My former colleague Larry Guth (now at MIT) visited us recently and gave a beautiful colloquium talk. The Department has recently deployed a video streaming service so we are able to share Larry’s talk with the world. We look forward to sharing other videos in the future.
Here is the video:
Unexpected applications of polynomials in combinatorics
by Larry Guth | MIT Time: 16:10 (Wednesday, Jan. 23, 2013) Location: BA6183, Bahen Center, 40 St George St Abstract:
In the last five years, several hard problems in combinatorics have been solved by using polynomials in an unexpected way. In some cases, the proofs are very short, and I will present a complete proof in the lecture. One of the problems is the joints problem. Given a set of lines in $R^3$, a joint is a point that lies in three non-coplanar lines. Given $L$ lines in $R^3$, how many joints can there be? Another problem is the distinct distance problem in the plane. If P is a set of points in the plane, the distance set of $P$ is the set of all distances from one point of $P$ to another. If $P$ is a set of $N$ points in the plane, how small can the distance set of $P$ be? The proofs involve studying a set of points in a vector space by finding a polynomial of controlled degree that vanishes at the points, and then using the geometry of the zero-set to understand the combinatorial properties of the points. The goal for the talk is to give an overview of this new method.
Suppose you’ve recorded a one hour lecture onto video and taken digital photographs of each blackboard used during the presentation. The video and photograph files contain metadata with time stamps: we know the time when each frame of video and photograph was captured. How should these files be blended to create a browser-based environment for studying the ideas in the lecture? My colleague Dror Bar-Natan has prototyped a remarkable software platform combining this data to support online courses and seminars.
The system is built atop MediaWiki (the software that runs Wikipedia) and allows metadata, such as comments and photographs, to be correlated in time to an embedded video stream. Comments can be attached to each frame of video. The system enables remote audience members to participate in the seminar frame-by-frame using their power to comment. In this way, the wClips system goes beyond playback by empowering the remote audience of the future to contribute to the past presentation.
The system was recently used to supplement a paper Dror is co-authoring with Zsuzsanna Dansco in the wClips seminar. Each section of the paper was discussed in detail in a seminar lecture captured on video. Each blackboard in the seminar series was photographed. Time links near the photographs move the video stream to the video time when that blackboard image was created. The sizes of the different elements can be changed by sliding the red line. In the final version of the paper, there will appear hyperlinks in each section pointing at the associated seminar presentation. Clicking on the photographs reveals another layer of comments highlighting the elements appearing on the blackboard.
The wClips system runs on a linux server using open source software so is inexpensive to deploy. Since it is based on MediaWiki, it has the potential to scale to support a massive audience. Getting the video content out of the camera and into the online course system involves some work but could be streamlined with the right scripts.
Members of the University of Toronto have access to a fantastic library. There are great resources like the Oxford English Dictionary, MathSciNet and Web of Knowledge and a whole lot more. To access these resources while away from campus, members of the UofT community can use the virtual private network service UTORvpn. That site tells you how to install the VPN service on your computer. When you turn on the VPN, your computer can access restricted library resources even though it is not located on the University network. On my computer, the active and inactive VPN service is indicated like this:
Suppose you wanted to look up the definition of the word syzygy. You could use google to get some information or you could drill down through the University of Toronto library looking for the Oxford English Dictionary to eventually find a more definitive result.
Researchers are busy. So, many will merely use google since the drilling down through the library takes too much time.
Customized Search at your fingertips
UofT’s restricted databases can be made available at your fingertips. Here is an example using Firefox. Right click (or control-mouse) the search field on the OED to reveal the option Add a Keyword for this Search
I chose to use the keyword oed.
Now, I can type into the URL bar, oed WORDTOLOOKUP, and I get the OED search results instantly.
The same thing can be done with AMS MathSciNet author search and other search engines.