Category Archives: Math

James Hartle R. I. P.

Jim Hartle, one of the smartest guys in my college class has died, and of Alzheimer’s disease, showing once again that intelligence does not absolutely protect against Alzheimer’s (although the more educated you are the less likely you are to get it [ Int. J. Epidemiol. Volume 49, Issue 4, August 2020, Pages 1163–1172 ].

He studied with John Wheeler as a Princeton undergraduate, got his PhD with Murray Gell-Mann and worked so extensively with Stephen Hawking that he was asked to speak at Hawking’s funeral.

My total person to person contact with Jim may have lasted 10 minutes (at my 50th reunion).  I knew all sorts of physics majors in the class, but he wasn’t one of them.  I knew about him only because I read our  50th reunion book, and found out how distinguished he was.  I found him relaxed, friendly and far from overbearing, like some of the physics majors I knew.
This was typical of just about everyone at the 50th.  A classmate’s wife (from Chile) described classmates at the 25th as a bunch of roosters.
We did correspond a bit, and he did send me the answer sheets to the problems in his book on Gravity (which I’ve never gone through preferring to get the math under my belt first rather than mouth various incantations which I didn’t understand).  Jim is the reason I started studying Math and Physics in earnest, hoping to have something intelligent to say to him at the next reunion.  He wasn’t present at the 55th,   COVID19 ended the 60th and now he’s gone.
Two more examples of brilliant men you might know of who died of Alzheimer’s are Daniel Quillen Harvard ’61 who won the Fields Medal and Claude Shannon.

The physics department at University of California Santa Barbara has an obituary describing much of his work

How far we’ve come from the McCulloch Pitts neuron

The McCulloch Pitts neuron was described in 1943.  It consists of a bunch of inputs (dendrites) some excitatory, some inhibitory, which are just summed (integrated) the results determining the output (whether the  axon of the neuron fired or didn’t).  Hooking them together could instantiate a variety of boolean functions and ultimately a Turing machine.

The McCulloch Pitts neuron really isn’t that far from the ‘neurons’ in neural nets which underlie the spectacular achievements of artificial intelligence (ChatGTP etc. etc.)   The neuron of the neural net is nothing more than a set of inputs, a set of weights, and an activation function. The neuron translates these inputs into a single output, which can then be picked up as input for another layer of neurons later on.

The major difference between the computation a linked bunch of neurons in the two models (McCulloch Pitts and neural net) is that given the same set of inputs in McCulloch Pitts you always get the same output, while in neural nets you don’t.  The difference is that the set of weights on the inputs to each neuron in the net which can be and are adjusted which depends on how close the output of the net is to the target (which in the case of ChatGTP is how accurately it predicts the next word in a sample of text).

There is a huge debate going on as to whether ChatGTP and similar neural nets understand what they are doing and whether they are/will become conscious.

So does ChatGTP explain how our brains do what they do?  Not at all.  Our neurons are doing far more than integrating input and firing.  This was brought home in a paper focused on something entirely different, the gamma oscillations of brain electrical activity (Neuron vol. 111 pp. 936 – 953 ’23).  People have been studying brain rhythms since Hans Berger discovered alpha rhythm just shy of a century ago.  The electroencephalogram (EEG) measures the various rhythms as they occur over the brain.  Back in the day when I was starting out in neurology (1967), it was one of the few diagnostic tools we had.  It wasn’t very good, and a cynical attending described it as useless but not worthless (because you could charge for it).

The gray matter of the surface of our brains (cerebral cortex) is gray because it is packed with the cell bodies of neurons — some 100,000 under each square millimeter of cortex.  Somehow they are wired together so that they can produce coherent rhythmic electrical activity as they fire.

The best place to study how a bunch neurons produce rhythms is the hippocampus, an area crucial in forming memories and one of the earliest places the senile plaques of Alzheimer’s disease show up.

Unlike the jumble of neurons in the cortex, the large neurons of the hippocampus are all lined up and oriented the same way like trees in a forest.  All the cell bodies lie in roughly the same layer, with the major dendrite (apical dendrite) going up like the trunk of a tree, and the ones near the cell body spreading out like the roots of a tree.

