Entropy Adventures

October 6 is now often called Mathocalypse event. Terry Tao calls it the start of Math 2.0 and sees a positive aspect that the research frontier of mathematics will expand significantly faster. I agree that there is little choice in that. At the moment, there is no realistic way to regulate AI due to international competition. I think that most creative mathematicians do regret the developments we see. Among the problems that have been solved allegedly (I do not understand the proof yet) is the proof that the Standard map on the two torus has positive entropy for large enough c. I originally wanted to talk today more about natural groups but decided to mourn a bit about the passing of one of my favorite problems in mathematics, one I have spent a lot of time on unsuccessfully but which had been a seed for many interesting learning adventures in math.

I have seen two talks claiming positive entropy for large c of the standard map. One was by Michael Herman at Caltech, around 1995 in a smaller setting. If I remember correctly there were only a handful of folks there. Herman presented with an overhead projector. Things were hard to read. I think Thomas Wolf had organized that special talk. I had met Herman a couple of times during that time as he was visiting Caltech for longer. He had no car and I chauffeured him around a couple of times, once to a brunch organized by some dynamical systems folks in China town in LA. The other talk was by Jakov Sinai in 1996 in a conference at UCLA. The later talk is actually interesting as I was unable to find any records of that legendary Quo Vadis conference in UCLA from 1996. It was a time, where the dynamical systems folks had a bit of a midlife crisis and did not know where to go, hence the title. The chaos hype driven by Mandelbrot or chaos hype driven by strange attractors had fizzeled out a bit and this is typical after an overhype. We have seen this in “catastrophe theory”, one of the most beautifuil areas in math, which however, due to overhype later were neglected. Everything had been reduced to catastrophe theory. Also for “fractals” or “chaos” the fate was clear. Everything was “chaos”, “factals” or “solitons”. Like catastrophe theory, the three topics belong to the most beautiful in mathematics but it was a bit tiring to see thousands of papers and hundreds of books on the subject. I gave once a math circle talk on “Fashion in mathematics” trying to explain the phenomenon a bit. Things have not changed. We culturally hype some things which are just ugly (like Navier Stokes) and pass over beautiful things (like the perfect number problem). Here is the announcement, from my email:

From dynamics@math.ucla.edu Fri Feb  9 20:50:11 1996

From: Dynamical Systems 1996 Conference  

To: knill@cco.caltech.edu 

Subject: Dynamical Systems Conference at UCLA 



                  DYNAMICAL SYSTEMS: Quo Vadis? 

              University of California, Los Angeles 

                       March 1 - 3, 1996 


Below is the schedule for the conference.  For more information (including
abstracts of talks, hotels and arrival information) and the latest updates,
please see our web page at http://www.math.ucla.edu/~dynamics
Alternatively you can finger dynamics@math.ucla.edu | more
Please make your travel arrangements soon if you have not done so already.

I myself spent maybe 15 years about this problem. It started in 1997, when writing my senior thesis. The last time I thought about this was maybe in 2014, when working on pseudo determinants. These are 28 years, but since 2000 research for me had more been more of a hobby and I did not intensively think about that problem since moving to Cambridge, only in two instances, when working with Lesieutre on KAM stuff in 2008/2009 and when working on Pseudo determinants 2012-2014. So, in total, 15 years is probably a good guess. The problem actually had been very rich. It led to connections with integrable systems, with Julia sets, with renormalization of dynamical systems, with dyadic integers, with subharmonic functions, with plurisubharmonic functions, with determinants, with von Neumann algebras, with spectral theory of Schroedinger operators, with Green functions with multi-linear algebra. It was a fantastic adventure illustrating that the journey is much more important than the actual goal.

Here is the list of participants (not UCLA) from that conference:

PARTIAL LIST OF PARTICIPANTS
============================
(This list does not include the participants from UCLA.)

