Abstract: Lagrangian fibrations play a crucial role in the study of hyper-Kaehler geometry and integrable systems. The P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Lagrangian fibrations and Hodge theory. In this talk, we will first discuss a compact version of this phenomenon, based on joint work with Andrew Harder, Zhiyuan Li, and Qizheng Yin. Then we will focus on interactions between compact and noncompact hyper-Kaehler geometry. Such connections lead to new progress on the P=W conjecture for Hitchin systems and character varieties. This is joint work with Mark de Cataldo and Davesh Maulik.
If time permits, I will further discuss the connection between P=W and Gopakumar-Vafa invariants for local curves.
Abstract: The Weil-Petersson metric for the moduli space or Riemann surfaces is Kaehler, incomplete with negative sectional curvature. The metric is a tool for understanding the geometry of the moduli space. Applications include a proof that the compactified moduli space is projective, a solution of the Nielsen Realization Problem and a CAT(0) geometry. The behavior of the sectional curvature is an ingredient in the Burns-Masur-Wilkinson proof that the geodesic flow is ergodic. A non trivial upper bound is required to study the mixing rate of the geodesic flow. I will present the first non trivial upper bound for the sectional curvature and discuss the optimal expected bound.
Abstract: Fingerprints are commonly understood as traits that uniquely identify an individual, an object, or a message, and can be exploited to detect and prevent impersonation, fraud, or unlawful duplication. In this talk we consider the intentional introduction of fingerprints to provide security in wireless communications. This addresses the fingerprint design, and its embedding into a communications waveform, so that it has several desired properties including stealth, security, and predictable performance. The framework draws on communications, signal processing, cryptographic hashing, and information theory, enabling control of performance trade-offs by design. Privacy and security analysis quantify the limited ability of an eavesdropper to detect and estimate the fingerprint or to impersonate a legitimate user. Fingerprints provide a message, and a secret codebook design is described that enables secure side-channel communications through fingerprint coding.
Abstract: In the lecture I will discuss recent joint work with Richard Bamler, which uses Ricci flow through singularities to construct deformations of spaces of metrics on 3-manifolds. We show that the space of metrics with positive scalar on any 3-manifold is either contractible or empty; this extends earlier work by Fernando Marques, which proved path-connectedness. We also show that for any spherical space form M, the space of metrics with constant sectional curvature is contractible. This completes the proof of the Generalized Smale Conjecture, and gives a new proof of the original Smale Conjecture for S^3.
Abstract: The collection of solutions of discrete parameter-dependent partial differential equations often takes the form of a low-rank object. We show that in this scenario, iterative algorithms for computing these solutions can take advantage of this structure to reduce both computational effort and memory requirements. Implementation of such solvers requires that explicit rank-compression computations be done to truncate the ranks of intermediate quantities that must be computed. We prove that when truncation strategies are used as part of a multigrid solver, the resulting algorithms retain "textbook" (grid-independent) convergence rates and we demonstrate how the truncation criteria affect convergence behavior. In addition, we show that these techniques can be used to construct efficient solution algorithms for computing the eigenvalues of parameter-dependent operators.
Abstract: How can two parties who are at a distance from one another carry out a fair coin flip? This problem has been an object of study for decades. Previous research has shown that coin flipping with positive bias (i.e., coin-flipping which permits a limited amount of cheating) is possible if the parties share a quantum information channel. Yet, proving quantum coin-flipping with bias tending to zero is mysteriously hard -- the only known protocol requires more than an exponential amount of communication.
This talk will be about a new result which untangles the mystery of why quantum coin-flipping is difficult. We will draw on some unexpected topics in mathematics (including the behavior of rational functions on complex numbers). Since this talk is aimed for a broad audience, the first half will be spent covering fundamental concepts and giving a detailed description of the central problem. The talk is based on https://arxiv.org/abs/1909.10103 .
Abstract: We consider the problem of data-driven forecasting of chaotic dynamical systems when the available data is from a sparse spatial sampling, i.e., the full state of the dynamical system cannot be observed directly. Recently, there have been several promising data-driven approaches to forecasting of chaotic dynamical systems using machine learning. Particularly promising among these are hybrid approaches that combine machine learning with a knowledge-based model, where a machine learning technique is used to correct the imperfections in the knowledge-based model. Such a hybrid approach is promising when a knowledge-based model is available but is imperfect due to incomplete understanding of the physical processes in the underlying dynamical system. However, previously proposed data-driven forecasting approaches assume knowledge of the full state of the dynamical system. We seek to relax this assumption by using a data assimilation technique along with Machine Learning in a novel technique that improves forecasts. We demonstrate that using partial measurements of the state of the dynamical system we can train a machine learning model to correct model error in an imperfect knowledge-based model.
Abstract: Dr. Franco received her PhD in Applied Mathematics at the University of Maryland under the advisement of Dr. Abram Kagan. Her thesis focused primarily on the asymptotic properties of semi-parametric estimators. She is now working as a Research Mathematical Statistician at the Center for Statistical Research and Methodology (CSRM) at the U.S. Census Bureau where she works on small area estimation and sampling. Her talk will include an overview of her academic and professional journey, as well as time for Q&A.
Abstract: Let $(X, \sigma)$ be a subshift and $\textrm{Aut}(X)$ be the automorphism group, the group of self conjugacies of $(X, \sigma)$. The mapping class group, denoted $\mathcal{M})(\sigma)$, is the group of self flow equivalences up to isotopy. We show that $\mathcal{M}(\sigma)$ is constrained in the case of low-complexity minimalsubshifts, similar to constraints on $\textrm{Aut}(X)$. We classify mapping class group for Sturmians, and also a generic set of codings of IETs.
Abstract: Wind tunnel tests are crucial to the design of tall structures. Scale models
are outfitted with pressure taps at many locations of interest, such as the
center of the roof. Each tap measures pressure at one location for the
duration of the test. Since the tap measurement is recorded at regular time
intervals, the data produced form a regular time series. Wind engineers are
typically concerned with very high and very low (suction) pressures.
Peaks-over-threshold (POT) extreme value models are one of two main approaches
used by wind engineers for these data. However, POT models require the choice
of a threshold, which can influence the final results, sometimes substantially,
because the threshold choice controls the data that enter into the analysis.
In this talk a method for combining results from multiple thresholds is
considered, thereby eliminating the need to choose only one. The focus is on
estimating the distribution of the maximum or minimum value in wind tunnel
tests. The new method is compared to several techniques for choosing a single
threshold using a large collection of pressure series from wind tunnel tests.
The comparison shows that choosing a single threshold underestimates the
uncertainty associated with predicting a future peak value.