Technology has marched on, and it is now possible to fashion electrodes, which can measure neuronal electrical activity along the trunk, and watch it in real time.

Figure 2b p. 941 shows that different parts of the trunk of the hippocampal  neurons show rhythmic activity at different frequencies at any given time.  Not only that, but as time passes each area of the trunk (apical dendrite) changes the frequency of its rhythmic activity.  This is light years away from the integrate and fire model of McCulloch Pitts, or the adjustment of weights on the inputs to the neurons of the neuronal net.

It shows that each of these neurons is a complex processor of information (a computer if you will).  Even though artificial intelligence has made great strides, it really isn’t telling us how the brain does what it does.

Finally if you want to see what genius looks like, check out the life of Walter Pitts —  — corresponding with Bertrand Russell about Principia Mathematica at age 12, studying with Carnap at the University of Chicago at 15, all while he was homeless.


What does (∂h/∂x)dx + (∂h/∂y)dy + (∂h/∂z)dz really mean?

We’ve all seen (∂h/∂x)dx + (∂h/∂y)dy + (∂h/∂z)dz many times and have used it to calculate without understanding what the various symbols  actually mean.  In my case, it’s just another example of mouthing mathematical incantations without understanding them, something I became very good at at young age — see for the gory details.

And now, finally, within a month of my 85th birthday, I finally understand what’s going on by reading only the first 25 pages of “Elementary Differential Geometry” revised second edition 2006 by Barrett O’Neill.

I was pointed to it by the marvelous Visual Differential Geometry by Tristan Needham, about which I’ve written 3 posts — this link has references to the other two —

He describes O’Neill’s book as follows.  “First published in 1966, this trail-blazing text pioneered the use of Forms at the undergraduate level.  Today more than a half-century later, O’Neill’s work remains, in my view the single most clear-eyed, elegant and (ironically) modern treatment of the subject available — present company excepted! — at the undergraduate level”

It took a lot of work to untangle the notation (typical of all works in Differential Geometry). There is an old joke “differential geometry is the study of properties that are invariant under change of notation” which is funny because it is so close to the truth (John M. Lee)

So armed with no more than calculus 101, knowing what a vector space is,  and a good deal of notational patience, the meaning of (∂h/∂x)dx + (∂h/∂y)dy + (∂h/∂z)dz (including what dx, dy and dz really are) should be within your grasp.

We begin with R^3, the set of triples of real numbers (a_1, a_2, a_3) where _ means that 1, 2, 3 are taken as subscripts). Interestingly, these aren’t vectors to O’Neill which will be defined shortly.  All 3 components of a triple can be multiplied by a real number c — giving (c*a_1, c*a_2, c*a_3). Pairs of triples can be added.  This makes R^3 into a vector space (which O’Neill calls Euclidean 3-space), the components of which are triples (which O’Neill calls points), but that is not how O’Neill defines a vector, which are pairs of points p = (p_1, p_2, p_3) and v = (v_1, v_2, v_3) — we’ll see why shortly.

A tangent vector to point p in R^3 is called a tangent vector to p (and is written v_p) and is defined as an ordered pair of points (p, v) where

p is the point of application of v_p (aka the tail of p)

v is the vector part of v_p (aka the tip of v_p)

It is visualized as an arrow whose tail is at p and whose tip (barb) is at  p + v (remember you are allowed to add points).  In the visualization of v_p, v does not appear.

The tangent space of R^3 at p is written T_pR^3 and is the set of vectors (p, v) such that p is constant and v varies over all possible points.

Each p in R^3 has its own tangent space, and tangent vectors in different tangent spaces can’t be added.

Next up functions.

A real value function on R^3 is written

f :  R^3 –> R^1 (the real numbers)

f : (a_1, a_2, a_3) |—> c (some real number)

This is typical of the way functions are written in more advanced math, with the first line giving the domain (R^3) of the function and the range of the function (R^1) and the second line giving what happens to a typical element of the domain on application of the function to it.

