The purpose of this paper is to demonstrate in a rather general setup how isoperimetric inequalities and lower bounds of the eigenvalues of the Laplacian can be derived from existence of a distance function with controllable

The ``principle of the fermionic projector'' provides a new mathematical framework for the formulation of physical theories and is a promising approach for physics beyond the standard model. The book begins with a brief review of relativity, relativistic quantum mechanics and classical gauge theories, with the emphasis on the basic physical concepts and the mathematical foundations. The external field problem and Klein's paradox are discussed and then resolved by introducing the so-called fermionic projector, a global object in space-time which generalizes the notion of the Dirac sea. The mathematical core of the book is to give a precise definition of the fermionic projector and to employ methods of hyperbolic differential equations for its detailed analysis. The fermionic projector makes it possible to formulate a new type of variational principles in space-time. The mathematical tools for the analysis of the corresponding Euler-Lagrange equations are developed. A particular variational principle is proposed which gives rise to an effective interaction showing many similarities to the interactions of the standard model. The main chapters of the book are easily accessible for beginning graduate students in mathematics or physics. Several appendices provide supplementary material which will be useful to the experienced researcher.

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In this work, we give a geometric interpretation to the Generative Adversarial Networks (GANs). The geometric view is based on the intrinsic relation between Optimal Mass Transportation (OMT) theory and convex geometry, and leads to a variational approach to solve the Alexandrov problem: constructing a convex polytope with prescribed face normals and volumes. By using the optimal transportation view of GAN model, we show that the discriminator computes the Wasserstein distance via the Kantorovich potential, the generator calculates the transportation map. For a large class of transportation costs, the Kantorovich potential can give the optimal transportation map by a close-form formula. Therefore, it is sufficient to solely optimize the discriminator. This shows the adversarial competition can be avoided, and the computational architecture can be simplified. Preliminary experimental results show the geometric method outperforms the traditional Wasserstein GAN for approximating probability measures with multiple clusters in low dimensional space.

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We present the first computational tool to help ordinary users create transforming pop-up books. In each transforming pop-up, when the user pulls a tab, an initial flat 2D pattern, i.e. a 2D shape with a superimposed picture, such as an airplane, turns into a new 2D pattern, such as a robot, standing up from the page. Given the two 2D patterns, our approach automatically computes a 3D pop-up mechanism that transforms one pattern into the other; it also outputs a design blueprint, allowing the user to easily make the final model. We also present a theoretical analysis of basic transformation mechanisms; combining these basic mechanisms allows more flexibility of final designs. Using our approach, inexperienced users can create models in a short time; previously, even experienced artists often took weeks to manually create them. We demonstrate our method on a variety of real world examples.

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Strong, electromagnetic, and weak forces were unified in the Standard Model (SM) with spontaneous gauge symmetry breaking. These forces were further conjectured to be unified in a simple Lie group gauge interaction in the Grand Unification (GUT). In this work, we propose a theory beyond the SM and GUT by adding new gapped Topological Phase Sectors consistent with the nonperturbative global anomaly cancellation and cobordism constraints (especially from the baryon minus lepton number B−L, the electroweak hypercharge Y, and the mixed gauge-gravitational anomaly). Gapped Topological Phase Sectors are constructed via symmetry extension, whose low energy contains unitary Lorentz invariant topological quantum field theories (TQFTs): either 3+1d non-invertible TQFT, or 4+1d invertible or non-invertible TQFT (short-range or long-range entangled gapped phase). Alternatively, there could also be right-handed "sterile" neutrinos, gapless unparticle physics, more general interacting conformal field theories, or gravity with topological cobordism constraints, or their combinations to altogether cancel the mixed gauge-gravitational anomaly. We propose that a new high-energy physics frontier beyond the conventional 0d particle physics relies on the new Topological Force and Topological Matter including gapped extended objects (gapped 1d line and 2d surface operators or defects, etc., whose open ends carry deconfined fractionalized particle or anyonic string excitations) or gapless conformal matter. Physical characterizations of these gapped extended objects require the mathematical theories of cohomology, cobordism, or category. Although weaker than the weak force, Topological Force is infinite-range or long-range which does not decay in the distance, and mediates between the linked worldvolume trajectories via fractional or categorical statistical interactions.

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New, “big data” sources allow measurement of city characteristics and outcomevariables at higher collection frequencies and more granular geographic scales thanever before. However, big data will not solve large urban social science questionson its own. Big urban data has the most value for the study of cities when it allowsmeasurement of the previously opaque, or when it can be coupled with exogenous shocksto people or place. We describe a number of new urban data sources and illustrate howthey can be used to improve the study and function of cities. We rst show how GoogleStreet View images can be used to predict income in New York City, suggesting thatsimilar imagery data can be used to map wealth and poverty in previously unmeasuredareas of the developing world. We then discuss how survey techniques can be improved tobetter measure willingness to pay for urban amenities. Finally, we explain how Internetdata is being used to improve the quality of city services.

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Einstein’s theory of general relativity is based on the desire to merge the newly developed theory of special relativity and Newton’s theory of gravity. He accomplished this daunting task in 1915. Most physicists consider this to be the most creative work in science in the history of mankind. This talk will explain some part of this theory.

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Let ϕ be a faithful normal semi-finite weight on a von Neumann algebra$M$. For each normal semi-finite weight ϕ on$M$, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operator$h$, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ($h$·). Conversely, each such$h$determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result that$M$is semi-finite if and only if Σ consists of inner automorphisms.

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