The equations of gauge theory lie at the heart of our understanding of particle physics. The Standard Model, which describes the electromagnetic, weak, and strong forces, is based on the Yang-Mills equations. Starting with the work of Donaldson in the 1980s, gauge theory has also been successfully applied in other areas of pure mathematics, such as low dimensional topology, symplectic geometry, and algebraic geometry.
More recently, Witten proposed a gauge-theoretic interpretation of Khovanov homology, a knot invariant whose origins lie in representation theory. Khovanov homology is a “categorification” of the celebrated Jones polynomial, in the sense that its Euler characteristic recovers this polynomial. At the moment, Khovanov homology is only defined for knots in the three-sphere, but Witten’s proposal holds the promise of generalizations to other three-manifolds, and perhaps of producing new invariants of four-manifolds.
This workshop will bring together researchers from several different fields (theoretical physics, mathematical gauge theory, topology, analysis / PDE, representation theory, symplectic geometry, and algebraic geometry), and thus help facilitate connections between these areas. The common focus will be to understand Khovanov homology and related invariants through the lens of gauge theory.
This workshop will include a poster session; a request for posters will be sent to registered participants in advance of the workshop.
Edward Witten will be giving two public lectures as part of the Green Family Lecture series:
March 6, 2017
From Gauge Theory to Khovanov Homology Via Floer Theory
The goal of the lecture is to describe a gauge theory approach to Khovanov homology of knots, in particular, to motivate the relevant gauge theory equations in a way that does not require too much physics background. I will give a gauge theory perspective on the construction of singly-graded Khovanov homology by Abouzaid and Smith.
March 8, 2017
An Introduction to the SYK Model
The Sachdev-Ye model was originally a model of quantum spin liquids that was introduced in the mid-1990′s. In recent years, it has been reinterpreted by Kitaev as a model of quantum chaos and black holes. This lecture will be primarily a gentle introduction to the SYK model, though I will also describe a few more recent results.
Open for submission now
Deadline for manuscript submissions: 31 August 2017
Deadline for manuscript submissions: 31 August 2017
Whereas Bayesian inference has now achieved mainstream acceptance and is widely used throughout the sciences, associated ideas such as the principle of maximum entropy (implicit in the work of Gibbs, and developed further by Ed Jaynes and others) have not. There are strong arguments that the principle (and variations, such as maximum relative entropy) is of fundamental importance, but the literature also contains many misguided attempts at applying it, leading to much confusion.
This Special Issue will focus on Bayesian inference and MaxEnt. Some open questions that spring to mind are: Which proposed ways of using entropy (and its maximisation) in inference are legitimate, which are not, and why? Where can we obtain constraints on probability assignments, the input needed by the MaxEnt procedure?
More generally, papers exploring any interesting connections between probabilistic inference and information theory will be considered. Papers presenting high quality applications, or discussing computational methods in these areas, are also welcome.
Dr. Brendon J. Brewer
Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.
Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed Open Access monthly journal published by MDPI.
No papers have been published in this special issue yet.
Back in December, you hopefully thoroughly immersed yourself in The Map of Physics, an animated video–a visual aid for the modern age–that mapped out the field of physics, explaining all the connections between classical physics, quantum physics, and relativity.
You can’t do physics without math. Hence we now have The Map of Mathematics. Created by physicist Dominic Walliman, this new video explains “how pure mathematics and applied mathematics relate to each other and all of the sub-topics they are made from.” Watch the new video above. You can buy a poster of the map here. And you can download a version for educational use here.
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We discuss properties of the "beamsplitter addition" operation, which provides a non-standard scaled convolution of random variables supported on the non-negative integers. We give a simple expression for the action of beamsplitter addition using generating functions. We use this to give a self-contained and purely classical proof of a heat equation and de Bruijn identity, satisfied when one of the variables is geometric.
Instagram filter used: Clarendon
Photo taken at: UCLA Math Sciences Building
There follows a discussion of flipping coins and the fact that frequencies have more random variation when the sample size is small, but he never stops to see if this is enough to explain the observation.
My intuition told me it did not, so I went and got some brain cancer data.
Jordan Ellenberg is called out a bit by Rick Durrett for one of his claims in the best seller How Not To Be Wrong: The Power of Mathematical Thinking.
I remember reading that section of the book and mostly breezing through that argument primarily as a simple example with a limited, but direct point. Durrett decided to delve into the applied math a bit further.
These are some of the subtle issues one eventually comes across when experts read others’ works which were primarily written for much broader audiences.
I also can’t help thinking that one paints a target on one’s back with a book title like that…
BTW, the quote of the day has to be:
Syndicated copies to:
… so I went and got some brain cancer data.
NIMBioS will host an Tutorial on Uncertainty Quantification for Biological Models
Uncertainty Quantification for Biological Models
Meeting dates: June 26-28, 2017
Location: NIMBioS at the University of Tennessee, Knoxville
Marisa Eisenberg, School of Public Health, Univ. of Michigan
Ben Fitzpatrick, Mathematics, Loyola Marymount Univ.
James Hyman, Mathematics, Tulane Univ.
