Ralph, you and I were dropping the word "manifold" on this site a year or so ago. It got me wondering if it is either necessary or helpful.

For example, what cases are we cutting out? What kind of situations are not manifolds?

- manifolds with singularities (this is what we really meant, I think, since aberrant days are remembered by all)

- algebraic spaces (recommended on MathOverflow as a precursor to understanding stacks = 2-sheaves)

-verdier duality (a condition which fails for non-manifolds)

In the 19th century, manifolds were always subsets of R^n (quoting bill fulton, representation theory). They might have holes or handles and they might have curvature. The curvature I think is what interests us in finance. Most term structures are curved, and order books are irregular as well.

All in all, I am still up in the air about whether this is a useful word to use in finance.



 In honor of our new spec and Larry, I hypothesize that like Pascal's Law, that a small move in a big market will cause probabilistically, a large move in a smaller market. I'll give 1,000 bucks to the personage that comes up with the best predictive relation on that theme.

Mr. Isomorphisms replies: 

I would start looking in smaller markets that are dependent on the larger one. Eg, look at a town where >50% of the workers are employed at one company. The decline of various parts of Detroit's supply chain could fit this story. Or Iceland/Scotland with their large banking sectors relative to within-country wealth.

Steve Ellison writes: 

Chair: "I hypothesize that like pascal's law, that a small move in a big market will cause probabilistically, a large move in a smaller market."

Let us first get an idea of which markets are big and small. Glancing at the futures contract listings in the newspaper, here are some rough estimates of open interest:

3-month Eurodollar 10,000,000
S&P 500 e-mini        3,100,000
10-year US Treasury 3,000,000
Crude oil                  1,300,000
Soybeans                   670,000
Euro                           465,000
Gold                           400,000
Wheat                         360,000
Cocoa                         175,000
Copper                        155,000
Silver                          150,000
Palladium                      31,000
Oats                               8,500
Lumber                           5,000

Next, I would like to introduce the bullwhip effect. Most goods have a "supply chain" that begins with raw materials, progresses to components and finished goods, and may include warehouses and stores. The bullwhip effect posits that, the farther "back" in the supply chain an operation is (farther away from the end customer), the greater the swings in production and inventory will be in response to fluctuations in end customer demand. Here is an article on the subject.

During the dot-com crash in October 2000, stocks briefly rallied on a glowing earnings report by JDS Uniphase. The CNBC screamer correctly noted that JDSU was a component supplier, farther back in the supply chain, and its record sales had probably just bloated the inventories of companies such as Cisco Systems that had already reduced forward guidance.

From this perspective, lumber seems a good candidate for a Pascal's Law study. It is a raw material that must go through additional processing before reaching most customers, and its futures market is very small. The S&P 500 is a large market that theoretically is based on the whole economy and should be more sensitive to final sales.

Using 3 months as a rough rule of thumb for total lead time in a typical supply chain, I compared quarterly net changes in the S&P 500 with net changes one quarter later in the lumber price.

LUMBER                        S&P 500
          Adjusted  Net              Adjusted   Net
Date      Close     change   Date    Close      change
                           09/30/11   1048.50
  12/30/11   291.1         12/30/11   1180.75  132.25
  03/30/12   283.0   -8.1  03/30/12   1337.50  156.75
  06/29/12   294.8   11.8  06/29/12   1297.50  -40.00
  09/28/12   311.3   16.5  09/28/12   1382.00   84.50
  12/31/12   391.4   80.1  12/31/12   1373.75   -8.25
  03/29/13   407.6   16.2  03/28/13   1521.50  147.75
  06/28/13   310.2  -97.4  06/28/13   1563.75   42.25
  09/30/14   355.7   45.5  09/30/13   1645.50   81.75
  12/31/13   365.7   10.0  12/31/13   1818.75  173.25
  03/31/14   334.4  -31.3  03/31/14   1849.25   30.50
  06/30/14   340.3    5.9  06/30/14   1944.50   95.25
  09/30/14   338.0   -2.3  09/30/14   1965.50   21.00

There were only 11 data points since 2012, not enough for significance. In a regression, the t score was -1.55, and the R squared was 0.21.

The quarterly changes in lumber were a little larger in percentage terms (about 8% mean absolute change) than in the S&P 500 (about 6% mean absolute change).

Victor Niederhoffer writes: 

Excellent. You win the prize. Everything a report should be. (A little weak on the predictivity but not your fault.) If it was easy, we would all be wealthy men. 



 On Friday the world's greatest mathematician, Alexander Grothendieck, died, in Saint-Girons, Ariege.

More in Le Monde and the NYT, but I liked this quotation from Recoltes et Semailles:

New tasks forever call him to new scaffoldings, driven as he is by a need that he is perhaps alone to fully respond to. He belongs out in the open. He is the companion of the winds and isn't afraid of being entirely alone in his task, for months or even years or, if it should be necessary, his whole life, if no-one arrives to relieve him of his burden. He, like the rest of the world, hasn't more than two hands — yet two hands which, at every moment, know what they're doing, which do not shrink from the most arduous tasks, nor despise the most delicate, and are never resistant to learning to perform the innumerable list of things they may be called upon to do. Two hands, it isn't much, considering how the world is infinite. Yet, all the same, two hands, they are a lot…

McLarty's lecture is the best philosophical (rather than mathematical) take I know of on Grothendieck's work on the Weil conjectures. In summary, with his topos-theoretic approach he built a space tailor made to his problem, from the simplest of bits–and then let the space itself do the work.

Richard Owen writes: 

One presumes he has reincarnated as an interior designer? It's a shame he went off his loop a touch in the last decades. Or maybe he achieved a type of sanity we're not smart enough to understand:

The windows and blinds are all closed in most of the rooms of this mansion, no doubt from fear of being engulfed by winds blowing from no-one knows where. And, when the beautiful new furnishings, one after another with no regard for their provenance, begin to encumber and crowd out the space of their rooms even to the extent of pouring into the corridors, not one of these heirs wish to consider the possibility that their cozy, comforting universe may be cracking at the seams. Rather than facing the matter squarely, each in his own way tries to find some way of accommodating himself, one squeezing himself in between a Louis XV chest of drawers and a rattan rocking chair, another between a moldy grotesque statue and an Egyptian sarcophagus, yet another who, driven to desperation climbs, as best he can, a huge heterogeneous collapsing pile of chairs and benches!

Makes me think of: "One of my old supervisors told me that Wilhelm Reich went through three developmental phases as a theorist. In the first, he was not crazy and was not very creative, in the second he was a little bit crazy and very creative, and in the third, he was very crazy and not very creative."


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