O’Neill assumes that all the functions on domain R^3 have continuous derivatives of all orders.  So the functions are smooth, differentiable or C^infinity — take your pick — they are equivalent.

The assumption of differentiability means that you have some mechanism for seeing how close two points are to each other.  He doesn’t say it until later, but this assumes the usual distance metric using the Pythagorean theorem — if you’ve taken calc. 101 you know what these are.

For mental visualization it’s better to think of the function as from R^2 (x and y variables — e.g,. the Euclidean plane) to the real numbers.  This is the classic topographic map, which tells how high over the ground you are at each point.

Now at last we’re getting close to (∂f/∂x)dx + (∂f/∂y)dy + (∂f/∂z)dz.

So now you’re on a ridge ascending to the summit of your favorite mountain.  The height function tells you how high your are where you’re standing (call this point p), but what you really want to know is which way to go to get to the peak.  You want to find a direction in which height is increasing.   Enter the directional derivative (of the height function)  Clearly height drops off on either side of the ridge and increases or decreases along the ridge.   Equally clearly there is no single directional derivative here (as there would be for a function g : R^1 –> R^1).  The directional derivative  depends on p (where you are) and v the direction you choose — this is why O’Neill defines tangent vectors by two points (p, and v)

So the directional derivative requires two functions

the height function h : R^3 –> R^1

the direction function f : p + t*v where t is in R^1.  This gives the a line through p going in direction v

So the directional derivative of  h at p is

d/dt  (h (p + t*v)) | _t = 0  ; take the limit of h (p + t*v)  as t approaches zero

Causing me a lot of confusion, O’Neill gives the directional derivative the following name v_p[h] — which gives you no sense that a derivative of anything is involved.  This is his equation

v_p[f] = d/dt  (h (p + t*v)) | _t = 0

Notice that changing p (say to the peak of the mountain) changes the directional derivative —  all of them point down.   This is why O’Neill defines tangent vectors using two points (p, v).

Now a few more functions and the chain rule and we’re about done.

x :    R^3                    –>  R^1

x : (v_1, v_2, v_3 ) |–>  v_1

similarly y :R^3 –> R^1 picks out the y coordinate of (v_1, v_2, v_3 )  e.g. v_2

Let’s look at p + t*v in coordinate form, remembering what p and v are that way

p + t*v  = ( p_1 + t * v_1, p_2 + t * v_2, p_3 + t * v_3)

Remember that we defined f = p  + t *v

so df/dt = d( p + t*v )/dt


df’/dt= d( p_1 + t * v_1, p_2 + t * v_2, p_3 + t * v_3)/dt = (v_1, v_2, v_3)

Let’s be definite about what h : R^3 –> R^1 actually is

h : (x, y, z) |—> x^2 * y^3 *z ^4 meaning you must use partial derivatives

so ∂h/∂x = 2 x * y^3 * z* 4,  etc.,

Look at v_p[h] = d/dt  (h (p + t*v)) | _t = 0 again

It’s really v_p[h] = d/dt (h (f (t))|_=0

so it’s time for the chain rule

d/dt (h (f (t)) = (dh/df ) * (df/dt)

dh/df in coordinates is really

(∂h/∂x, ∂h/∂y,∂h/∂z)

df/dt in coordinates is really

(v_1, v_2, v_3)

But the chain rule is applied to each of the three terms

so what you have is d/dt (h (f (t))  = (∂h/∂x * v_1,  ∂h/∂y * v_2, ∂h/∂z * v_3)

I left one thing out.  The |_=0

So to do this you need to plug in the numbers (evaluating everything at p) and sum so what you get is

v_p[h] = ∂h/∂x * v_1 +  ∂h/∂y * v_2 +  ∂h/∂z * v_3

We need one more definition. Recall that the tangent space of R^3 at p is written T_pR^3 and is the set of vectors (p, v) such that p is constant and v varies over all possible points.

