The schedule below will be updated as speakers are confirmed.
|9/26/2018||Xiao-Gang Wen (MIT)||Title: A classification of low dimensional topological orders and fully extended TQFTs
Abstract: In this talk, I will review the recent progress on classification of gapped phases of quantum matter (ie topological orders) in 1,2, and 3 spatial dimensions for boson systems. In 1-dimension, there is no non-trivial topological orders. In 2-dimensions, the topological orders are classified by modular tensor category theory. In 3-dimensions, the topological orders are classified by a simple class of braided fusion 2-categories. The classification of topological orders may correspond to a classification of fully extended unitary TQFTs.
|10/03/2018||Richard Schoen (Stanford)||Title: Perspectives on the scalar curvature
Abstract: This will be a general talk concerning the role that the scalar curvature plays in Riemannian geometry and general relativity. We will describe recent work on extending the known results to all dimensions, and other issues which are being actively studied.
|10/10/2018||Justin Solomon (MIT)||Title: Correspondence and Optimal Transport for Geometric Data Processing
Abstract: Correspondence problems involving matching of two or more geometric domains find application across disciplines, from machine learning to computer vision. A basic theoretical framework involving correspondence along geometric domains is optimal transport (OT). Dating back to early economic applications, the OT problem has received renewed interest thanks to its applicability to problems in machine learning, computer graphics, geometry, and other disciplines. The main barrier to wide adoption of OT as a modeling tool is the expense of optimization in OT problems. In this talk, I will summarize efforts in my group to make large-scale transport tractable over a variety of domains and in a variety of application scenarios, helping transition OT from theory to practice. In addition, I will show how OT can be used as a unit in algorithms for solving a variety of problems involving the processing of geometrically-structured data.
|10/17/2018||Jeremy England (MIT)||Title: Wisdom of the Jumble
Abstract: There are certain, specific behaviors that are particularly distinctive of life. For example, living things self-replicate, harvest energy from challenging environmental sources, and translate experiences of past and present into actions that accurately anticipate the predictable parts of their future. What all of these activities have in common from a physics standpoint is that they generally take place under conditions where the pronounced flow of heat sharpens the arrow of time. We have therefore sought to use thermodynamics to understand the emergence and persistence of life-like phenomena in a wide range of messy systems made of many interacting components.
In this talk I will discuss some of the recent insights we have gleaned from studying emergent fine-tuning in disordered collections of matter exposed to complexly patterned environments. I will also point towards future possible applications in the design of new, more life-like ways of computing that have the potential to either be cheaper or more powerful than existing means.
|10/31/2018||Moon Duchin (Tufts)||Title: Exploring the (massive) space of graph partitions
Abstract: The problem of electoral redistricting can be set up as a search of the space of partitions of a graph (representing the units of a state or other jurisdiction) subject to constraints (state and federal rules about the properties of districts). I’ll survey the problem and some approaches to studying it, with an emphasis on the deep mathematical questions it raises, from combinatorial enumeration to discrete differential geometry to dynamics.
|11/14/2018||Dusa McDuff (Columbia)||Title: The virtual fundamental class in symplectic geometry
Abstract: Essential to many constructions and applications of symplectic geometry is the ability to count J-holomorphic curves. The moduli spaces of such curves have well understood compactifications, and if cut out transversally are oriented manifolds of dimension equal to the index of the problem, so that they a fundamental class that can be used to count curves. In the general case, when the defining equation is not transverse, there are various different approaches to constructing a representative for this class, We will discuss and compare different approaches to such a construction e.g. using polyfolds or various kinds of finite dimensional reduction. Most of this is joint work with Katrin Wehrheim.
|11/19/2018||Xiaoqin Wang (Johns Hopkins)||Title: Computational Principles of Auditory Cortex
Abstract: Auditory cortex is located at the top of a hierarchical processing pathway in the brain that encodes acoustic information. This brain region is crucial for speech and music perception and vocal production. Auditory cortex has long been considered a difficult brain region to study and remained one of less understood sensory cortices. Studies have shown that neural computation in auditory cortex is highly nonlinear. In contrast to other sensory systems, the auditory system has a longer pathway between sensory receptors and the cerebral cortex. This unique organization reflects the needs of the auditory system to process time-varying and spectrally overlapping acoustic signals entering the ears from all spatial directions at any given time. Unlike visual or somatosensory cortices, auditory cortex must also process and differentiate sounds that are externally generated or self-produced (during speaking). Neural representations of acoustic information in auditory cortex are shaped by auditory feedback and vocal control signals during speaking. Our laboratory has developed a unique and highly vocal non-human primate model (the common marmoset) and quantitative tools to study neural mechanisms underlying audition and vocal communication.
|11/28/2018||Robert Haslhofer (University of Toronto)||Title: Recent progress on mean curvature flow
Abstract: A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow is the most natural evolution in extrinsic geometry and shares many features with Hamilton’s Ricci flow from intrinsic geometry. In the first half of the talk, I will give an overview of the well developed theory in the mean convex case, i.e. when the mean curvature vector everywhere on the surface points inwards. Mean convex mean curvature flow can be continued through all singularities either via surgery or as level set solution, with a precise structure theory for the singular set. In the second half of the talk, I will report on recent progress in the general case without any curvature assumptions. Namely, I will describe our solution of the mean convex neighborhood conjecture and the nonfattening conjecture, as well as a general classification result for all possible blowup limits near spherical or cylindrical singularities. In particular, assuming Ilmanen’s multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed. This is joint work with Kyeongsu Choi and Or Hershkovits.
|12/5/2018||Robert McCann (University of Toronto)||Title: Displacement convexity of Boltzmann’s entropy characterizes positive energy in general relativity
Abstract: Einstein’s theory of gravity is based on assuming that the fluxes of a energy and momentum in a physical system are proportional to a certain variant of the Ricci curvature tensor on a smooth 3+1 dimensional spacetime. The fact that gravity is attractive rather than repulsive is encoded in the positivity properties which this tensor is assumed to satisfy. Hawking and Penrose (1971) used this positivity of energy to give conditions under which smooth spacetimes must develop singularities. By lifting fractional powers of the Lorentz distance between points on a globally hyperbolic spacetime to probability measures on spacetime events, we show that the strong energy condition of Hawking and Penrose is equivalent to convexity of the Boltzmann-Shannon entropy along the resulting geodesics of probability measures. This new characterization of the strong energy condition on globally hyperbolic manifolds also makes sense in (non-smooth) metric measure settings, where it has the potential to provide a framework for developing a theory of gravity which admits certain singularities and can be continued beyond them. It provides a Lorentzian analog of Lott, Villani and Sturm’s metric-measure theory of lower Ricci bounds, and hints at new connections linking gravity to the second law of thermodynamics.
Preprint available at http://www.math.toronto.edu/mccann/papers/GRO.pdf
|12/12/2018||Zhiwei Yun (MIT)||Title: Shtukas: what and why
Abstract: This talk is of expository nature. Drinfeld introduced the notion of Shtukas and the moduli space of them. I will review how Shtukas compare to more familiar objects in geometry, how they are used in the Langlands program, and what remains to be done about them.
|1/30/2019||Richard Freeman (Harvard)||Title: TBA|
|2/6/2019||Ulrich Mueller (Princeton)||Title: TBA|
|2/13/2019||Christian Santangelo (UMass Amherst)||Title: TBA|
|2/20/2019||Michael Woodford (Columbia)||Title: TBA|
|2/27/2019||Ian Martin (LSE)||Title: TBA|
For information on previous CMSA colloquia, click here.