Monthly Archives: January 2014

Joel David Hamkins’ 1000th MO Answer is Coming

joel

Update (May 2014): The second MO contributor to answer 1000 questions is another distinguished mathematician (and a firend) Igor Rivin.

Joel David Hamkins’ profile over MathOverflow reads: “My main research interest lies in mathematical logic, particularly set theory, focusing on the mathematics and philosophy of the infinite. A principal concern has been the interaction of forcing and large cardinals, two central concepts in set theory. I have worked in group theory and its interaction with set theory in the automorphism tower problem, and in computability theory, particularly the infinitary theory of infinite time Turing machines. Recently, I am preoccupied with the set-theoretic multiverse, engaging with the emerging field known as the philosophy of set theory.”

Joel is a wonderful MO contributor, one of those distinguished mathematicians whose arrays of MO answers in their areas of interest draw coherent deep pictures for these areas that you probably cannot find anywhere else. And Joel is also a very highly decorated and prolific MO contributor, whose 999th answer appeared today!!

Here is a very short selection of Joel’s answers. To (MO founder) Anton Geraschenko’s question What are some reasonable-sounding statements that are independent of ZFC? Joel answered; “If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y.” Joel gave a very thorough answer to  my question on Solutions to the Continuum Hypothesis; His 999th answer is on the question Can an ultraproduct be infinite countable? (the answer is yes! but this is a large cardinal assumption.) Update: Joel’s 1000th answer on a question about logic in mathematics and philosophy was just posted.

Joel also wrote a short assay, the use and value of MathOverflow over his blog. Here it is:

The principal draw of mathoverflow for me is the unending supply of extremely interesting mathematics, an eternal fountain of fascinating questions and answers. The mathematics here is simply compelling.

I feel that mathoverflow has enlarged me as a mathematician. I have learned a huge amount here in the past few years, particularly concerning how my subject relates to other parts of mathematics. I’ve read some really great answers that opened up new perspectives for me. But just as importantly, I’ve learned a lot when coming up with my own answers. It often happens that someone asks a question in another part of mathematics that I can see at bottom has to do with how something I know about relates to their area, and so in order to answer, I must learn enough about this other subject in order to see the connection through. How fulfilling it is when a question that is originally opaque to me, because I hadn’t known enough about this other topic, becomes clear enough for me to have an answer. Meanwhile, mathoverflow has also helped me to solidify my knowledge of my own research area, often through the exercise of writing up a clear summary account of a familiar mathematical issue or by thinking about issues arising in a question concerning confusing or difficult aspects of a familiar tool or method.

Mathoverflow has also taught me a lot about good mathematical exposition, both by the example of other’s high quality writing and by the immediate feedback we all get on our posts. This feedback reveals what kind of mathematical explanation is valued by the general mathematical community, in a direct way that one does not usually get so well when writing a paper or giving a conference talk. This kind of knowledge has helped me to improve my mathematical writing in general.

So, thanks very much mathoverflow! I am grateful.

Thanks very much, Joel,  for your wonderful mathoverflow answers and questions!

JDH

Amazing: Peter Keevash Constructed General Steiner Systems and Designs

KONICA MINOLTA DIGITAL CAMERA

Here is one of the central and oldest problems in combinatorics:

Problem: Can you find a collection S of q-subsets from an n-element set X set so that every r-subset of X is included in precisely λ sets in the collection?

A collection S  of this kind are called a design of parameters (n,q,r, λ),  a special interest is the case  λ=1, and in this case S is called a Steiner system.

For such an S to exist n should be admissible namely {{q-i} \choose {r-i}} should divide \lambda {{n-i} \choose {r-i}} for every 1 \le i \le r-1.

There are only few examples of designs when r>2. It was even boldly conjectured that for every q r and λ if n is sufficiently large than a design of parameters  (n,q,r, λ) exists but the known constructions came very very far from this.   … until last week. Last week, Peter Keevash gave a twenty minute talk at Oberwolfach where he announced the proof of the bold existence conjecture. Today his preprint the existence of designs, have become available on the arxive.

Brief history

The existence of designs and Steiner systems is one of the oldest and most important problems in combinatorics.

1837-1853 – The existence of designs and Steiner systems was asked by Plücker(1835), Kirkman (1846) and Steiner (1853).

1972-1975 – For r=2 which was of special interests, Rick Wilson proved their existence for large enough admissible values of n.

1985 -Rödl proved the existence of approximate objects (the property holds for (1-o(1)) r-subsets of X) , thus answering a conjecture by Erdös and Hanani.

1987  – Teirlink proved their existence for infinitely  many values of n when r and q are arbitrary and  λ is a certain large number depending on q and r but not on n. (His construction also does not have repeated blocks.)

2014 – Keevash’s  proved the existence of Steiner systems for all but finitely many admissible  values of n for every q and r. He uses a new method referred to as Randomised Algebraic Constructions.

Update: Just 2 weeks before Peter Keevash announced his result I mentioned the problem in my lecture in “Natifest” in a segment of the lecture devoted to the analysis of Nati’s dreams. 35:38-37:09.

Update: Some other blog post on this achievement: Van Vu Jordan Ellenberg, The aperiodical . A related post from Cameron’s blog Subsets and partitions.

Update: Danny Calegary pointed out a bird-eye similarity between Keevash’s strategy and the strategy of the  recent Kahn-Markovic proof of the Ehrenpreis conjecture http://arxiv.org/abs/1101.1330 , a strategy used again by Danny and Alden Walker to show that random groups contain fundamental groups of closed surfaces http://arxiv.org/abs/1304.2188 .

Many Short Updates

Things in Berkeley and later here in Jerusalem were very hectic so I did not blog much since mid October. Much have happened so let me give brief and scattered highlights review.

Two “real analysis” workshops at the Simons Institute – The first in early October was on Functional Inequalities in Discrete Spaces with Applications and the second in early December was on Neo-classical methods in discrete analysis. Many exciting lectures! The links lead to the videotaped  lectures. There were many other activities at the Simons Institute also in the parallel program on “big data” and also many interesting talks at the math department in Berkeley, the CS department and MSRI.

inequality

To celebrate the workshop on inequalities, there were special shows in local movie theaters

My course at Berkeley on analysis of Boolean functions – The course went very nicely. I stopped blogging about it at weak 7. Just before a lecture on MRRW upper bounds for binary codes, a general introductory lecture on social choice, and then several lectures by Guy Kindler (while I was visiting home) on the invariance principle and majority is stablest theorem.  The second half of the course covered sharp threshold theorems, applications for random graphs, noise sensitivity and stability, a little more on percolation and a discussion of some open problems.

nams

Back to snowy Jerusalem, Midrasha, Natifest, and Archimedes. I landed in Israel on Friday toward the end of the heaviest  snow storm in Jerusalem. So I spent the weekend with my 90-years old father in law before reaching Jerusalem by train. While everything at HU was closed there were still three during-snow mathematics activities at HU. There was a very successful winter school (midrasha) on analytic number theory which took place in the heaviest storm days.  Natifest was a very successful conference and I plan to devote to it a special post, but meanwhile, here is a link to the videotaped lectures and a picture of Nati with Michal, Anna and Shafi. We also had a special cozy afternoon event joint between the mathematics department and the department for classic studies  where Reviel Nets talked about the Archimedes Palimpses.

royal

The story behind Reviel’s name is quite amazing. When he was born, his older sister tried to read what was written in a pack of cigarettes. It should have been “royal” but she read “reviel” and Reviel’s parents adopted it for his name.

archimedes