## Excerpts from Alain Aspect’s talk:

“Quantum phenomena do not occur in a Hilbert space. They occur in a laboratory.”
-Asher Peres

“Truth and clarity are complemetary”
-Neils Bohr

Categories: Uncategorized

## Legendrian Knot Theory: Lecture 1

-Joan Licata. IAS

Knot theory is an elegant union of topology and geometry. We all know what a knot is.  Let’s look at the mathematical definition.

Defn. A knot is a smooth embedding $K: S^1 \rightarrow \mathbb{R}^3$

We are only interested in classes of equivalent knots. By equivalent I mean that knots obtained on translating or stretching each other should all be equivalent. This naturally leads to defining an isotopy.

Defn.  $K_0, K_1$ are isotopic if there is a homotopy  $K: S^1X [0,1] \rightarrow \mathbb{R}^3$ such that

• $\forall t, H(S^1Xt)$is an embedding
• $H(S^1X0)=K_0$
• $H(S^1X1)=K_1$

This simply means that you can go from one knot to the other by a smooth transformation. Think of ‘t’ as some sort of a parametrisation for the ‘path’ between the two knots.

Projections

On paper, knots are represented by 2-D diagrams called knot diagrams. Of course, we need to show the over and undercrossings to completely describe a knot. This is called a knot diagram. A projection is a diagram without this information. It is the ‘shadow’ of a knot.

Here is a common knot called ‘trefoil’ and its realisation as a physical model.

Fig 1: Isotopic knots: From Left to Right: a, b, c.

Fig 2: Models of the trefoil corresponding to a and b in Fig 1

In Fig 1, a and b are isotopic. One could deform a to get b without cutting the string. c is the Knot projection of b.

So far we have spoken of general knots. Now we come to a very important additional structure that makes Legendrian knots so special.

The standard contact structure

Defn. The standard contact structure $\xi_{std}$ on $\mathbb{R}^3$ is the 2 plane field such that at $(x,y,z)$,

• the normal vector is $\left[ \begin{array}{c} -y \\ 0\\1\end{array} \right]$
• (equivalent defn) The plane is spanned by $\left[ \begin{array}{c} 0 \\1\\ 0\end{array} \right]$ and $\left[ \begin{array}{c} 1\\ 0\\y\end{array} \right]$

By a ‘plane field’ we mean that there is a plane associated with each point is space, just like an electric field associates a vector with each point in space. Try sketching these planes in $\mathbb{R}^3$.  The planes don’t change along X! It will always be convenient to be an observer on the -Y axis, very far away from the origin, such that we are facing the X-Z plane. This is how the plane field looks.

Fig 3: Standard contact structure

Now we come to the definition on a Legendrian knot.

Defn. K is Legendrian (wrt $\xi_{std}$ if at every point on K, the tangent vector to K lies in $\xi_{std}$.

Note that this will qualify only certain special knots as Legendrian, of the infinite possibilities. Two Legendrian knots are isotopic if one can be deformed into another while always preserving the Legendrian condition.

See figure 4 for an  example of a Legendrian knot showing the contact structure.

Fig 4: Example of a Legendrian knot showing the contact structure

Defn The projection of a Legendrian knot on the XZ plane is called the front projection.

It would seem that projection would lose out a lot of information (in the Y direction) and make it impossible to reconstruct the knot simply by looking at its shadow. But Legendrian knots are special.  It  turns out that a projection  is enough to reconstruct a Legendrian knot. Let us see why.

Consider a point P on a the front projection of Legendrian knot K. This corresponds to the point P on the actual knot K. Consider the line L tangent to K at P. This line, by definition, must belong to the XZ plane. Moreover, the slope of the line, dz/dx, is nothing but the y-coordinate of P! Therefore the information lost by projecting is retrieved from the slope.