Marcy Barge             Montana St    umsfmbar@newton.math.montana.edu
Peter Baxendale         USC           baxendal@math.usc.edu
Bill Beckner            Texas         beckner@math.utexas.edu
Bjorn Birnir            UCSB          birnir@math.ucsb.edu
Lejonia Bunimovich      Ga Tech       bunimovh@math.gatech.edu
James T. Campbell       Memphis       campbelj@hermes.msci.memst.edu
Nikolai Chernov         Alabama       chernov@vorteb.math.uab.edu
David DeLatte           N Texas       delatte@unt.edu
Jack Feldman            UC-Berkeley   feldman@math.berkeley.edu
Jacek Graczyk           Caltecg       jacek@cco.caltech.edu
Eugene Gutkin           USC           egutkin@math.usc.edu
Jenny Harrison          UC-Berkeley   harrison@math.berkeley.edu
Nicolai Haydn           USC           nhaydn@math.usc.edu
Stefano Isola           Bologna       isola@math.usc.edu
Michael V. Jakobson     Maryland      mvy@math.umd.edu
Yunping Jiang           CUNY-Queens   yunqc@yunping.math.qc.edu
Svetlana Jitomirskaya   UC-Irvine     szhitomi@uci.edu
Konstantin Khanin       Princeton     khanin@math.Princeton.EDU
Alexander Khibnik       Cornell       khibnik@cam.cornell.edu
Bruce Kitchens          IBM           kitch@watson.ibm.com
Abel Klein              UC-Irvine     aklein@uci.edu
Oliver R. Knill         Caltech       knill@cco.caltech.edu
Hans Koch               Texas         koch@math.utexas.edu
Eric Kostelich          Arizona St    kostelich@asu.edu
Samuel Kurcbart         UCSD          kurcbart@hypatia.ucsd.edu
Nathan Kutz             Princeton     kutz@math.Princeton.EDU
Michel L. Lapidus       UC-Riverside  lapidus@math.ucr.edu
Yuri Latushkin          Missouri      yuri@msindy7.cs.missouri.edu
Mark Levi               Rensselaer    levim@rpi.edu
Hector Lomeli           Colorado      lomeli@newton.colorado.edu
Mikhail Lyubich         Stony Brook   mlyubich@math.sunysb.edu
Dan Mauldin             N Texas       mauldin@dynamics.math.unt.edu
Ian Melbourne           Houston       ism@math.uh.edu
Tom Milac               UC-Irvine     tmilac@math.uci.edu
Michal Misiurewicz      IUPUI         mmisiure@math.iupui.edu
Sheldon E. Newhouse     Michigan St   sen@math.msu.edu
Alec Norton             Texas         alec@math.utexas.edu
Yakov Pesin             Penn State    pesin@math.psu.edu
Mark Pollicott          Manchester    mp@ma.man.ac.uk
Yves Pomeau             Arizona       pomeau@lps.ens.fr                  
Serge Preston           Portland St   serge@mth.pdx.edu
Charles Pugh            UC-Berkeley   pugh@math.berkeley.edu
Charles Radin           Texas         radin@math.utexas.edu
Nicolai Reshetikhin     UC-Berkeley   reshetik@math.berkeley.edu
Senya Shlosman          UC-Irvine     shlosman@math.uci.edu
Slobodan Simic          UI-Chicago    simic@math.uic.edu
John Smillie            Cornell       smillie@math.cornell.edu
Boris Solomyak          Washington    solomyak@math.washington.edu
Nils Svanstedt          UCSB          nils@cardinal.math.ucsb.edu
Grzegorz Swiatek        Penn State    swiatek@math.psu.edu
Edriss Titi             UC-Irvine     etiti@math.uci.edu
Tom Taylor              Arizona St    attjt@acvax.inre.asu.edu
Serge Troubetzkoy       Alabama       troubetz@math.uab.edu
Mariusz Urbanski        N Texas       urbanski@dynamics.math.unt.edu
Kirill L. Vaninsky      IAS           vaninsky@IAS.EDU
Don Wang                Vanderbilt    wang@math.vanderbilt.edu
Robert F. Williams      Texas         bob@math.utexas.edu
Maciej P. Wojtkowski    Arizona       maciejw@math.arizona.edu
Jack Xin                Arizona       xin@math.arizona.edu

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This conference is funded jointly by the National Science Foundation and UCLA.
Organizers:  L. Carleson, R. Perez-Marco and L.-S. Young