and resultant forces. In conventional IB methods, Eulerian and Lagrangian variables are linked through integral equations with Dirac delta function kernels, and these singular kernels are replaced by regularized delta functions when the equations are discretized for computer simulation. This talk will focus on three related extensions of the IB method. I first detail an IB approach to structural models that use the framework of large-deformation nonlinear elasticity. I will focus on efficient numerical methods that enable finite element structural models in large-scale simulations, with examples focusing on models of the heart and its valves. Next, I will describe an extension of the IB framework to simulate soft material failure using peridynamics, which is a nonlocal structural mechanics formulation. Numerical examples demonstrate constitutive correspondence with classical mechanics for non-failure cases along with essentially grid-independent predictions of fluid-driven soft material failure. Finally, I will introduce a reformulation of the IB large-deformation elasticity framework that enables accurate and efficient fluid-structure coupling through a version of the immersed interface method, which is a sharp-interface IB-type method. Computational examples demonstrate the ability of this methodology to simulate a broad range of fluid-structure mass density ratios without suffering from artificial added mass instabilities, and to facilitate subgrid contact models. I will also present biomedical applications of the methodology, including models of clot capture by inferior vena cava filters.*This colloquium is co-sponsored by the AIM Seminar Series and Michigan Institute for Computational Discovery and Engineering (MICDE).*

Boyce Griffith’s talk is available to view online here.

**January 24, 2024****Nicholas Boffi, New York University****Title:** On Flows and Diffusions: From the Many-body Fokker-Planck Equation to Stochastic Interpolants**Abstract:** Given a stochastic differential equation, its corresponding Fokker-Planck equation is generically intractable to solve because its high dimensionality prohibits the application of standard numerical techniques. In this talk, I will exploit an analogy between the Fokker-Planck equation and modern generative models from machine learning to develop an algorithm for its solution in high dimension. The method enables the computation of previously intractable quantities of interest, such as the entropy production rate of active matter systems, which quantifies the magnitude of nonequilibrium effects. I will then highlight how insight from the Fokker-Planck equation facilitates the development of a new class of generative models known as stochastic interpolants, which generalize state of the art diffusion models in several key ways that can be leveraged to improve practical performance. Along the way, I will argue that methods from machine learning offer a compelling solution for many fascinating high-dimensional mathematical problems that are currently out of reach with more traditional computational tools.

Nicholas Boffi’s talk is available to view online here.

*FALL SEMESTER 2023*

**November 10, 2023****Don Estep, Simon Fraser University****Title:** Cars, Steaks, and Hurricanes: A General Bayesian Approach to Inverse Problems**Abstract:** The inverse problem of determining information about the state of a physical system from observations of its behavior is fundamental to scientific inference and engineering design. Frequently, this can be formulated as computing a probability measure on physical characteristics of a system from observed data on the output of a model of system behavior. In abstract terms, this is the empirical stochastic inverse problem for a random vector on a probability space with an unknown probability measure. Over the last fifteen years, collaborators and I have developed a general Bayesian approach to the formulation and solution of this problem. Our approach has a solid theoretical foundation that avoids alterations of the model like regularization as well as unrealistic and limiting assumptions about prior knowledge of system characteristics, allows for numerical solution by a novel importance sampling approach, and provides a platform to address critical issues arising in the practical application to scientific and engineering problems. I will lay out the theoretical and computational foundation of our approach with the details motivated by practical applications including optimizing car mileage, cooking steaks, hurricane storm surge forecasting, and forecasting COVID surges. Time permitting, I will discuss the relationship with common Bayesian statistics.

Don Estep’s talk is available to view online here.

**November 9, 2023****MCAIM Series on Algorithm Startups: A Conversation with Olivia Walch, Arcascope****Abstract:** Many do not realize the potential of mathematical and algorithmic ideas to have a societal impact through the commercial space. Mathematicians often receive advice tailored for other areas, such as building physical objects, consulting, or general software development, that may not apply. Additionally, startups with sophisticated mathematical, statistical, or engineering algorithms may encounter unique problems. In this new colloquium series, startup CEOs will discuss the story of their companies from seed to IPO stages and offer advice on building successful businesses based on mathematics or algorithms.

