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Register a new userModels from the 19th Century used for Visualizing Optical Phenomena and Line Geometry
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David E. Rowe
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The main focus of this paper is on models of quartic surfaces, especially so-called complex surfaces. These are special fourth-degree surfaces that Julius Plucker introduced in the 1860s for visualizing the local structure of a quadratic line complex. Pluckers complex surfaces turned out to be closely related to Kummer surfaces and both of these types of quartics are examples of caustic surfaces, which arise in geometrical optics. Indeed, Kummer surfaces represent a natural generalization of the wave surface, first introduced by Augustin Fresnel to explain double refraction in biaxial crystals.
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David Mumford made groundbreaking contributions in many fields, including the pure mathematics of algebraic geometry and the applied mathematics of machine learning and artificial intelligence. His work in both fields influenced my career at several key moments.
Scientific research is and was at all times a transnational (global) activity. In this respect, it crosses several borders: national, cultural, and ideological. Even in times when physical borders separated the scientific community, scientists kept their minds open to the ideas created beyond the walls and tried to communicate despite all the obstacles. An example of such activities in the field of physics is the travel in the year 1838 of a group of three scientists through the Western Europe: Andreas Ettingshausen (professor at the University of Vienna), August Kunzek (professor at the University of Lviv) and P. Marian Koller (director of the observatory in Chremsminster, Upper Austria). 155 years later a vivid scientific exchange began between physicists from Austria and Ukraine, in particular, between the Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine in Lviv and the Institute for Theoretical Physics of Johannes Kepler University Linz. This became possible due to the programs financed by national institutions, but it had its scientific background in already knotted historic scientific networks, when Lviv was an international center of mathematics and in Vienna the School of Statistical Thought arose. Due to the new collaboration, after the breakup of the Soviet Union, Ukraine became the first country to join the Middle European Cooperation in Statistical Physics (MECO) founded in the early 1970s with the aim of bridging the gap between scientists from the Eastern and Western parts of Europe separated by the iron curtain.
Transformer architectures show significant promise for natural language processing. Given that a single pretrained model can be fine-tuned to perform well on many different tasks, these networks appear to extract generally useful linguistic features. A natural question is how such networks represent this information internally. This paper describes qualitative and quantitative investigations of one particularly effective model, BERT. At a high level, linguistic features seem to be represented in separate semantic and syntactic subspaces. We find evidence of a fine-grained geometric representation of word senses. We also present empirical descriptions of syntactic representations in both attention matrices and individual word embeddings, as well as a mathematical argument to explain the geometry of these representations.
We present some episodes from the history of interactions between geometry and physics over the past century.
Eta Carinae (Eta Car) is one of the most massive binary stars in the Milky Way. It became the second-brightest star in the sky during its mid-19th century Great Eruption, but then faded from view (with only naked-eye estimates of brightness). Its eruption is unique among known astronomical transients in that it exceeded the Eddington luminosity limit for 10 years. Because it is only 2.3 kpc away, spatially resolved studies of the nebula have constrained the ejected mass and velocity, indicating that in its 19th century eruption, Eta Car ejected more than 10 M_solar in an event that had 10% of the energy of a typical core-collapse supernova without destroying the star. Here we report the discovery of light echoes of Eta Carinae which appear to be from the 1838-1858 Great Eruption. Spectra of these light echoes show only absorption lines, which are blueshifted by -210 km/s, in good agreement with predicted expansion speeds. The light-echo spectra correlate best with those of G2-G5 supergiant spectra, which have effective temperatures of ~5000 K. In contrast to the class of extragalactic outbursts assumed to be analogs of Eta Cars Great Eruption, the effective temperature of its outburst is significantly cooler than allowed by standard opaque wind models. This indicates that other physical mechanisms like an energetic blast wave may have triggered and influenced the eruption.
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