Observe the way the panes twist as one moves along the Y axis in Fig 3. A line on the +Y side of space will be seen having a positive slope in the front projection, a line on the -Y side will be seen to have a negative slope. Hence, just looking at the tangent lines of our front projection, we can tell which how the strands are oriented, which strands are in front and which go behind. You must have noticed that there seems to be a problem when the slope goes to infinity, i.e. for vertical tangents. It’s not really a problems since these appear as cusps in our projection.

Fig 5: Reconstructing a Legendrian knot. From left to right: a, b, c.

Observe figure 5. Let’s start with c. How do we know which of the strands goes behind and which is in the front? Use the thumb rule that the more negative slope is in the front (remember that our observer is at $-\infty$ on the Y axis, facing the XZ plane. Now you can easily see how a and b follow.

In the next lecture I will write about knot invariants.

Categories: Math Tags: ,

## Sudoku as a coloured graph

February 28, 2012 5 comments

Prereqs: Just the definition of a graph – vertices, edges and adjacent vertices.

2 years back I gave an exposition on counting the number of unique sudokus, which are not related to each other by the usual symmetry transformations like permuting cells, rows, etc. No more on that. You can read everything you want to know about this on http://www.afjarvis.staff.shef.ac.uk/sudoku/.  By the way, the number is 6670903752021072936960. Yes, it’s large enough to be comforted that you have enough unique sudokus to solve all your life. Unless you’re a computer.

Today I was thinking about how people can build a “well formed” sudoku (that is, one with a unique solution). People, mind you, not a sudoku-builder program that uses incremental or decremental generation to come up with a valid sudoku.

What is the minimum number of elements one needs to specify for the sudoku to have a unique solution?

This is not at all an easy question to answer. You may answer it with some work for a $4*4$ sudoku perhaps, but as you increase the size the question becomes way harder.  After thinking and searching for whatever literature I could find about this, I stumbled upon the most elegant solution I’ve seen in a while (Alright, I haven’t seen much. Granted. But this got me intrigued.) I shall discuss below the solution as proposed by Herzberg and Ram Murty in their paper.

Trivia: The minimum number is atmost 17. Nobody knows if a sudoku with 16 entries to start with can have a unique solution.

Basic graph theory in a nutshellI will not be writing about this but you can go to the hyperlink if you need a refresher.

_________________________________________

A sudoku as a coloured graph:

A coloured graph is a set of vertices and edges, with an addition variable called “colour” which each vertex is assigned. How do we view a sudoku as a coloured graph?

Consider the usual $9*9$ sudoku. Number all the cells from 1 through 81. These are your vertices. Now connect them as follows: each vertex is connected to all vertices in the same row, column and square ( by square I mean the $3*3$ squares that the sudoku is made up of.) Introduce 9 colours (Girls have the added advantage for visualization here as there are more complex colours on their palette – fuschia, turquoise and what not). Now assign each cell a colour out of these 9. To solve a sudoku, you need to assign these colours such that no two connected vertices have the same colour!

Let’s state this formally.

Proper colouring of a graph

A $\lambda-$ colouring of a graph G is a map f from the vertex set of G to {$1,2, \ldots, \lambda$} . This is called a proper colouring if f(x) $\neq$  f(y) whenever x and y are adjacent in G (Adjacent means that the two vertices of the graph are connected by an edge). So a sudoku puzzle is basically an incomplete colouring( called a partial colouring) which the solver needs to complete.

To quote the experts,

“A Sudoku puzzle corresponds to a partial coloring and the question is whether this partial coloring can be completed to a total proper coloring of the graph”

Does that make you

I hope it does, because otherwise it would imply I have a bad taste in exciting math problems. Anyway, back to work.

_______________________________

A regular graph is one in which the degree of each vertex ( i.e. the number of vertices it is connected to)  is the same. Any $n^2*n^2$ sudoku is a regular graph of degree $3n^2-2n-1$

Now for the punchline – the theorems which answer our question. I will simply state the 2 brilliant theorems here. You can read their equally brilliant proofs in the original paper.