Dr. Walch is the CEO of Arcascope, which makes science-backed apps to help people fix their sleep and circadian rhythms. She received her Ph.D in Applied Mathematics from the University of Michigan in 2016 and studied the mathematics of sleep for the last ten years. Outside of sleep, she was co-editor of Political Geometry, a volume on the mathematics of gerrymandering, with Moon Duchin. Her webcomic, Imogen Quest, is available online, as is my mini-comic You Can Try Again. Learn more on her personal web site.

The conversation with Olivia Walch is available to view online here.*Co-sponsored by MCAIM, Michigan Innovation Partnerships, and the Van Loo Symposium.*

**October 25, 2023****Manas Rachh, Flatiron Institute****Title: **Edge Effects at Insulator Interfaces, Acoustic Simulations, and Modeling Optics in the Miniwasp Eye**Abstract:** The numerical simulation of wave scattering processes plays a critical role in chip and antenna design, radar cross section determination, biomedical imaging, wireless communications, and the development of new meta-materials and better waveguides to name a few. In order to enable design by simulation for problems arising in these applications, automatically adaptive solvers which resolve the complexity of the geometry and the input data, and combining these tools with neural networks to navigate the complicated design-optimization landscape play a critical role. On the adaptive solvers side, this has been made possible through the development of high-order integral equation methods which rely on well-conditioned integral representations, efficient quadrature formulas, and coupling to fast algorithms. In this talk, I will discuss the development and application of these tools for a) studying edge effects at insulator interfaces where waves tend to propagate at the interface of two insulating media (and are evanescent in the bulk); b) an industry collaboration with Meyer Sound for optimizing the design of horn-loaded speakers; and c) studying light propagation in an insect eye.

**October 11, 2023****MCAIM Series on Algorithm Startups: A Conversation with Eric Schadt, Icahn School of Medicine at Mount Sinai****Abstract:** Many do not realize the potential of mathematical and algorithmic ideas to have a societal impact through the commercial space. Mathematicians often receive advice tailored for other areas, such as building physical objects, consulting, or general software development, that may not apply. Additionally, startups with sophisticated mathematical, statistical, or engineering algorithms may encounter unique problems. In this new colloquium series, startup CEOs will discuss the story of their companies from seed to IPO stages and offer advice on building successful businesses based on mathematics or algorithms.

Dr. Schadt is an expert on the generation and integration of very large-scale sequence variation, molecular profiling and clinical data in disease populations for constructing molecular networks that define disease states and link molecular biology to physiology. Dr. Schadt was founder and CEO of Sema4, a predictive health company creating tools to better diagnose, treat, and prevent disease. Prior to joining Mount Sinai in 2011, he was Chief Scientific Officer at Pacific Biosciences. Previously, Dr. Schadt was Executive Scientific Director of Genetics at Rosetta Inpharmatics, a subsidiary of Merck & Co., and before Rosetta, Dr. Schadt was a Senior Research Scientist at Roche Bioscience. He received his B.A. in applied mathematics and computer science from California Polytechnic State University, his M.A. in pure mathematics from University of California, Davis, and his Ph.D. in bio-mathematics from University of California, Los Angeles (requiring Ph.D. candidacy in molecular biology and mathematics).

The conversation with Eric Schadt is available to view online here.*Co-sponsored by MCAIM, the Department of Computational Medicine and Bioinformatics (DCMB), Michigan Innovation Partnerships, and the Van Loo Symposium*

**October 6, 2023****Rishi Sonthalia**, UCLA**Title:** From Classical Regression to the Modern Regime: Surprises for Linear Least Squares Problems**Abstract:** Linear regression is a problem that has been extensively studied. However, modern machine learning has brought to light many new and exciting phenomena due to overparameterization. In this talk, I briefly introduce the new phenomena observed in recent years. Then, building on this, I present recent theory work on linear denoising. Despite the importance of denoising in modern machine learning and ample empirical work on supervised denoising, its theoretical understanding is still relatively scarce. One concern about studying supervised denoising is that one might not always have noiseless training data from the test distribution. It is more reasonable to have access to noiseless training data from a different dataset than the test dataset. Motivated by this, we study supervised denoising and noisy-input regression under distribution shift. We add three considerations to increase the applicability of our theoretical insights to real-life data and modern machine learning. First, we assume that our data matrices are low-rank. Second, we drop independence assumptions on our data. Third, the rise in computational power and dimensionality of data have made it essential to study non-classical learning regimes. Thus, we work in the non-classical proportional regime, where data dimension $d$ and number of samples N grow as d/N = c + o(1).

