Katrina Honigs, a post-doc at the University of Utah, recently posted a fabulous essay about the time she sought out and met Alexander Grothendieck in early 2012. You should definitely read it:
“When I looked up from my reverie, I realized that a figure with a large white beard wearing a brown robe over his clothes had appeared utterly silently quite nearby on my left. I was so startled that I felt like I was outside myself watching the scene. My eyes widened, I jumped involuntarily, my heart pounded. I thought, wildly, “Oh no, what if I have accidentally come to the home of the wrong hermit?”. But as soon as I looked at his face, I recognized it from photos I had seen. Oddly enough, it was not the more recent photos where I saw the resemblance, but to a much older black and white photo of him when he was young. “It’s you!” I managed, idiotically. Grothendieck stood impassively. In one hand, he held a short pitchfork loosely at his side. It reminded me of his doodle of devils with pitchforks around the Grothendieck–Riemann–Roch formula. His free hand rose, brandishing an admonitory finger. “Il ne faut pas entrer,” he said, advancing slowly toward me. I tried to form some sentences about visiting, but Grothendieck did not react. He continued to walk slowly toward me, wagging his finger, telling me that I shouldn’t be in here disturbing him, and asking me how I got in here.”
Hat-tip to this blog post in the Scientific American for pointing this out.
defined over the rationals the rational points of 
where $T$ is the torsion subgroup. Since this was proved in the 1920’s by Mordell people numerous people have work to describe, which groups can appear in this context. For example, in the 1970’s Mazur proved that the torsion subgroup 
is one of 15 finite groups! Understanding the rank on the other hand has proven to be more difficult. There have been numerous conjectures — parity principal, boundedness, etc. — but very little unconditional progress on any of theses. Thus, many have spent time creating heuristics for the the rank should behave.Prof. Matchett Wood’s talk focused not on a new heuristic for the rank of elliptic curves (joint work with Jennifer Park, Bjorn Poonen and John Voight.)Since we want to count elliptic curves only up to isomorphism when I say elliptic curve I mean the vanishing set of an equation of the form 
where 
and if 
is a prime such that 
then 
. Let 
be the set of all such elliptic curves. If we are going to do some sort of counting we better be working with finite sets, and so we will partition the set of all elliptic curves by their naive heights. Recall the naive height of an elliptic curve 
is equal to the maximum of 
and $27 |B|^2$. (Don’t worry about the constants out front.) With all of this in mind we let:
is equal to 
. I will not go into the specifics of specifics of their model. However, the idea is that consider co-kernels of specific families of random integer matrices, and is motivated by the work of 
.
over a the rational numbers how many rational points does 
over 
satifying some additional conditions then:
.
is free. (A ring spectrum, at least in my mind is a cohomology theory, which spits out rings. Although by Brown Representability you might be supposed to think about this as a sequence of topological spaces. Not really sure on this…) This definition is intended to mimic definition of skew field as an associative ring all of whose modules are free.
![f\in \mathbb{R}[x_0,\cdots,x_n]](https://s0.wp.com/latex.php?latex=f%5Cin+%5Cmathbb%7BR%7D%5Bx_0%2C%5Ccdots%2Cx_n%5D&bg=eeeeee&fg=666666&s=0&c=20201002)
we say that:
is non-negative iff 
for all $latex x\in S:=\mathbb{R}^{n+1}.
for some $;atex g_i\in S$.
is non-negative if and only if there exists 
such that:
we might need to check for the above theorem. It turns out a 
then there exists 
.
is will be a real projective sub-variety. By real sub-variety we mean that 
is defined by real polynomials. Moreover we require that 
be Zariski dense in 
be the coordinate ring associated to 
is:
such that 
.
be a curve of degree 
and arithmetic genus 
. If $f\in R_2j$ is non-negative then there exists a sum of squares $h\in R_2k$ such that $fh$ is a sum of squares such that:
.
and 
a line bundle on 
. Now 
defines an embedding of 
, and we will let 
be the coordinate ring associated to 
on 
to the 
-module:
.
is finitely generated as an 
. In fact, the multiplication map 
is surjective. Taking the kernel of this map we get the following complex:
.
then:
is a free resolution of 
for some range of $p$? With this question in mind we say that 
iff 
for all 
.
fails 
.
if there is a degree 
map from 
. Thus, this conjecture explicitly connects the geometry of a curve to the its syzygies. In this light it has a very similar flavor to the Green-Lazarsfeld Secant Conjecture discussed in my post on Day 3.
where 
. We let 
be the homogenous coordinate ring of 
is a free resolution then each $F_i$ is free and so is of the form 
were 
is 
. Recall this is a resolution as graded modules so we need our maps to respect the grading hence the twisting. These 
are called the Betti numbers of 
.) The general idea behind the (GLS) conjecture is that these the level of ampleness of 
if and only if 
-very ample.
-secant $\latex p$-plane. (Not sure exactly why…) Regardless the big point of the talk was the following theorem:
of birational transformations of a variety 
. The general theme of this has been that as 
gets larger the complexity of 
is a finite index subgroup with 
and 
embeds into 
then $\text{dim}(V)\geq n-1$. Moreover in the case of equality 
is 
. (Alternatively it is the group of automorphisms of 
.
is isomorphic to 
if and only if 
. (
)
be a compact group and 
be the complexification of 
or 
). Suppose 
. Hence $G^c$ acts on 
. Now the motivating question of GIT is: How can we parameterize the orbits of 
? Put different how can we construct the moduli space of 
is a somewhat reasonable thing to consider. Moreover 
is stable iff 
is closed.
.
is a Hermitian metric on 
we may define a function 
by 
. This turns out to be invariant under $G$ and so descends to a function 
. Finally we have that
attains a minimum.
at two points is isomorphic to 
blown up at one point.
Yellow Pigs 