### Ravi Kunjwal

Shared publicly -Scientific publishing requires patience, even after a paper has been formally accepted.

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Ravi Kunjwal

Worked at Perimeter Institute for Theoretical Physics

Attends The Institute of Mathematical Sciences ("Matscience"), Chennai

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Scientific publishing requires patience, even after a paper has been formally accepted.

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The AdS/MERA correspondence has been making the rounds of the blogosphere with nice posts by Scott Aaronson and Sean Carroll, so let's take a look at the topic here at Quantum Frontiers. The questi...

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I decided to add my name to a petition by, as of this writing, 81 MIT faculty, calling on MIT to divest its endowment from fossil fuel companies. (My co-signatories include Noam Chomsky, so I guess there's something we agree about!) There's also a wider petition signed by nearly 3500 MIT ...

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"Do you want to impress me with your moral backbone? Then go and find a group that almost all of your Facebook friends still consider it okay, even praiseworthy, to despise and mock, for moral failings that either aren’t failings at all or are no worse than the rest of humanity’s. (I promise: once you start looking, it shouldn’t be hard to find.) Then take a public stand for that group."

Yesterday was a historic day for the United States, and I was as delighted as everyone else I know. I've supported gay marriage since the mid-1990s, when as a teenager, I read Andrew Hodges' classic biography of Alan Turing, and burned with white-hot rage at Turing's treatment.

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"The puzzle becomes a bit more interesting when you learn that you can't find a single equation that defines distributive lattices: you need 2. "

In math the rules of a game are called

I'm not sure, but I have a candidate. A

And in 1970 someone solved it:

Before I go into details, I should say a bit about lattices. The concept of a lattice is far from pointless - there are lattices all over the place!

For example, suppose you take integers, or real numbers. Let x ∨ y be the

Or, suppose you take statements. Let p ∨ q be the statement "p or q", and let p ∧ q be the statement "p and q". Then the 6 axioms here hold!

For example, consider the axiom p ∧ (p ∨ q) = p. If you say "it's raining, and it's also raining or snowing", that means the same thing as "it's raining" - which is why people don't usually say this.

The two examples I just gave obey other axioms, too. They're both

p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r)

and the rule with ∧ and ∨ switched:

p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r)

But nondistributive lattices are also important. For example, in quantum logic, "or" and "and" don't obey these distributive laws!

Anyway, back to the main story. In 1970, Ralph McKenzie proved that you can write down a single equation that is equivalent to the 6 lattice axioms. But it was an equation containing 34 variables and roughly 300,000 symbols! It was too long for him to actually bother writing it down. Instead, he proved that you

Later this work was improved. In 1977, Ranganathan Padmanabhan found an equation in 7 variables with 243 symbols that did the job. In 1996 he teamed up with William McCune and found an equation with the same number of variables and only 79 symbols that defined lattices. And so on...

The best result I know is by McCune, Padmanbhan and Robert Veroff. In 2003 they discovered that this equation does the job:

(((y∨x)∧x)∨(((z∧(x∨x))∨(u∧x))∧v))∧(w∨((s∨x)∧(x∨t))) = x

They also found another equation, equally long, that also works.

That is

How did they find these equations? They checked about a half a trillion possible axioms using a computer, and ruled out all but 100,000 candidates by showing that certain non-lattices obey those axioms. Then they used a computer program called OTTER to go through the remaining candidates and search for proofs that they are equivalent to the usual axioms of a lattice.

Not all these proof searches ended in success or failure... some took too long. So, there could still exist a single equation, shorter than the ones they found, that defines the concept of lattice.

Here is their paper:

• William McCune, Ranganathan Padmanabhan, Robert Veroff, Yet another single law for lattices, http://arxiv.org/abs/math/0307284.

By the way:

When I said "it's a pointless puzzle, as far as I can tell", that's not supposed to be an insult. I simply mean that I don't see how to connect this puzzle - "is there a single equation that does the job?" - to themes in mathematics that I consider important. It's always possible to learn more and change ones mind about these things.

The puzzle becomes a bit more interesting when you learn that you can't find a single equation that defines

By contrast, "varieties of semigroups where every element is idempotent" can always be defined using just a single equation. This was rather shocking to me.

However, I still don't see any point to reducing the number of equations to the bare minimum! In practice, it's better to have a larger number of

#spnetwork arXiv:math/0307284 #lattice #variety

#bigness

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Work

Occupation

Graduate student (Theoretical Physics) at IMSc, Chennai.

Employment

- Perimeter Institute for Theoretical PhysicsVisiting Graduate Fellow
- The Institute of Mathematical SciencesGraduate student ("Senior Research Fellow")

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Gender

Male

Looking for

Friends, Networking

Story

Tagline

i came, i saw, and i still wonder

Education

- The Institute of Mathematical Sciences ("Matscience"), ChennaiTheoretical Physics, 2010 - present
- St. Stephen's College, DelhiPhysics, 2007 - 2010
- St. Thomas' College, Dehradun2002 - 2007
- St. John Bosco College, Lucknow1993 - 2002

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