For this setting, we derive general test error expressions for both denoising and noisy-input regression and study when overfitting the noise is benign, tempered, or catastrophic. We show that the test error exhibits double descent under general distribution shifts, providing insights for data augmentation and the role of noise as an implicit regularizer. We also perform experiments using real-life data, matching the theoretical predictions with under 1% MSE error for low-rank data.

Rishi Sonthalia’s talk is available to view online here.

**September 28, 2023****Caroline Terry, Ohio State University****Title:** Measuring Combinatorial Complexity Via Regularity Lemmas**Abstract:** A major theme in combinatorics is understanding the structure of graphs with forbidden subgraphs. This can be phrased by asking, given some local combinatorial restriction in a graph, what are the global implications? Are there special local restrictions which yield very strong information about global structure? These kinds of questions are also studied in model theory, but with a focus on the infinite setting.

Many tools have been developed in combinatorics to study global structure in finite graphs. One such tool is called Szemer\'{e}di’s regularity lemma, which gives a structural decomposition for any large finite graph. Beginning with work of Alon-Fischer-Newman, Lov\'{a}sz-Szegedy, and Malliaris-Shelah, it has been shown over the last 15 years that regularity lemmas can be used to detect structural dichotomies in graphs, and that these dichotomies always have deep connections to model theory. In this talk, I present extensions of this type of result to arithmetic regularity lemmas, which are analogues of graph regularity lemmas, tailored to the study of combinatorial problems in finite groups. This work uncovered tight connections between tools from additive combinatorics, and ideas from the model theoretic study of infinite groups.

**September 20, 2023****Sanjukta Krishnagopal, UC Berkeley & UCLA****Title:** The Theory and Application of Networks: From Mathematical Machine Learning to Simplicial Complexes**Abstract:** Networks are ubiquitous in nature and appropriate for mathematical investigation of various systems. In this talk I will discuss some aspects at the intersection of mathematics, machine learning, and networks to introduce interdisciplinary methods with wide application. First, I will discuss some recent advances in mathematical machine learning for prediction on graphs. Machine learning is often a black box. Here I will present some exact theoretical results on the dynamics of weights while training graph neural networks using graphons – a limiting function of a graph with infinitely many nodes. Next, I will use these ideas to present a new method for early prediction of disease subtype, characterized by dynamic co-evolution of multiple variables, with remarkable success in prediction of Parkinson’s subtype five years in advance. Then, I will discuss some work on higher-order models of graphs: simplicial complexes – that can capture simultaneous many-body interactions. I will present some results on spectral theory of simplicial complexes, as well as introduce a mathematical framework for studying the topology and dynamics of multilayer simplicial complexes using Hodge theory, applied to brain connectome data. Finally, I will discuss applications of such interdisciplinary methods to studying bias in society, opinion dynamics, hate speech propagation in social media, and extreme mountaineering.

Sanjukta Krishnagopal’s talk is available to view online here.

**September 5, 2023****Xiuyuan Cheng, Duke University****Title:** Gaussian Kernelized Graph Laplacian: Bi-Stochastic Normalization and Eigen-Convergence**Abstract:** Eigen-data of graph Laplacian matrices are widely used in data analysis and machine learning, such as dimension reduction by spectral embedding. Many graph Laplacian methods start by building a kernelized affinity matrix from high-dimensional data points, which may lie on some unknown low-dimensional manifolds embedded in the ambient space. When clean manifold data are corrupted by high dimensional noise, it can negatively influence the performance of graph Laplacian methods. In this talk, we first introduce the use of bi-stochastic normalization to improve the robustness of graph Laplacian to high-dimensional outlier noise, possibly heteroskedastic, with a proven convergence guarantee under the manifold data setting. Next, for the important question of eigen-convergence (namely the convergence of eigenvalues and eigenvectors to the spectra of the Laplace-Beltrami operator), we show that choosing a smooth kernel function leads to improved theoretical convergence rates compared to prior results. The proof is by analyzing the Dirichlet form convergence and constructing candidate approximate eigenfunctions via convolution with the manifold heat kernel. When data density is non-uniform on the manifold, we prove the same rates for the density-corrected graph Laplacian. The theory is supported by numerical results. Joint work with Boris Landa and Nan Wu.