Ralph Smith, Mathematics, North Carolina State Univ.
Clayton Webster, Computational and Applied Mathematics (CAM), Oak Ridge National Laboratory; Mathematics, Univ. of Tennessee
Mathematical modeling and computer simulations are widely used to predict the behavior of complex biological phenomena. However, increased computational resources have allowed scientists to ask a deeper question, namely, “how do the uncertainties ubiquitous in all modeling efforts affect the output of such predictive simulations?” Examples include both epistemic (lack of knowledge) and aleatoric (intrinsic variability) uncertainties and encompass uncertainty coming from inaccurate physical measurements, bias in mathematical descriptions, as well as errors coming from numerical approximations of computational simulations. Because it is essential for dealing with realistic experimental data and assessing the reliability of predictions based on numerical simulations, research in uncertainty quantification (UQ) ultimately aims to address these challenges.
Uncertainty quantification (UQ) uses quantitative methods to characterize and reduce uncertainties in mathematical models, and techniques from sampling, numerical approximations, and sensitivity analysis can help to apportion the uncertainty from models to different variables. Critical to achieving validated predictive computations, both forward and inverse UQ analysis have become critical modeling components for a wide range of scientific applications. Techniques from these fields are rapidly evolving to keep pace with the increasing emphasis on models that require quantified uncertainties for large-scale applications. This tutorial will focus on the application of these methods and techniques to mathematical models in the life sciences and will provide researchers with the basic concepts, theory, and algorithms necessary to quantify input and response uncertainties and perform sensitivity analysis for simulation models. Concepts to be covered may include: probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, quantification of model discrepancy, adaptive surrogate model construction, high-dimensional approximation, random sampling and sparse grids, as well as local and global sensitivity analysis.
This tutorial is intended for graduate students, postdocs and researchers in mathematics, statistics, computer science and biology. A basic knowledge of probability, linear algebra, and differential equations is assumed.
Application deadline: March 1, 2017
To apply, you must complete an application on our online registration system:
- Click here to access the system
- Login or register
- Complete your user profile (if you haven’t already)
- Find this tutorial event under Current Events Open for Application and click on Apply
Participation in NIMBioS tutorials is by application only. Individuals with a strong interest in the topic are encouraged to apply, and successful applicants will be notified within two weeks after the application deadline. If needed, financial support for travel, meals, and lodging is available for tutorial attendees.
The application process is now closed.
Summary Report. TBA
Live Stream. The Tutorial will be streamed live. Note that NIMBioS Tutorials involve open discussion and not necessarily a succession of talks. In addition, the schedule as posted may change during the Workshop. To view the live stream, visit http://www.nimbios.org/videos/livestream. A live chat of the event will take place via Twitter using the hashtag #uncertaintyTT. The Twitter feed will be displayed to the right of the live stream. We encourage you to post questions/comments and engage in discussion with respect to our Social Media Guidelines.
MyScript MathPad is a mathematic expression demonstration that lets you handwrite your equations or mathematical expressions on your screen and have them rendered into their digital equivalent for easy sharing. Render complex mathematical expressions easily using your handwriting with no constraints. The result can be shared as an image or as a LaTeX* or MathML* string for integration in your documents.
This looks like something I could integrate into my workflow.Syndicated copies to:
A mathematical model could lead to a new approach to the study of what is possible, and how it follows from what already exists.
Innovation is one of the driving forces in our world. The constant creation of new ideas and their transformation into technologies and products forms a powerful cornerstone for 21st century society. Indeed, many universities and institutes, along with regions such as Silicon Valley, cultivate this process.
And yet the process of innovation is something of a mystery. A wide range of researchers have studied it, ranging from economists and anthropologists to evolutionary biologists and engineers. Their goal is to understand how innovation happens and the factors that drive it so that they can optimize conditions for future innovation.
This approach has had limited success, however. The rate at which innovations appear and disappear has been carefully measured. It follows a set of well-characterized patterns that scientists observe in many different circumstances. And yet, nobody has been able to explain how this pattern arises or why it governs innovation.
Today, all that changes thanks to the work of Vittorio Loreto at Sapienza University of Rome in Italy and a few pals, who have created the first mathematical model that accurately reproduces the patterns that innovations follow. The work opens the way to a new approach to the study of innovation, of what is possible and how this follows from what already exists.
The notion that innovation arises from the interplay between the actual and the possible was first formalized by the complexity theorist Stuart Kauffmann. In 2002, Kauffmann introduced the idea of the “adjacent possible” as a way of thinking about biological evolution.
I know he discusses some of this in At Home in the Universe.
The adjacent possible is all those things—ideas, words, songs, molecules, genomes, technologies and so on—that are one step away from what actually exists. It connects the actual realization of a particular phenomenon and the space of unexplored possibilities.
But this idea is hard to model for an important reason. The space of unexplored possibilities includes all kinds of things that are easily imagined and expected but it also includes things that are entirely unexpected and hard to imagine. And while the former is tricky to model, the latter has appeared close to impossible.
What’s more, each innovation changes the landscape of future possibilities. So at every instant, the space of unexplored possibilities—the adjacent possible—is changing.
