Here follows some links that I find very useful, starting with arxiv.org
which is a depository of almost all maths and phys preprints since 1992.
It's not easy to think of something that's not there.
For almost anything that's not on arxiv.org, one can use google scholar,
which is very useful in finding all papers that refer toa certain paper.
Project Euclid includes papers that are not available on arxiv.org,
including those that predate 1992.
Not Even Wrong regularly posts links to conferences, books and papers
on modern mathematical physics. Peter Woit provides an antidote to
hype related to string theory as a unified theory of all fundamental
interactions, but here we focus only on mathematical physics.
Not Even Wrong
Prof Takasaki's web site provides an introduction to integrable models.
Prof K Takasaki's integrable models web page
A number of research institutes post videos of lectures on modern mathematical
Online lectures, IAS, Princeton, NJ, USA
Online lectures, KITP, Santa Barbara, CA, USA
Online lectures, Perimeter Institute, Waterloo, Canada
and N=4 SYM scattering amplitudes,
Online lectures, PI, Waterloo
Euler Institute lecture series
In the following TASI lectures, I recommend M Aganagic's 2007
lectures on topological strings, as well as all lectures in 2015
TASI lecture videos
The Graduate Physics lectures are almost exclusively
dedicated to modern developments in mathematical physics.
Graduate Physics lectures
A number of lecture series are posted on youtube.com
6D conformal field theory workshop
The 29th (2012) Jerusalem Winter School in Theoretical Physics
A number of maths and phys departments have channels on
youtube.com where the post videos of their lecture series.
The following are the ones that I know.
Stanford Institute for Theoretical Physics
Institute for Advanced Studies, Princeton
Department of Mathematics, Princeton
Department of Mathematics, Harvard
Department of Mathematics, U of Illinois
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