*WINTER SEMESTER 2023*

**April 12, 2023Gunther Uhlmann, University of WashingtonTitle: **The Calderon Problem: 40 Years Later

**Abstract:**Calderon’s inverse problem asks whether can one determine the conductivity of a medium by making voltage and current measurements at the boundary. This question arises in several areas of applications including medical imaging and geophysics. I will report on some of the progress that has been made on this problem since Calderon proposed it, including recent developments on similar problems for nonlinear equations and nonlocal operators.

Gunther Uhlmann’s talk is available to view online here.

**April 5, 2023Sharon Glotzer, University of Michigan Chemical EngineeringTitle:** Packings, Tilings and Assemblies of Shapes

**Abstract:**Packings and tilings of shapes have been of interest to mathematicians, physicists, and puzzlers for millenia. Certain shapes can pack densely to tile space in 2D or 3D. Others cannot tile space, and thus the arrangements that allow them to pack most densely (but with packing fraction necessarily less than one) are of interest for fundamental as well as practical reasons. At lower packing fractions than those that produce densest packings, statistical thermodynamics permits the ordering of many shapes into periodic and aperiodic structures by self-assembly. For some shapes, these ordered assemblies are structurally identical to the shape’s (putative) densest packing, but more interesting, and possibly more common, is when the assemblies and densest packings are different. Here we discuss the packing, tiling and self-assembly of polyhedra and polygons, including the curious case of the tetrahedron, where the simplest 3D shape assembles into one of the most complex ordered structures – a quasicrystal.

Sharon Glotzer’s talk is available to view online here.

**February 8, 2023Naoki Masuda, University at BuffaloTitle:** Modeling and Analyzing Temporal Networks and Dynamics on Them

**Abstract:**Two salient features of empirical temporal (i.e., time-varying) network data are the time-varying nature of network structure itself and heavy-tailed distributions of inter-contact times. Both of them can strongly impact dynamical processes occurring on networks, such as contagion processes, synchronization dynamics, and random walks. In the first part of the talk, I introduce theoretical explanation of heavy-tailed distributions of inter-contact times by state-dynamics modeling approaches in which each node is assumed to switch among a small number of discrete states in a Markovian manner and the nodes’ states determine time-dependent edges. This approach is interpretable, facilitates mathematical analyses, and seeds various related mathematical modeling, algorithms, and data analysis (e.g., theorizing on epidemic thresholds, random walks on metapopulation models, inference of mixtures of exponential distributions, new Gillespie algorithms, embedding of temporal network data), some of which we will also discuss. The second part of the talk is on modeling of temporal networks by static networks that switch from one to another at regular time intervals. This approach facilitates analytical understanding of diffusive and epidemic dynamics on temporal networks as well as an efficient algorithm for containing epidemic spreading as convex optimization. Finally, I will touch upon some of my interdisciplinary collaborations including those on static networks.

Naoki Masuda’s talk is available to view online here.

**FALL SEMESTER 202***2*

**November 9, 2022Philipp Schoenhoefer, University of Michigan Chemical EngineeringTitle:** Hard Particle Self-Assembly From the Perspective of Geometric Frustration