The set of all tangent spaces over R^3 is written (TR^3)

Finally on p. 24 O’Neill defines what you’ve all been waiting for :  dh

dh : TR^3 –> R^1

dh : p  ——>  v_p[h] = ∂h/∂x * v_1 +  ∂h/∂y * v_2 +  ∂h/∂z * v_3

One last bit of manipulation — what is dx (and dy and dz)?

we know that  the function x is defined as follows

x :    R^3                    –>  R^1

x : (v_1, v_2, v_3 ) |–>  v_1

so dx = (dx/dx, dx/dy, dx/dz)|_=0

is just  v_1

so at (very) long last we have

dh : TR^3 –> R^1

dh : p  ——>  v_p[h] = ∂h/∂x * dx +  ∂h/∂y * dy +  ∂h/∂z * dz

Remember ∂h/∂x, ∂h/∂y,  ∂h/∂z are all evaluated at p = (p_1, p_2, p_3)

So it’s a (fairly) simple matter to apply dh to any point p in R^3 and any direction  (v_1, v_2, v_3) in R^3 to get the directional derivative

Amen. Selah.

What AlphaZero ‘knows’ about Chess

I’m a lousy chess player.  When I was 12, my 7 year old brother beat me regularly.  Fortunately chess ability doesn’t correlate with intelligence, as another pair of brothers will show.   The younger brother could beat the older one at similar ages, but as the years passed, the older brother became a Rhodes scholar, while the severely handicapped younger brother (due to encephalitis at one month of age) is currently living in a group home.

A fascinating paper [ Proc. Natl. Acad. Sci vol. 119 e2206625119 ’22 ] opens the black box of AlphaZero, a neural net which is currently the world champion, to see what it ‘knows’ about chess as it relentlessly plays itself to build up expertise.

The paper is highly technical and I don’t claim to understand all of it (or even most of it), but it’s likely behind a paywall so you’ll have to content yourself with this unless you subscribe ($235/year for the online edition). The first computer chess machines used a bunch of rules developed by expert chess players.  Neural nets require training.  For picture classification they required thousands and thousands of pictures, and feed back about whether they got it right or wrong.  Then the probability of firing between elements of the net (neurons) was adjusted up if the answer was correct, down otherwise.  This is supervised learning.

Game playing machines are unsupervised, they just play thousands and millions of games against themselves (AlphaZero played one million).  Gradually they get better and better, so they beat humans and earlier rule based machines.  A net that has played 32,000 games beats the same net that has played 16,000 games 100 games out of 100 games.  However the 128,000 beats 64,000 only 64 times.

They they had a world chess champion (V.K.) analyze how the machines were playing.

Between 16,000 and 32,000 plays the net began to understand the relative values of the pieces (anything vs. pawns, queen vs. rook etc. etc.)

Between 32,000 and 64,000 king safety appeared

Between 64,000 and 128,00 games which attack was most likely to succeed appeared.

Showing that there is no perfect strategy, separate 1,000,000 runs of the machine settled on two variants of the (extremely popular) Ruy Lopez opening.

They studied recorded human games (between experts or they wouldn’t have been recorded) in the past 500 years.  Initially most people played the same way, with variants appearing as the years passed.  The neural net was just the opposite, trying lots of different things initially and subsequently settling on few approaches.

All in all, a fascinating look inside the black box of a neural net.

History repeats itself

The stories of young and not so young Russians running for the exits to escape the Tsar’s (Putin’s) army, resonates with me as both my grandfathers did exactly that in the 1880’s – 1890’s.  What really brought it home was the fact we just found out this year that my mother’s father’s last name was not what we’ve thought it was. He was adopted by another family with many children and given their name to avoid conscription.  My father’s father got out because he was the first born son, and in line for a lifetime in the Tsar’s army.

Ah Russia !  The gift that keeps on giving.