Theorem 1: Let G be a finite graph with v vertices. Let C be a partial colouring of t vertices of G using $d_0$ colours. Let $p_{G,C}$($\lambda$) be the number of ways of completing this colouring by using $\lambda$ colours to obtain a proper colouring. Then $p_{G,C}$($\lambda$) is a monic polynomial in $\lambda$ with integer coefficients of degree v-t for $\lambda \geq d_0$

Implications: The number of ways of completing our $9*9$ sudoku is $p_{G,C}$(9). A unique solution is quaranteed if and only if $p_{G,C}$(9)=1.

____________________________________________

The minimal number of colours required to properly colour the vertices of a graph G is called the chromatic number of G and denoted $\chi(G)$

____________________________________________

Theorem 2: Let G b a graph with chromatic number $\chi(G)$ and C be a partial colouring of G using only $\chi(G)-2$ colours. If the partial coloring can be completed to a total proper coloring of G, then there are at least two ways of extending the colouring.

Implications: If C is a partial colouring of G that can be uniquely completed to a proper total colouring, then C must use at least $\chi(G)-1$ colours. So for our $9*9$ sudoku, at least 8 colours must be used in the given cells for the sudoku to be “well formed”

Note that with these theorems you can make statements about sudokus of all sizes!

____________________________________________

References:

Categories: Math Tags: , ,

## Writing ‘The history of love’

November 16, 2011 1 comment

(Princeton Public Library Distinguished Lecture Series)

People often describe reading as a means of escape. I read to arrive.

Introduction:

“Nicole Krauss is the author of the international bestsellers Great House, a finalist for the National Book Award and the Orange Prize, and The History of Love, which won the William Saroyan International Prize for Writing, France’s Prix du Meilleur Livre Ėtranger, and was short-listed for the Orange, Médicis, and Femina prizes. Her first novel, Man Walks Into a Room, was a finalist for the Los Angeles Times Book Award for First Fiction. In 2007, she was selected as one of Granta’s Best Young American Novelists, and in 2010 The New Yorker named her one of the 20 best writers under 40. Her fiction has been published in The New Yorker, Harper’s, Esquire, and Best American Short Stories, and her books have been translated into more than thirty-five languages. She lives in Brooklyn, New York.” Krauss majored in English from Stanford and did her Masters in Art history.

_______________________________________________________________________________________

The history of love (THOL):

“A long-lost book reappears, mysteriously connecting an old man searching for his son and a girl seeking a cure for her widowed mother’s loneliness.

Leo Gursky is just about surviving, tapping his radiator each evening to let his upstairs neighbor know he’s still alive. But life wasn’t always like this: sixty years ago, in the Polish village where he was born, Leo fell in love and wrote a book. And though Leo doesn’t know it, that book survived, inspiring fabulous circumstances, even love. Fourteen-year-old Alma was named after a character in that very book. And although she has her hands full—keeping track of her brother, Bird (who thinks he might be the Messiah), and taking copious notes on How to Survive in the Wild—she undertakes an adventure to find her namesake and save her family. With consummate, spellbinding skill, Nicole Krauss gradually draws together their stories.

This extraordinary book was inspired by the author’s four grandparents and by a pantheon of authors whose work is haunted by loss—Bruno Schulz, Franz Kafka, Isaac Babel, and more. It is truly a history of love: a tale brimming with laughter, irony, passion, and soaring imaginative power.”

_____________________________________________________________________________________________

(I shall write in first person from here on, using Krauss’ words as far as possible). I have also supplemented what she said in the lecture with lines from some of her earlier interviews.

____________________________________________________________________________________________

When I started writing THOL, I was a young author who had just published her first novel. Many questions bothered me.

How many people is enough people? How do you measure the impact of your writings on others? Why should one continue to write if one doesn’t know if it matters to others?

I wanted to write a book which very few people would read but which would have an impact on them and connect them together.

