Yanbei Chen (California Institute of Technology; Simons Investigator)
Chen reviewed the discovery of gravitational waves made at LIGO and the future of gravitational wave astronomy, including international ground-based observatories and space-science projects, such as LISA. He described the status of numerical general relativity and its applications to gravitational wave astronomy. He also reported on tests of modified theories of gravity via LIGO observations and discussed several quantum mechanics aspects underlying the LIGO operation.
Amit Singer (Princeton University; Simons Investigator; Algorithms and Geometry PI)
Singer talked about algorithmic aspects arising in the analysis of data from cryo-electron microscopy. The mathematical ideas concerning efficient reconstruction of 3D shapes from random 2D slices go back to the work of Goncharov and Vainshtein from 1986, but it was only recently that the computational tools reached the necessary level to be applied to these problems, especially in the presence of a high level of noise. Modern approaches are based on statistical learning approaches and led to breakthrough advances in our ability to recover fine-grained 3D structures of proteins, including their different configurations. The method of moments brings new mathematical insight and may further lead to reconstructions of even smaller proteins.
Nigel Cooper (University of Cambridge; Simons Investigator)
Cooper reviewed the theory of topological phases of matter and discussed the role of topological invariants in the context of condensed matter systems. Symmetry principles underlying novel states of matter were outlined. Last but not least, he spoke about the cold atom realizations of certain new topological phases and the rich set of dynamical effects that these quantum mechanical systems can display.
Michael Wolf (Rice University; Simons Fellow in Math)
Wolf discussed the simplest instance of a deep mathematical theory, the nonabelian Hodge theory, relating character varieties with moduli spaces of Higgs bundles on curves. His example starts with the monodromy of a flat connection on a G-bundle, realized by concrete, ordinary differential equations and leads to energy-minimizing maps, which simplify in the high-energy asymptotic regime. The limiting objects are described synthetically, displaying a unity between the gauge-theoretic and geometric-topological perspectives on these representation spaces.
Rachel Rosen (Columbia University; Origins of the Universe PI)
Rosen began her talk with a down-to-earth survey of the basic constituents of elementary particles of the standard model, deriving them from Poincare invariance of special relativity. She then explained an extension of general relativity to a massive nonlinear theory for interacting spin-two particles and discussed the high-energy behavior of such a theory. Rosen concluded her talk with a description of the fate of new types of black hole solutions that she discovered in this theory.
Lucy Colwell (University of Cambridge; Simons Investigator in MMLS)
Colwell began her lecture with remarkable examples of recent clinical applications of the structural theory of proteins (e.g., to the development of effective Ebola vaccines). The combination of deep-learning techniques with advanced lab experiments allows us to rapidly design proteins with desirable structural properties.
Scott Aaronson (University of Texas at Austin; Simons Investigator and It from Qubit PI)
Aaronson talked about a surprising new connection between a problem in quantum computation and techniques from differential privacy, a field which provides a rigorous framework for thinking about issues of privacy. The measurement of quantum states without complete collapse of the wave function in the context of quantum computation and differential privacy, in particular on establishing an effective dictionary between soft measurement in quantum computation and the field of differential privacy. It was also exciting to hear about the quantum supremacy, established by Google.
Michael Brenner (Harvard University, Simons Investigator)
Brenner talked about applications of deep-learning methods to numerical solutions of nonlinear PDEs. The main idea is to attach weights to constants appearing in numerical schemes, which are ‘learned’ by deep learning. He showed striking examples of applications of this new technique to the advection of a passive scalar by a turbulent flow.
Thursday, October 17
9:30 AM Yanbei Chen | Gravitational Wave Science: from Black Hole Physics to Macroscopic Quantum Mechanics 11:00 AM Amit Singer | Mathematics of Cryo-Electron Microscopy 1:30 PM Nigel Cooper | Topology and Dynamics in Quantum Matter 3:00 PM Michael Wolf | Limits of Geometric Structures on Surfaces 4:30 PM Rachel Rosen | Black Holes for Massive Gravitons
Friday, October 18
9:30 AM Lucy Colwell | Using Evolutionary Sequence Variation to Build Predictive Models of Protein Structure and Function 11:00 AM Scott Aaronson | Gentle Measurement of Quantum States and Differential Privacy 1:00 PM Michael Brenner | Machine Learning for PDEs
University of Texas at Austin
Gentle Measurement of Quantum States and Differential Privacy
Aaronson will discuss a recent connection between two seemingly unrelated problems: how to measure a collection of quantum states without damaging them too much (‘gentle measurement’) and how to provide statistical data without leaking too much about individuals (‘differential privacy,’ an area of classical CS). This connection leads to, among other things, a new protocol for ‘shadow tomography’ of quantum states (that is, answering a large number of questions about a quantum state given few copies of it).
Scott Aaronson has established fundamental theorems in quantum computational complexity and inspired new research directions at the interface of theoretical computer science and the study of physical systems.
Machine Learning for PDE’s
Brenner will discuss several ways in which machine learning can be used for solving and understanding the solutions of nonlinear partial differential equations.
Michael Brenner is the Michael T. Cronin Professor of Applied Mathematics and Applied Physics at the Harvard School of Engineering and Applied Sciences. His research uses mathematics to examine a wide variety of problems in science and engineering, ranging from understanding the shapes of whale flippers, bird beaks and fungal spores, to answering ordinary questions about daily life, such as why a droplet of fluid splashes when it collides with a solid surface.