Here’s another example which is part of an old post

Hitler’s gifts (and Russia’s gift)

In the summer of 1984 Barack Obama was at Harvard Law, his future wife was a Princeton undergraduate, and Edward Frenkel, a 16 year old mathematical prodigy, was being examined for admission to Moscow State University. He didn’t get in because he was Jewish. His blow by blow description of the 5 hour oral exam designed to exclude him on pp. 28 – 38 of his book “Love & Math” is as painful to read as it must have been for him to write.

Harvard recognized his talent, and made him a visiting professor at age 21, later enrolling him in grad school so he could get a PhD. He’s now a Stanford prof.

Here’s a link to the full post  –


Book Review: Proving Ground, Kathy Kleiman

Proving Ground is a fascinating book about the 6 women who programmed the first programmable computer, the ENIAC (Electronic Numerical Integrator And Computer).  Prior to this, the women were computers as the term was used in the 1940s for people who sat in front of calculating machines and performed lengthy numerical computations solving differential equations to find the path of an artillery shell one bloody addition/subtraction/multiplication/division at a time.  When World War II started and when the man were off in the army, the search was on for  women with a mathematical background who could do this.

A single trajectory took a day to calculate, and each trajectory had to be separately calculated for different wind currents, air temperature, speed and weight of the shell.  The computations were largely done at the Moore School of Engineering at Penn and were way too slow (although accurate) to produce the numbers of trajectories the army needed.

Enter Dr. John Mauchley who had an idea of how to do this using vacuum tubes, and a brilliant 23 year old engineer, J. Presper Eckert, who could instantiate it. The army committed money to building the machine, which came in 42 monster boxes 8 feet tall, 2 feet wide and what looks like 4 feet deep.

6 of the best and brightest computers of trajectories were recruited to figure out how to wire the boxes together to mimic the trajectory calculations they had already been doing.  So, if you’ve ever done any programming, you’ll know that having a definite target to mimic with software makes life much easier.

Going a bit deeper, if you’ve done any programming in machine language, you know about registers, the addition and logical unit, hard wired memory, alterable memory.

Here’s what the 6 women were given by Dr. Eckert (without ever seeing the monster boxes)

l. A circuit diagram of each box, showing how this vacuum tube activated that vacuum tube etc. etc. The 42 boxes contained 18,000 vacuum tubes.  Vacuum tubes and transistors are similar in that their utility is that they only conduct electricity in one direction and can be turned on and off.

2. A block diagram — which showed how the functions of a unit or system interrelate

3. A logical diagram — places for dials switches, plug and cables on the front of the 42 units.

So given this, the 6 had to figure out what each unit did, and how to wire them together to mimic the trajectory calculations they had been doing.

They did it, and initially without being able to enter the room with the boxes (because they didn’t have the proper security clearance).  Eventually they got it and were able to figure out how to wire the boxes together.

If that isn’t brilliant enough, because the calculations were still taking too long, they invented parallel programing.

For those of you who know computing, that should be enough to make you thirst for more detail.

The book contains a lot of sociology.  The women were treated like dirt by the higher ups (but not by Mauchley or Eckert).  When the time came to show ENIAC off to the brass (both academic and military), they were tasked with serving coffee and hanging up coats.  When Kleiman found pictures of them with ENIAC and asked who they were, she was told they were ‘refrigerator ladies’ — whose function was similar to the barely clothed models draped over high powered automobiles to sell them.

I’ll skip the book’s sociology for some sociology of my own.  The book has biographies and much fascinating detail about all 6 women.  I grew up near Philly, and know the buildings at Penn where this was done (I went to Penn Med). Two of the 6 were graduates of Chestnut Hill College, a small Catholic school west of Philly.  The girl across the street went there.  Her mother was born in County Donegal and cleaned houses.  Her father dropped out of high school at 16 to support his widowed mother.  No social services between the two world wars, wasn’t that terrible etc. etc.  Her father worked in a lumberyard, yet the two of them sent both children to college, and owned their own home (eventually free of debt).  The Chestnut Hill grad I know became an editor at Harcourt Brace, her brother became a millionaire insurance executive.  It would be impossible for two working class people to do this today where I grew up (or probably in most places).