I am influenced by Bruno Schulz and knowing that there were lost manuscripts of one of my favourite author which I would never be able to read was incredibly painful to me. Almost everyone in the novel is a writer of some kind or another. Some of their books have never been read, some have been lost, some are written in journals, some published under the wrong name. And yet, being readers as well as writers, they’re all held together by the invisible threads that tie together those whose lives have been changed in some way by a certain book written sixty years ago.

What kind of a writer did I want to be?

I was ready to pose questions despite not having all the answers and to lose myself in the different strands of the book.

The fine line of distinction between personal and autobiographical:

The 14 year old Alma was initially too close to me for me to be unhindered regarding the development of her character. Writing should be intimate but not too autobiographical, since it leads to loss of freedom.

____________________________

Idea of structure:

I’m very interested in structure, how multiple stories are assembled in different ways; that is what memory does as well. I’ve always thought of novels as containers of memory. The idea is to juxtapose these fragments and create a work of art that could never have been made from those pieces in that order.

I was a poet for for several years of my life, and now here I am, as a lowly novelist. ‘Stanza’ is the Italian word for room. Each stanza of a poem is liek a room, which you can improve to perfection. But when you close the door, you finish the poem, it’s over!

Novels on the other hand, are houses, as opposed to these perfect rooms.

In a house, something or the other is always broken. The door is stuck, the roof is bad, there’s leakage..and so on. Novels, just like houses, are inherently imperfect. It’s upto you to define and decide the form of the novel and I found this immensely exciting.

_____________________________

Memory as a creative act:

My grandparents were Jews who were forced to leave Europe. Stories hat I’ve heard from them gave me this sense of nostalgia, the feeling that you can never go back. We empathize with people because we can remember our own experiences. But what if you don’t remember anything? Like the protagonist of ‘Man walks into a room’ who is found wandering in the Nevada desert with no memory of his previous life.

We forget vast portions of our lives. Instead we remember just a few, discrete moments which we string together to construct a narrative about ourselves. In a way, you are all fiction writers! Good luck!

We alter our past to make our lives bearable and to have a sense of self and coherence. THOL is a celebrations of this act of imagination to create a sense of self. Like Leo, a survivor of catastrophies, says in THOL:

Truth is the thing I invented, so that I could live.

________________________________

Novels and history, Jewish memories:

I feel novels tell us as much about a culture as do history books.  For the Jewish community which has been physically separated for so many years, stories have been critical in holding the community together. Writing novels is an effort to rewrite history in a somewhat bearable way.

________________________________

Writing a novel is getting a structural  blueprint of how your mind works and I would recommend it  to everyone just for that.

Categories: Literature Tags: ,

## Discrete Quantum Gravity

October 19, 2011 1 comment

-Stanley Gudder, University of Denver

Q. GR deals with smooth functions on spacetime. QM deals with self adjoint operators on a Hilbert space. How are they related?

A. GR and QM both have bad singularities. Maybe we shouldn’t be looking at the continuum but the discrete picture!

We discuss the causal set approach to unify gravity and quantum mechanics. Let us begin with light cones. We know about the future and past lightcones of an event, say a.

Source: Wikipedia

Introduction and definitions

We talk about the causal structure $(M,<)$ of a Lorentzian spacetime $(M,g)$. $M$ is a partially ordered set(Poset).

For $a, b \in M$, we say $a< b$ if b is in the causal future of a.

In the discrete situation, the smallest length is the Planck length $l_p \sim 1.6 \times 10^{-35}$ and the smallest time interval is $l_t \sim 5.4 \times 10^{-43}$

We call a finite poset a ‘causet

$\mathbb{P}_m$ = All causets of cardinality m.

$\mathbb{P} = \bigcup_m \mathbb{P}_m$

If $a, we call $a$ the ancestor and $b$ the successor.

If $a, we call $a$ the parent and $b$ the child if
$!\exists c \ni a

$a$ is maximal if $!\exists b \ni a

$X \in \mathbb{P}_m, Y \in \mathbb{P}_{m+1}$

X produces Y if Y is obtained from X by adjoining a single element to X that is maximal in Y. We call X the producer and Y the offspring.

