New NSF awards will bring together cross-disciplinary science communities to develop foundations of data science

Article from https://www.nsf.gov/news/news_summ.jsp?cntn_id=242888

News Release 17-079

New NSF awards will bring together cross-disciplinary science communities to develop foundations of data science

TRIPODS awards are NSF’s first major investment toward Harnessing the Data Revolution, one of ’10 Big Ideas for Future NSF Investments’

August 24, 2017

The National Science Foundation (NSF) today announced $17.7 million in funding for 12 Transdisciplinary Research in Principles of Data Science (TRIPODS) projects, which will bring together the statistics, mathematics and theoretical computer science communities to develop the foundations of data science. Conducted at 14 institutions in 11 states, these projects will promote long-term research and training activities in data science that transcend disciplinary boundaries.

“Data is accelerating the pace of scientific discovery and innovation,” said Jim Kurose, NSF assistant director for Computer and Information Science and Engineering (CISE). “These new TRIPODS projects will help build the theoretical foundations of data science that will enable continued data-driven discovery and breakthroughs across all fields of science and engineering.”

Technological advances and unprecedented access to computing infrastructure have resulted in an explosion of data from different sources. The availability of these data — their volume and variety, and the speed at which they are collected — is transforming research in all fields of science and engineering. Through Harnessing the Data Revolution, one of the “10 Big Ideas for Future NSF Investments,” the foundation seeks to support fundamental research in data-driven science and engineering; shape a cohesive, federated, national-scale approach to research data infrastructure; and develop a 21st century data-capable workforce.

The TRIPODS awards will enable data-driven discovery through major investments in state-of-the-art mathematical and statistical tools, better data mining and machine learning approaches, enhanced visualization capabilities and more. These awards will build upon NSF’s long history of investments in foundational research, contributing key advances to the emerging data science discipline, and supporting researchers to develop innovative educational pathways to train the next generation of data scientists.

“TRIPODS will accelerate the development of modern foundations of data science through a truly transdisciplinary collaboration between mathematicians, statisticians and theoretical computer scientists, while also creating opportunity for fundamental development to occur in finding solutions to important data science challenges in the domain sciences,” said Jim Ulvestad, NSF acting assistant director for Mathematical and Physical Sciences (MPS).

TRIPODS is a partnership between NSF’s CISE and MPS directorates. NSF’s Established Program to Stimulate Competitive Research (EPSCoR) also co-funded one of the projects.

A portfolio supporting another of NSF’s Big Ideas, Growing Convergent Research, contributed $1.1 million to the new TRIPODS awards, co-funding three of them. Convergence is the integration of knowledge, techniques and expertise from multiple fields to address scientific and societal challenges. To build an ecosystem that truly supports convergent science, NSF seeks to strategically invest in research projects and programs that are motivated by intellectual opportunities and important societal problems. The goal is that everyone, not just scientists and engineers, will benefit from the convergence of the physical sciences, biological sciences, computing, engineering and the social and behavioral sciences.

The TRIPODS Phase I awards announced today will support the development of small collaborative institutes. A future TRIPODS Phase II is planned to support a smaller number of larger institutes. Phase II will select awardees through a second competitive proposal process from among the Phase I institutes, as well as any new collaborative partners Phase I awardees bring on board.

The award titles, principal investigators and institutions for the TRIPODS Phase I projects are listed below:

-NSF-

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Lecture: The model-existence and amalgamation spectra of $\mathcal{L}_{\omega_1,\omega}$-sentences

Given on 2/6 at the University of Notre Dame

Notre Dame- 2018 02 06-no pause

An example why Scott’s Representation Theorem does not preserve infinite conjunctions and disjunctions

 

Consider the Lebesgue measurable subsets of $[0,1]$ module the null sets. I.e. $A,B\subset [0,1]$ are equivalent is they agree
a.e. This is a $\sigma$-algebra. Consider the set of ultrafilters on this $\sigma$-algebra.

Also, consider the family of sets $\{[0,\frac{1}{n}]|n\in\omega\}$ (each set is identified with its equivalent class. E.g.
$[0,\frac{1}{2}]$ means the equivalent class of $[0,\frac{1}{2}]$). This collection satisfies the finite intersection property.
So, there is an ultrafilter $U$ that extends this collection. For every $n\in\omega$, let $X_n$ be the set of all
ultrafilters that contain $[0,\frac{1}{n}]$. By Scott’s Representation Theorem each $[0,\frac{1}{n}]$ maps to $X_n$.