“Though the creative power of the adjacent possible is widely appreciated at an anecdotal level, its importance in the scientific literature is, in our opinion, underestimated,” say Loreto and co.
Nevertheless, even with all this complexity, innovation seems to follow predictable and easily measured patterns that have become known as “laws” because of their ubiquity. One of these is Heaps’ law, which states that the number of new things increases at a rate that is sublinear. In other words, it is governed by a power law of the form V(n) = knβ where β is between 0 and 1.
Words are often thought of as a kind of innovation, and language is constantly evolving as new words appear and old words die out.
This evolution follows Heaps’ law. Given a corpus of words of size n, the number of distinct words V(n) is proportional to n raised to the β power. In collections of real words, β turns out to be between 0.4 and 0.6.
Another well-known statistical pattern in innovation is Zipf’s law, which describes how the frequency of an innovation is related to its popularity. For example, in a corpus of words, the most frequent word occurs about twice as often as the second most frequent word, three times as frequently as the third most frequent word, and so on. In English, the most frequent word is “the” which accounts for about 7 percent of all words, followed by “of” which accounts for about 3.5 percent of all words, followed by “and,” and so on.
This frequency distribution is Zipf’s law and it crops up in a wide range of circumstances, such as the way edits appear on Wikipedia, how we listen to new songs online, and so on.
These patterns are empirical laws—we know of them because we can measure them. But just why the patterns take this form is unclear. And while mathematicians can model innovation by simply plugging the observed numbers into equations, they would much rather have a model which produces these numbers from first principles.
Enter Loreto and his pals (one of which is the Cornell University mathematician Steve Strogatz). These guys create a model that explains these patterns for the first time.
They begin with a well-known mathematical sand box called Polya’s Urn. It starts with an urn filled with balls of different colors. A ball is withdrawn at random, inspected and placed back in the urn with a number of other balls of the same color, thereby increasing the likelihood that this color will be selected in future.
This is a model that mathematicians use to explore rich-get-richer effects and the emergence of power laws. So it is a good starting point for a model of innovation. However, it does not naturally produce the sublinear growth that Heaps’ law predicts.
That’s because the Polya urn model allows for all the expected consequences of innovation (of discovering a certain color) but does not account for all the unexpected consequences of how an innovation influences the adjacent possible.
The upshot of the whole thing:
So Loreto, Strogatz, and co have modified Polya’s urn model to account for the possibility that discovering a new color in the urn can trigger entirely unexpected consequences. They call this model “Polya’s urn with innovation triggering.”
The exercise starts with an urn filled with colored balls. A ball is withdrawn at random, examined, and replaced in the urn.
If this color has been seen before, a number of other balls of the same color are also placed in the urn. But if the color is new—it has never been seen before in this exercise—then a number of balls of entirely new colors are added to the urn.
Loreto and co then calculate how the number of new colors picked from the urn, and their frequency distribution, changes over time. The result is that the model reproduces Heaps’ and Zipf’s Laws as they appear in the real world—a mathematical first. “The model of Polya’s urn with innovation triggering, presents for the first time a satisfactory first-principle based way of reproducing empirical observations,” say Loreto and co.
The team has also shown that its model predicts how innovations appear in the real world. The model accurately predicts how edit events occur on Wikipedia pages, the emergence of tags in social annotation systems, the sequence of words in texts, and how humans discover new songs in online music catalogues.
Interestingly, these systems involve two different forms of discovery. On the one hand, there are things that already exist but are new to the individual who finds them, such as online songs; and on the other are things that never existed before and are entirely new to the world, such as edits on Wikipedia.
Loreto and co call the former novelties—they are new to an individual—and the latter innovations—they are new to the world.
Curiously, the same model accounts for both phenomenon. It seems that the pattern behind the way we discover novelties—new songs, books, etc.—is the same as the pattern behind the way innovations emerge from the adjacent possible.
That raises some interesting questions, not least of which is why this should be. But it also opens an entirely new way to think about innovation and the triggering events that lead to new things. “These results provide a starting point for a deeper understanding of the adjacent possible and the different nature of triggering events that are likely to be important in the investigation of biological, linguistic, cultural, and technological evolution,” say Loreto and co.
We’ll look forward to seeing how the study of innovation evolves into the adjacent possible as a result of this work.
Ref: arxiv.org/abs/1701.00994: Dynamics on Expanding Spaces: Modeling the Emergence of Novelties
AMS Open Math Notes is a repository of freely downloadable mathematical works in progress hosted by the American Mathematical Society as a service to researchers, teachers and students. These draft works include course notes, textbooks, and research expositions in progress. They have not been published elsewhere, and, as works in progress, are subject to significant revision. Visitors are encouraged to download and use these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.
h/t to Terry Tao for the notice.Syndicated copies to:
Retired UCI math professor Steven Roman has just started making a series of Group Theory lectures on YouTube. No prior experience in group theory is necessary. He’s the author of the recent Fundamentals of Group Theory: An Advanced Approach. 
He hopes to eventually also offer lectures on ring theory, fields, vector spaces, and module theory in the near future.