What is dx really?

“Differential geometry is the study of properties that are invariant under change of notion”  — Preface p. vii of “Introduction to Smooth Manifolds” J. M. Lee second edition.  Lee says this is “funny primarily because it is so close to the truth”.

Having ascended to a substantial base camp for the assault on the Einstein Field equations (e.g. understanding the Riemann curvature tensor), I thought I’d take a break and follow Needham’s advice about another math book “Elementary Differential Geometry” 2nd edition (revised 2006) by Barrett O’Neill.  “First published in 1966, this trail-blazing text pioneered the use of Forms at the undergraduate level.  Today, more than a half-century later, O’Neill’s work remains, in my view, the single most clear-eyed, elegant and (ironically) modern treatment of the subject available — present company excepted! — at the undergraduate level.”

Anyone taking calculus has seen plenty of dx’s — in derivatives, in integrals etc. etc..  They’re rarely explained.  O’Neill will get you there in just the first 24 pages.  One more page and you’ll understand

df =  (partial f/partial x) * dx + (partial f/partial y) * dy + (partial f/partial z) * dz

which you’ve doubtless seen before primarily as (heiroglyphics) before you moved on.

Is it easy?  No, not unless you read definitions and internalize them immediately.  The definitions are very clearly explained.

His definition of vector is a bit different — two points in Euclidean 3-space (written R^3 which is the only space he talks about in the first 25 pages).  His 3-space is actually a vector space in which point can be added and scalar multiplied.

You’ll need to bone up on the chain rule from calculus 101.

A few more definitions — natural coordinate functions, tangent space to R^3 at p, vector field on R^3, pointwise principle, natural frame field, Euclidean coordinate function (written x_i, where i is in { 1, 2, 3 } ), derivative of a function in direction of vector v (e.g. directional derivative), operation of a vector on a function, tangent vector, 1-form, dual space. I had to write them down to keep them straight as they’re mixed in paragraphs containing explanatory text.


at long last,

differential of x_i (written dx_i)

All is not perfect.  On p. 28 you are introduced to the alternation rule

dx_i ^ dx_j = – dx_j ^ dx_i with no justification whatsoever

On p. 30 you are given the formula for the exterior derivative of a one form again with no justification.  So its back to mumbling incantations and plug and chug

Understanding the Riemann curvature tensor is like doing a spinal tap

Back in the day when I was doing spinal taps, I spent far more time setting them up (positioning the patient so that the long axis of the spinal column was parallel to the floor and the vertical axis of the recumbent patient was perpendicular to the floor) than actually doing the tap.  Why? because then, all I had to do was have the needle parallel to the floor, with no guessing about how to angle it when the patient had rolled (usually forward into the less than firm mattress of the hospital bed).

So it is with finally seeing what the Riemann curvature tensor actually is, why it is the way it is, and why the notation describing it is such a mess.  Finally on p. 290 of Needham’s marvelous book “Visual Differential Geometry and Forms” the Riemann curvature tensor is revealed in all its glory.  Understanding it takes far less time than understanding the mathematical (and geometric) scaffolding required to describe it, a la spinal taps.

Over the years while studying relativity, I’ve seen it in one form or other (always algebraic) without understanding what the algebra was really describing.

Needham will get you there, but you have to understand a lot of other things first. Fortunately almost all of them are described visually, so you see what the algebra is trying to describe.  Along the way you will see what the Lie bracket of two vector fields is all about along with holonomy.  And you will really understand what curvature is.  And Needham will give you 3 ways to understand parallel transport (which underlies everything — thanks Ashutosh)

Needham starts off with Gauss’s definition of curvature of a surface — the angular excess of a triangle, divided by its area.

Here is why this definition is enough to show you why the surface of a sphere is curved.   Go to the equator.  Mark point one, then point two 1/4 of the way around the sphere.  Form longitudes (perpendiculars) there and extend them as great circles toward the North pole. You now have a triangle containing 3 right angles, (clearly an angular excess from Euclid who states that the sum the angles of a triangle is two right angles).  The reason, of course, is because the sphere is curved.