**Abstract:**It is well established that the appearance and properties of self-assembled structures are affected by the geometry of their constituents. This is especially true for hard polyhedrally shaped particles, which interact solely via excluded volume to form a plethora of entropically stabilized crystal structures. Yet, a priori prediction of these structures is non-trivial for anything but the simplest of space-filling shapes, such as cubes, especially when the thermodynamically preferred structure differs from the densest packing structure. By sufficiently curving space, however, we can eliminate the geometric constraints that prevent polyhedra from forming locally dense packings and theoretically create tessellations for all regular polyhedra. Using Monte Carlo simulations, we show that most hard polyhedra belonging to the family of Platonic solids can self-assemble into space-filling crystal structures when constrained to the surface of a hypersphere. By increasing the hypersphere radius to gradually flatten space, we introduce geometric frustration that prevents the particles from tessellating the hypersphere, and inevitably introduces defects. Lastly, we compare systems assembled in curved and flat space by applying different local environment metrics and show that all the observed assemblies of Platonic shapes in Euclidean space can be interpreted as shadows of tessellations and defects on the hypersphere.

**October 26, 2022Nima Arkani-Hamed, Institute for Advanced StudyTitle:** Spacetime, Quantum Mechanics and Positive Geometry At Infinity

**Abstract:**The past decade has seen the emergence of surprising new connections between the real-world physics of elementary particle

scattering processes, and simple new mathematical structures in combinatorics, algebra and geometry. These ideas provide, in a number

of examples, a different starting point for conceptualizing physics, where the fundamental principles of spacetime and quantum mechanics are not taken as primary, but instead emerge from a more primitive mathematical rubric. In this talk I will illustrate these ideas in their simplest settings, showing in a variety of examples how basic particle scattering processes are autonomously determined by “positive geometries”, which are natural generalization of simplices and polygons. The rules of spacetime and quantum mechanics emerge from the combinatorial geometry seen in the boundary structure of these spaces. This talk will be entirely self-contained; no previous knowledge of either the relevant physics or mathematics will be assumed or needed.

Nima Arkani-Hamed’s talk is available to view online here.

**October 12, 2022Ben Rossman, Duke UniversityTitle:** Symmetric Models of Computation

**Abstract:**Many well-studied Boolean functions {0,1}^n → {0,1} are naturally invariant under some group P acting on coordinates 1,…,n. For example, Graph Connectivity on m-vertex graphs, viewed as a function {0,1}^(m choose 2) → {0,1}, is invariant under the action of Sym(m). It is interesting to investigate the complexity of P-invariant functions in symmetric models of computation, where each state of a computation is itself a P-invariant object. In this talk, I will discuss results concerning two symmetric models: P-invariant Boolean circuits (joint work with William He) and the

*choiceless*computation model of Blass-Gurevich-Shelah 1999.

**October 5, 2022Robert Wald, The University of ChicagoTitle:** Quantum Field Theory in Curved Spacetime

**Abstract:**Quantum field theory in curved spacetime is a theory wherein matter is treated fully in accord with the principles of quantum field theory but gravity is treated classically in accord with general relativity. It is not expected to be an exact theory of nature, but it should provide a good approximate description in circumstances where the quantum effects of gravity itself do not play a dominant role, and it has provided us with fundamental insights into phenomena such as those involving black holes. This talk will give an introduction to quantum field theory in curved spacetime, with emphasis on the conceptual and mathematical issues arising in the formulation of the theory.

Robert Wald’s talk is available to view online here.

**WINTER SEMESTER 202***2*

**April 13, 2022Abhay Ashtekar, The Pennsylvania State UniversityTitle:** Emergence of General Relativity from a (Diffeomorphism Invariant) Gauge Theory

**Abstract:**Space-time metric and Einstein’s equations lie at the foundation of general relativity. Somewhat surprisingly, one can start with an SU(2)-gauge theory, write down the simplest equations one can, without making reference to any background field, and show that the Riemannian geometry and Einstein’s equations emerge by setting up an appropriate dictionary. Equations of the `fundamental’ gauge theory are simple low order polynomials. Complexity of Einstein’s equations can be traced back to the fact that the explicit dictionary from the gauge theory to general relativity is rather intricate. The gauge theory framework provides new insights into general relativity, including an interesting interplay between the Lie algebra of volume-preserving diffeomorphisms and the ‘integrable’ sector of (anti-)self-dual solutions to Einstein’s equations. It also leads to a new infinite dimensional Lie algebra that generalizes the Lie algebra of the diffeomorphism group, opening up directions for new mathematical work. I will make a special eort to make the summary of these structures and results accessible to mathematicians as well as physicists.

