New publication by Sze-Man Ngai:
Abstract: For Laplacians defined by measures on a bounded domain in Rn, we prove analogs of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Polya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.
New research by Alex Teytelboym et al. presented at the 2018 Big Data Conference was recently published in Science Advances. The article, “Interpreting Economic Complexity,” explores patterns related the economic complexity index (ECI) and product complexity index (PCI). The paper can be found here.
Abstract: Two network measures known as the economic complexity index (ECI) and product complexity index (PCI) have provided important insights into patterns of economic development. We show that the ECI and PCI are equivalent to a spectral clustering algorithm that partitions a similarity graph into two parts. The measures are also closely related to various dimensionality reduction methods, such as diffusion maps and correspondence analysis. Our results shed new light on the ECI’s empirical success in explaining cross-country differences in gross domestic product per capita and economic growth, which is often linked to the diversity of country export baskets. In fact, countries with high (low) ECI tend to specialize in high-PCI (low-PCI) products. We also find that the ECI and PCI uncover specialization patterns across U.S. states and U.K. regions.
arXiv:1812.11968 [pdf, ps, other]
New Higher Anomalies, SU(N) Yang-Mills Gauge Theory and ℂℙN−1 Sigma Model
Zheyan Wan, Juven Wang
Subjects: High Energy Physics – Theory (hep-th)
We hypothesize a new and more complete set of anomalies of quantum field theories (QFTs) and then give an eclectic proof. First, we propose a set of 4d ‘t Hooft higher anomalies of time-reversal symmetric SU(N)-Yang-Mills (YM) gauge theory at θ=π, specifically at N=2,3,4 and others, by enlisting all possible 5d cobordism invariants (higher symmetry-protected topological states) and selecting the matched terms. Second, we propose a set of 3d ‘t Hooft anomalies of ℂℙN−1-sigma model obtained via compactifying YM theory on a 2-torus, and select the matched 3d cobordism invariants. Based on algebraic/geometric topology, QFT analysis, manifold generator dimensional reduction, condensed matter inputs and additional physical criteria, we derive a correspondence between 5d and 3d new invariants, thus broadly prove a more complete anomaly-matching between 4d YM and 2d ℂℙN−1 models via a 2-torus reduction. We formulate a higher-symmetry analog of “Lieb-Schultz-Mattis theorem” to constrain the low-energy dynamics. Continue reading New Publications by Juven Wang
Abstract: We prove existence of all possible bi-axisymmetric near-horizon geometries of 5-dimensional minimal supergravity. These solutions possess the cross-sectional horizon topology S3, S1×S2, or L(p,q) and come with prescribed electric charge, two angular momenta, and a dipole charge (in the ring case). Moreover, we establish uniqueness of these solutions up to an isometry of the symmetric space G2(2)/SO(4).
Abstract: The modeling of multivariate time series in an agnostic manner, without assumptions about underlying theoretical structure is traditionally conducted using Vector Auto-Regressions. They are well suited for linear and state-independent evolution. A more general methodology of Multivariate Recurrent Neural Networks allows to capture non-linear and state-dependent dynamics. This paper takes a range of small- to large-scale Long Short-Term Memory MRNNs and pits them against VARs in an application to US data on GDP growth, inflation, commodity prices, Fed Funds rate and bank reserves. Even in a small-sample regime, MRNN outperforms VAR in forecasting: its out-of-sample predictions are about 20% more accurate. MRNN also fares better in interpretability by means of impulse response functions: for instance, a temporary shock to the Fed Funds rate variable generates system dynamics that are more plausible according to conventional
In Fall 2018, CMSA will focus on a program that aims at recent mathematical advances in describing shape using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems.
The Center of Mathematical Sciences and Applications at Harvard University welcomes applications for a two-year Postdoctoral position at the intersection of mathematics and economics, with a particular preference for finance. The area of research interests is understood broadly (for example, they may include but are not limited to asset pricing and corporate finance, macro-finance and monetary economics, operations research and financial engineering , economic theory and game theory, industrial organization and market design, econometrics and machine learning). That said, there is some preference for candidates working with modern empirical methods in addition to theory.
Ravi Jagadeesan, an economic design fellow at CMSA and frequent collaborator with Center Affiliate Scott Kominers, is the recipient of the 2019 AMS-MAA-SIAM Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student. Ravi’s “fundamental contributions across several topics in pure and applied mathematics, including algebraic geometry, statistical theory, mathematical economics, number theory, and combinatorics” have qualified him for this prize. Jagadeesan is currently a PhD student in Business Economics at Harvard. He graduated from Harvard with an A.B. in Mathematics and an A.M. in Statistics in Spring 2018. In his response he thanked the Center of Mathematical Sciences and Applications for support.