Now, notice that the intersection $\bigcap_n [0,\frac{1}{n}]$ is the empty set.
If Scott’s Representation theorem was respecting countable conjunctions, the intersection $\bigcap_n [0,\frac{1}{n}]=\emptyset$
would map to the intersection $\bigcap_n X_n=\emptyset$. But the intersection of the $X_n$’s is not empty as it contains the
ultrafilter $U$. Contradiction.

Bottom Line: What goes wrong is that in the above example, the Boolean Algebra given by Scott’s Representation theorem is not
$\sigma$-complete.

The experience of mathematical beauty and its neural correlates

See the full article at https://doi.org/10.3389/fnhum.2014.00068

Original Research ARTICLE

Front. Hum. Neurosci., 13 February 2014 |

The experience of mathematical beauty and its neural correlates

newprofile_default_profileimage_new.jpgSemir Zeki1*, newprofile_default_profileimage_new.jpgJohn Paul Romaya1, newprofile_default_profileimage_new.jpgDionigi M. T. Benincasa2 and Thumb_24.jpgMichael F. Atiyah3

  • 1Wellcome Laboratory of Neurobiology, University College London, London, UK
  • 2Department of Physics, Imperial College London, London, UK
  • 3School of Mathematics, University of Edinburgh, Edinburgh, UK

Many have written of the experience of mathematical beauty as being comparable to that derived from the greatest art. This makes it interesting to learn whether the experience of beauty derived from such a highly intellectual and abstract source as mathematics correlates with activity in the same part of the emotional brain as that derived from more sensory, perceptually based, sources. To determine this, we used functional magnetic resonance imaging (fMRI) to image the activity in the brains of 15 mathematicians when they viewed mathematical formulae which they had individually rated as beautiful, indifferent or ugly. Results showed that the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely field A1 of the medial orbito-frontal cortex (mOFC), as the experience of beauty derived from other sources.

Introduction

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty”

Bertrand Russell, Mysticism and Logic (1919).

The beauty of mathematical formulations lies in abstracting, in simple equations, truths that have universal validity. Many—among them the mathematicians Bertrand Russell (1919) and Hermann Weyl (Dyson, 1956; Atiyah, 2002), the physicist Paul Dirac (1939) and the art critic Clive Bell (1914)—have written of the importance of beauty in mathematical formulations and have compared the experience of mathematical beauty to that derived from the greatest art (Atiyah, 1973). Their descriptions suggest that the experience of mathematical beauty has much in common with that derived from other sources, even though mathematical beauty has a much deeper intellectual source than visual or musical beauty, which are more “sensible” and perceptually based. Past brain imaging studies exploring the neurobiology of beauty have shown that the experience of visual (Kawabata and Zeki, 2004), musical (Blood et al., 1999; Ishizu and Zeki, 2011), and moral (Tsukiura and Cabeza, 2011) beauty all correlate with activity in a specific part of the emotional brain, field A1 of the medial orbito-frontal cortex, which probably includes segments of Brodmann Areas (BA) 10, 12 and 32 (see Ishizu and Zeki, 2011 for a review). Our hypothesis in this study was that the experience of beauty derived from so abstract an intellectual source as mathematics will correlate with activity in the same part of the emotional brain as that of beauty derived from other sources.

Plato (1929) thought that “nothing without understanding would ever be more beauteous than with understanding,” making mathematical beauty, for him, the highest form of beauty. The premium thus placed on the faculty of understanding when experiencing beauty creates both a problem and an opportunity for studying the neurobiology of beauty. Unlike our previous studies of the neurobiology of musical or visual beauty, in which participating subjects were neither experts nor trained in these domains, in the present study we had, of necessity, to recruit subjects with a fairly advanced knowledge of mathematics and a comprehension of the formulae that they viewed and rated. It is relatively easy to separate out the faculty of understanding from the experience of beauty in mathematics, but much more difficult to do so for the experience of visual or musical beauty; hence a study of the neurobiology of mathematical beauty carried with it the promise of addressing a broader issue with implications for future studies of the neurobiology of beauty, namely the extent to which the experience of beauty is bound to that of “understanding.”

See the full article at https://doi.org/10.3389/fnhum.2014.00068