Abhay Ashtekar ‘s talk is available to view online here.

**March 30, 2022Hayden Schaeffer, Carnegie Mellon UniversityTitle:** Concentration and Conditioning of Random Feature Matrices

**Abstract:**Being able to write down a mathematical model, i.e. a system of governing equations, for complex physical systems is a fundamental problem in science and engineering and is seeing assistance and automation from the machine learning perspective. As data-driven scientific discovery increases in popularity so does the need for rigorous algorithmic and theoretical work. In this talk, I will discuss a sparse random feature method with applications to learning equations from data. I will provide an overview of our theoretical results on the concentration of these random feature matrices, the connections to generalization and complexity bounds, and the design and applications of the method. Examples and applications to high-dimensional modeling and dynamics will be included.

**March 16, 2022Dmitry Chelkak, École normale superiéure, ParisTitle:** Planar Ising Model: Convergence Results on Regular Grids and S-Embeddings of Irregular Graphs Into R^{2,1}

**Abstract:**In the first part of the talk we briefly discuss convergence results obtained during the last fifteen years for the critical Ising model on the square lattice. This research program was started by Smirnov in the mid-2000s and is based on the discrete holomorphicity of fermionic observables. Smirnov’s ideas were later developed by a number of authors including the speaker, similar results were also obtained for the near-critical model and for the Z-invariant model on rhombic lattices. However, until very recently it was unclear what a generalization of these techniques for irregular graphs should look like. In the second part of the talk we discuss a new tool: the so-called s-embeddings of planar graphs carrying the Ising model into the Minkowski space R^{2,1}. These embeddings can be thought of as a certain analogue of classical Tutte’s harmonic embeddings and, among other things, naturally lead to an appearance of quasi-conformal mappings in the planar Ising model context.

Dmitry Chelkak’s talk is available online here.

**FALL SEMESTER 2021**

**November 10, 2021Jacob Bedrossian, University of MarylandTitle:** Hydrodynamic Stability at High Reynolds Number

**Abstract:**The stability of equilibria solutions of the incompressible Euler and Navier-Stokes equations at high Reynolds number has been studied since the 1800s with the work of Kelvin, Rayleigh, Reynolds and others. However, only in recent years have we started to get a firm mathematical understanding of this field, even for the deceptively simple case of shear flows and vortices. I will outline some of the many recent advances in the area, including inviscid damping, enhanced dissipation, subcritical transition, vortex axi-symmetrization, and the local well-posedness of vortex filaments.

Jacob Bedrossian’s talk is available to view online here.

**November 3, 2021****Chris Rycroft, Harvard UniversityTitle:** Uncovering the Rules of Crumpling with a Data-Driven Approach

**Abstract:**When a sheet of paper is crumpled, it spontaneously develops a network of creases. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility. Recent experiments have shown that when a sheet is repeatedly crumpled, the total crease length grows logarithmically [1]. This talk will offer insight into this surprising result by developing a correspondence between crumpling and fragmentation processes. We show how crumpling can be viewed as fragmenting the sheet into flat facets that are outlined by the creases, and we use this model to reproduce the characteristic logarithmic scaling of total crease length, thereby supplying a missing physical basis for the observed phenomenon [2].

This study was made possible by large-scale data analysis of crease networks from crumpling experiments. We will describe recent work to use the same data with machine learning methods to probe the physical rules governing crumpling. We will look at how augmenting experimental data with synthetically generated data can improve predictive power and provide physical insight [3]

*[1] O. Gottesman et al., Commun. Phys. 1, 70 (2018).*

*[2] J. Andrejevic et al., Nat. Commun. 12, 1470 (2021).*

*[3] J. Hoffmann et al., Sci. Advances 5, eaau6792 (2019)*

Chris Rycroft’s talk is available to view online here.

**September 15, 2021Gérard Ben Arous, The Courant Institute , New York UniversityTitle:** 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**

**December 2, 2020****Josselin Garnier, Ecole Polytechnique, FranceTitle: **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/abs/2009.03704

**October 21, 2020 ****Monica Valluri, University of Michigan Dept. AstronomyTitle: **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.