A path in $\mathbb{P}$ is a string
$\omega= \omega_1 \omega_2 \ldots \ni \omega_i \in \mathbb{P}_i, \omega_{i+1} \in \mathbb{P}_{i+1}$

Each such path represents a ‘universe’. Every $l_t$ unit of time we have a new element in the path. Note that we have 2 notions of time here, the ‘chronological time’ and the ‘geometric time’.

An n-path is
$\omega= \omega_1 \omega_2 \ldots \omega_n$

$\Omega = \{\omega|\omega= \omega_1 \omega_2 \ldots \}$

$\Omega_n = \{\omega|\omega= \omega_1 \omega_2 \ldots \omega_n \}$

$Cylinder(\omega_0) = \{\omega \in \Omega| \omega= \omega_0 \ldots\}$

For $A \subseteq \Omega_n, cyl(A)= \bigcup_{\omega \in A} cyl(\omega)$

$a_n = \{cyl(A): A \subseteq \Omega_n\}$

We have the heirarchy
$a_1 \subseteq a_2 \subseteq a_3\subseteq \ldots$

$C(\Omega) = \bigcup a_n.$ This is an algebra of the subsets of $\Omega.$

If $X \rightarrow Y$ in $r$ isomorphic ways,
we write $m(X \rightarrow Y) = r$, where $m$ is the multiplicity.

Source: DQG by Stan Gudder

Classical Sequential Growth Processes

$C= (C_0, C_1, \ldots)$
$C_i \geq 0$ are the coupling constants.

For $r\leq s \in \mathbb{N}$ define
$\lambda_c(s,r) = \displaystyle\sum\limits_{k=0}^s \begin{pmatrix} s-r\\k-r\end{pmatrix} C_k$

$X \in \mathbb{P}_m, Y \in \mathbb{P}_{m+1}, X \rightarrow Y$

We define the transition probability as

$p_c(X \rightarrow Y) = m(X \rightarrow Y) \frac{\lambda_c(\alpha, \pi)}{\lambda_c(m, 0}$

where
$\alpha$ is the number of ancestors
$\pi$ is the number of elements adjoined to $X$ to get $Y$.

It is not obvious that this is a probability but this can be shown.

For the part of the talk from here onward, I will just sketch the outline here. We can define a corresponding quantum sequential growth process which leads to a theory of quantum gravity. I would encourage interested reader to read the original papers listed below.

Further readings and references-

## Photonic properties of non-crystalline solids

October 15, 2011 2 comments

- Paul Steinhardt, Princeton

Photonic crystals are semi-conductor devices for light, that is, with them we can get allowed and forbidden frequency bands. They are not the topic of discussion of this talk. Instead we ask ourselves,

Q. What do we know about the photonic properties of non-crystalline solids?

Q. Why talk about non-crystalline solids?

Schroedinger – Maxwell analogy

Schroedinger equation Maxwell’s Equations
Quantum Classical
“Scalar” (spin neglected) Vector. TM + TE modes. Different band gaps for each mode.
Complicated by e-e interaction Linear. No $\gamma - \gamma$ interaction.
Fundamental scale No fundamental scale limit. If the problem is solved for one scale, it is solved for all. We can choose whichever scale is convenient to work with experimentally.( Ignoring absorption, which is frequency dependent and can’t be scaled any way we want.)
Massive quanta. Parabolic dispersion Massless quanta. Linear dispersion
Atomic/molecular structures Can design continuous structures

2 distinct scattering mechanisms:

• Bragg scattering: ‘large scale’ resonance of the array
• Mie scattering: ‘small scale’ resonance of scatterers
To get the largest band gap, we find conditions such that Bragg and Mie scattering reinforce each other. The experimental challenge is that the TE and TM modes prefer different dielectric patterns to achieve an optimum bandgap. The search is for “complete band gaps“, which are the overlap of TE and TM, so that they are band gaps for both polarization modes. The optimum configuration was found to be cylinders connected by a trivalent network of walls.
In general the band gap for TE < TM. The complete gap is, obviously, narrower than both. The figure of merit of the band gap is delta(f)/f, where delta(f) is the width of the band gap.
The major disadvantage found was that the band gap is anisotropic, which meant less symmetry. ( Symmetry beyond 6-fold is forbidden). There has been no theoretical framework to optimize band gaps. Most of the results were achieved by brute force simulations on computer or by running experiments.
Non-crystalline solids:
Until 1984 solids were thought to be either crystalline or disordered. Today we know of quasicrystals and many new classes of disordered solids.
Q. Why even consider non-crystalline solids?
• Higher symmetry may give wider band gaps, at least for some dielectric constant ratios.
• New types of structures, different modes and defects can find new applications.

Octogonal quasicrystal

In their pape rin 1984, Paul Steinhardt and D. Levine talk about quasicrystals,  the natural extension of the notion of a crystal to structures with quasiperiodic, rather than periodic, translational order. In their paper in Nature in 2005,  W. Man et al show that quasicrystals can have optical bandgaps rendering them useful to technology and also talk about the experimental measurement of the photonic properties of icosahedral quasicrystals.
It was found that quasicrystals beat diamond, the previous record holder, in terms of symmetry and wider band gaps. To find a more evenly distributed network, Delaunay tesselation was used.
Q. What if we applied it to isotropic disordered point pattern such as Poisson or hyperuniform?
Type Figure of merit
Crystal 19.3%
Quasicrystal 16.5%
Disordered 10.2% (isotropic rings – Bragg pattern)

We see that have a tradeoff between band gap and isotropy.

## Optics, Self-Assembly, and Evolution of Aperiodic Color Producing Structures from Birds and Insects

October 15, 2011 1 comment

- Richard Prum, Yale.

“I know what you are! You are a bird-watcher in q-space!”

When we think of colors in nature, the things that comes to mind are pigments and dyes. However, nature makes extensive use of nano structures to produce the vibrant hues that captivate us. These structural colors make use of interference patterns of light and hence depends on the angle of observation, unlike those produced by pigments. This is known as Iridescence. In most birds these are made by melanin-keratin nano-structures in feather barbules.

The structures are periodic in 1D, 2D or less commonly, 3D.

Velvet Asity

The male Velvet Asity, or Philepitta castanea, produces the brilliant green and blue colors from by the hexagonal array of parallel collagen fibers in its caruncle tissue. Since this is periodic, we can use Bragg’s law to describe the interference.

But in some birds quasi-ordered arrays were found. A simple description using Bragg’s law would no longer work. This was found in mammals too, like the vervet monkey.

The speaker presented a 2D fourier analysis of spatial variation in the refractive index of the feather barb of Cotinga Maynana. The colors are non-iridescent under diffusive light illumination like in nature, but iridescent under artificial directional lighting often used in the laboratory.

There also exist  channel-type nanostructures consisting of beta-keratin bars and air channels in elongate and tortuous forms. Sphere-type nanostructures consist of spherical air cavities in a beta-keratin matrix.

Q. How does spongy medullary keratin self-assemble?

A. Phase separation by Spinodal decomposition (SD)

SD is an unstable phase separation. Phase separation in spongy cells is “arrested” at a specific size to produce a specific color.

Phase separation in 3D is seen in butterflies, which get their colors from mathematical objects called gyroids.

A gyroid is a minimal surface, meaning its mean curvature vanishes. Minimal surfaces include, but are not limited to, surfaces of minimum area subject to various constraints.  The planes of a gyroid’s surface never intersect. Gyroids also contain no straight lines, and can never be divided into symmetrical parts

In butterflies this is formed by hydrophilic-hydrophobic interactions in copolymers.

Further readings and references :

Categories: Biology, Condensed matter