Ever since I met a classmate 12 years ago at a college reunion who was a relativist working with Hawking, I decided to try to learn relativity so I’d have something intelligent to say to him if we ever met again (COVID19 stopped all that although we’re still both alive).

Now that I understand what the math of relativity is trying to describe, I may be able to understand general relativity.

Be prepared for a lot of work, but do start with Needham’s book.  Here are some links to other things I’ve written about it.  It will take some getting used to as it is very different from any math book you’ve ever read (except Needham’s other book).

12 July 21 —

4 Dec 21 —

7 Mar 22 —

27 June 22 —

17 July 22 —

Apologies for another posting delay

Hopefully the post on the paper I’m so impressed with will be out in the next few days.  I’ve been clearing away the underbrush in Needham’s Visual Differential Geometry and Forms before the final push on the Einstein field equation and Riemannian geometry.

Apologies for the delay

Here’s a clue for you all to think about — what effects does proline have on (1) the alpha helix (2) the beta pleated sheet?

Bye bye stoichiometry

I’m republishing this old post from 2018, to refresh my memory (and yours) about liquid liquid phase separation before writing a new post on one of the most interesting papers I’ve read in recent years.  The field has exploded since this was written.

Until recently, developments in physics basically followed earlier work by mathematicians Think relativity following Riemannian geometry by 40 years.  However in the past few decades, physicists have developed mathematical concepts before the mathematicians — think mirror symmetry which came out of string theory — You may skip the following paragraph, but here is what it meant to mathematics — from a description of a 400+ page book by Amherst College’s own David A. Cox

Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is the first completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kahler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem. This title features: numerous examples worked out in detail; an appendix on mathematical physics; an exposition of the algebraic theory of Gromov-Witten invariants and quantum cohomology; and, a proof of the mirror theorem for the quintic threefold.

Similarly, advances in cellular biology have come from chemistry.  Think DNA and protein structure, enzyme analysis.  However, cell biology is now beginning to return the favor and instruct chemistry by giving it new objects to study. Think phase transitions in the cell, liquid liquid phase separation, liquid droplets, and many other names (the field is in flux) as chemists begin to explore them.  Unlike most chemical objects, they are big, or they wouldn’t have been visible microscopically, so they contain many, many more molecules than chemists are used to dealing with.

These objects do not have any sort of definite stiochiometry and are made of RNA and the proteins which bind them (and sometimes DNA).  They go by any number of names (processing bodies, stress granules, nuclear speckles, Cajal bodies, Promyelocytic leukemia bodies, germline P granules.  Recent work has shown that DNA may be compacted similarly using the linker histone [ PNAS vol.  115 pp.11964 – 11969 ’18 ]

The objects are defined essentially by looking at them.  By golly they look like liquid drops, and they fuse and separate just like drops of water.  Once this is done they are analyzed chemically to see what’s in them.  I don’t think theory can predict them now, and they were never predicted a priori as far as I know.

No chemist in their right mind would have made them to study.  For one thing they contain tens to hundreds of different molecules.  Imagine trying to get a grant to see what would happen if you threw that many different RNAs and proteins together in varying concentrations.  Physicists have worked for years on phase transitions (but usually with a single molecule — think water).  So have chemists — think crystallization.

Proteins move in and out of these bodies in seconds.  Proteins found in them do have low complexity of amino acids (mostly made of only a few of the 20), and unlike enzymes, their sequences are intrinsically disordered, so forget the key and lock and induced fit concepts for enzymes.

Are they a new form of matter?  Is there any limit to how big they can be?  Are the pathologic precipitates of neurologic disease (neurofibrillary tangles, senile plaques, Lewy bodies) similar.  There certainly are plenty of distinct proteins in the senile plaque, but they don’t look like liquid droplets.

It’s a fascinating field to study.  Although made of organic molecules, there seems to be little for the organic chemist to say, since the interactions aren’t covalent.  Time for physical chemists and polymer chemists to step up to the plate.