California Institute of Technology
Gravitational Wave Science: From Black Hole Physics to Macroscopic Quantum Mechanics
Since the Advanced Laser Interferometer Gravitational-wave Observatory (LIGO) started operation in September 2015, gravitational waves from more than 12 pairs merging binary black holes have been observed. In August 2017, the neutron-star binary merger GW170817 was observed by both gravitational-wave and electromagnetic telescopes. Since April 2019, compact binaries are being detected at a rate of around once per week. From these initial events, we have tested basic properties of gravitational waves and black holes and have made the connection between neutron star mergers and gamma-ray bursts. Next, we will not only gather more population statics and make better measurements, but will also be testing more subtle predictions of general relativity, detecting events from the more distant/earlier universe, and search for deviations from general relativity and exotic phenomena. These ambitious goals will be achieved by new generations of ground-base detectors, as well as space-based detectors. Gravitational-wave detectors are at the frontier of precision measurement physics. Macroscopic test masses in Advanced LIGO are already being monitored continuously at a level of only several times the Heisenberg Uncertainty. Improving detector sensitivity requires considering measurement-induced back action, as well as the quantum coherence between the test masses and the measuring devices. This provides us with the opportunity to study the quantum mechanical behaviors of macroscopic objects.
Yanbei Chen made major contributions to understanding the noise of laser-interferometer gravitational-wave detectors that arise from quantum fluctuations of light and matter. He proposed conceptual interferometer designs that can achieve better sensitivity, also formulating a vision for experimentally testing quantum mechanics and quantum measurement theory on macroscopic objects. Chen made important contributions to gravitational-wave data-analysis strategies and works on using gravitational-wave observations to test the predictions of general relativity in strong gravity and to study the structures of black holes.
University of Cambridge
Topology and Dynamics in Quantum Matter
Quantum many-body systems arise in a wide range of physical settings from quark-gluon plasmas to electrons in semiconductor devices. They are known to give rise to very rich and complex forms of collective behavior, such as superconductivity and the quantum Hall effect. Much progress has been made in understanding equilibrium phases of quantum matter, through concepts of symmetry breaking and, more recently, via topological classifications. However, recent developments of experimental platforms are allowing the far-from-equilibrium dynamics of quantum many-body systems to be explored in detail. Cooper will describe some of the new theoretical issues that arise, focusing on the constraints imposed by topology on dynamical evolution.
University of Cambridge
Using Evolutionary Sequence Variation to Build Predictive Models of Protein Structure and Function
The evolutionary trajectory of a protein through sequence space is constrained by its function. A central challenge across the biological sciences is to predict the functional properties of a protein from its sequence, and thus (i) discover new proteins with specific required functionality and (ii) better understand the functional effect of changes within protein coding genes. The explosive growth in the number of available protein sequences raises the possibility of using the natural variation present in homologous protein sequences to infer these constraints and thus identify residues that control different protein phenotypes. Because in many cases phenotypic changes are controlled by more than one amino acid, the mutations that separate one phenotype from another may not be independent, requiring us to build models that take into account the correlation structure of the data. Models that have this feature are capable of (i) inference of residue pair interactions accurate enough to predict all atom 3-D structural models and predictions of (ii) binding interactions between different proteins and (iii) accurate annotation of sequence domains with as low as 20 percent identity to the training set.
Lucy Colwell has demonstrated that the 3-D structure of proteins can be determined from large sequence alignments. Her current research develops methods for relating phenotype to genotype, using large data sets from high throughput biological experiments, focusing mainly on proteins, small molecules and nucleic acids.
Mathematics of Cryo-Electron Microscopy
Single particle cryo-EM is becoming an increasingly popular technique for determining 3-D molecular structures at high resolution. We will discuss the mathematical principles for reconstruction using cryo-EM and then focus on computational challenges, in particular, reconstruction of small molecules and heterogeneity analysis.
Amit Singer is one of the leaders in the mathematical analysis of noisy data provided by cryo-EM.
Black Holes for Massive Gravitons
What would happen to a black hole if the graviton had a mass? In this talk, Rosen will review the status of black hole solutions for theories in which the gravitational force is mediated by a massive spin-2 particle. She will present the arguments that such black holes must necessarily be time dependent and will discuss the implications for black hole mechanics and for observations.
Rachel A. Rosen’s research focuses on quantum field theory and its applications to particle physics, gravitational physics and condensed matter systems. She is best known for her contributions to massive gravity, a theory in which the graviton — the spin-2 particle that transmits the gravitational force — has a mass. She has also helped to develop new techniques for studying various states of matter, from ordinary fluids to an exotic quantum liquid that could exist in the cores of certain very dense stars.
Limits of Geometric Structures on Surfaces
The study of surfaces lies at the crossroads of many subjects, and this reflects in there being a variety of geometries — hyperbolic, complex projective and affine spherical, among others — that a surface can admit. We discuss how these moduli spaces of geometric structures may by parametrized by holomorphic objects associated to variational problems and then focus on describing the singular objects that emerge when we allow the geometric surfaces to degenerate. Along the way, we meet a number of constructions central to this subject.
Wolf’s research focuses on deformations of geometric structures on surfaces, typically with applications to and from conformal optimization problems. The work often blends complex analytic quantities with synthetic constructions that reflect the qualitative features of the solutions to the extremal problems. A recent principal interest is in higher rank Teichmuller theory, which applies gauge theory to the study of representations of surface groups in Lie groups. He has past contributions to complex projective geometry as well as to classical minimal surface theory, where he and collaborators found the first embedded complete minimal surface in space with infinite total curvature but finite topology since the eighteenth-century helicoid.