Apologies for the long silence. It’s been a very hectic past few months, between working on multiple research projects and papers, applying to graduate schools, beginning a senior thesis, and increased involvement in Divest Harvard, where I’ve been coordinating the alumni wing of the campaign. I hope to have more to say about the first item in the next few weeks. In the meantime, here’s a talk that I gave that relates to the last. 

I recently attended the 30th anniversary event of the Center for Excellence in Education, as an alum of the Research Science Institute, which was my first experience being a (however small) part of a mathematical community, and incidentally where I began blogging about mathematics. CEE offered attending alumni the chance to present short talks about topics of their choice. My talk, whose title is that of this post, is included below; the talk was also videotaped, and the video has been posted online. Here is the text.

It is great to be here. I was RSI ’09, and it was one of the best summers of my life. I would like to thank the Center for Excellence in Education for making that experience possible, and for organizing today’s very enjoyable events.

Like most of you here today, I am a scientist — or rather, a scientist-in-training. I am a scientist because I think discovering new things is stimulating and exciting. Yet I want to make the case that making discoveries is not enough for the world we live in — and that we have an ethical obligation to do more, to transcend the traditional scientific position of neutrality.

Much has been said about the ethics of science, ranging from physicists’ work on nuclear weapons to the treatment of animals. But the question we face today isn’t a question about the ethics of science itself: it’s a question about what happens when science speaks and yet no one listens. How can science make itself heard? And should it?

As you may surmise, I am referring to the climate crisis. Decades after scientists have understood the role of fossil fuels in the warming of our planet, the world’s annual carbon dioxide emissions continue their steady growth. I know everyone here has heard a lecture about polar bears at some point in their lives. I don’t wish to repeat that — because it’s too abstract, and it overlooks the absolutely fundamental human rights dimensions of the crisis. Climate change, simply put, threatens hundreds of millions of lives, and my generation’s future. I believe that it represents one of the critical issues that future generations will judge us on — just as we judge previous generations by their positions on civil rights, or on slavery.

My generation is obviously not the first to take climate change seriously. Many people have been valiantly fighting climate change for decades, developing cleaner energy technologies and lobbying our political system — some of you may be among them. But these efforts have been insufficient, and for a clear reason: powerful forces stand in the way. And foremost among those forces is the fossil fuel industry.

Why is that? According to the IPCC and others, the world has a “carbon budget,” comprising some 565 gigatons of carbon dioxide that can be burned to have a 80% chance of at most two degrees warming, the upper limit that the international community has set for global warming. This “carbon budget” leaves us with a limited time window, roughly thirty years at our present rate, in which to transition to a low-carbon future. It’s no secret that the world is not on track to make that transition. This is, in fact, a huge understatement. The proven reserves of the world’s fossil fuel companies amount to 2,795 gigatons. At this point, there is no expectation — in the markets or otherwise — that they won’t all be burned, leaving almost no chance for a stable future. It’s clear that no industry wants to have to write off the vast majority of their assets — which makes the motivations for fossil fuel industry’s well-documented campaigns to block climate change legislation all the more evident.

That’s why thousands of students at universities across the country, and across the world, have been calling on their schools to divest from fossil fuel companies — along with activists at numerous local governments and religious institutions. I’ve been proud to have been one of them, through the Divest Harvard campaign. Since last fall, we’ve been putting pressure on our administration to divest by cultivating a groundswell of support from students, faculty, and alumni. We’ve had considerable success: for example, in a referendum last fall, 72\% of Harvard undergraduates voted for a resolution calling for divestment from fossil fuels. Nationally, so far, seven universities have divested, along with several religious institutions and local governments. We don’t expect this to be an easy or quick victory, but then again, climate change is complicated.

What is divestment? Divestment is the removal of one’s investments from a particular firm or industry, often for ethical reasons. As a tool for social change, it has illustrious precedent. After pressure from students and faculty, numerous universities, notably UC Berkeley, divested from South Africa in the 1980s, in addition to pension funds and state governments. This has been credited with helping to end the apartheid regime.

All the same, many of you are probably wondering about the connection between divestment and stopping the climate crisis. It’s admittedly true that divestment itself is no substitute for better solar panels and better policies. Divestment is, instead a tool, to stigmatize an industry whose very business model necessitates catastrophic warming. And as a tool it has enormous promise. A recent Oxford University study showed that previous divestment campaigns, such as divestment from apartheid South Africa, were highly effective in bringing about necessary restrictive legislation. That report, moreover, found concluded that fossil fuel divestment is growing much faster than any of the previous campaigns analyzed.

There are many questions that have been raised, by people generally in support of action on climate change, on divestment. It is not, after all, the type of technique traditionally used by the environmental movement. Classical environmentalism has focused on individual responsibility and moral suasion. Important as that is, it suffers from a fatal flaw: there is no way putting on a sweater can bring about a political solution on climate change. And we really do need a political solution on the climate crisis.  The challenge is, after all, to convince a hugely profitable industry to write off the majority of its assets.

The most common counterargument against divestment observes that we are all complicit in the world’s dependence on fossil fuels. Nonetheless, I believe that it is the fossil fuel industry that has made it impossible for us not to be complicit: it has prevented the political action that would allow us meaningful alternatives. Given the size of its reserves, this was only rational on their part.

But another common counterargument, which we often hear both from scientists and researchers and from university administrators, states a position of neutrality. Scientists and researchers, especially those who study climate change, are reluctant to do anything that might be seen as politicizing their work. After all, we’ve been told that it’s our goal to make the discoveries, not to legislate.
Money managers claim that endowments and pension funds should maintain a neutrality to best ensure returns.

The problem with that is that climate change is an existential threat. It is not a political issue, and wanting a stable future is not a special interest. There is no neutral ground for us, as scientists, and there is no neutral ground for institutions — like our universities — that will be directly affected by climate change.

My generation is not, obviously, the first to understand the seriousness of the climate crisis. But members of my generation, at least the ones I’ve talked to, have a certain urgency in confronting the climate crisis — an intensity matched, perhaps, by the seriousness of the problem. Members of my generation tend to see climate change as more than a technical fix to be solved with engineering wizardry, but instead as a profound ethical issue. Though we didn’t cause the problem, we are, after all, the ones who will inherit a warming planet. In calling for divestment, our hope is that we can bring about a world that decides to keep four-fifth’s of the fossil fuel reserves in the ground.

As scientists and scientists-in-training, I believe we have a special obligation to confront the climate crisis. But I do not believe neutral research and education, the role that we and our universities traditionally play, can suffice: all the solar energy research in the world cannot help if we elect to keep burning coal anyway. I believe that there is a place for grassroots social activism on this, in which we can play a role.

The chasm between political organizing and scientific research is often vast. But the example of James Hansen, among others, suggests that it may be bridged at the highest levels of both. I hope many of you will consider bridging it yourself, whether by telling your alma mater that you won’t donate until it divests or by writing a letter to your senator explaining why you support a carbon tax.

Like most of you, I went to RSI because I wanted to solve hard problems. This may be the hardest problem the world has ever seen. I hope we can work together on it. Thank you.

In the previous post, we introduced the Fano scheme of a subscheme of projective space, as the Hilbert scheme of planes of a certain dimension on that subscheme. In this post, I’d like to work out an explicit example, of the 27 lines on a smooth cubic surface in \mathbb{P}^3; as we’ll see, the Fano scheme is 27 reduced points, and the count can be made with a little calculation on the Grassmannian. Although the calculation is elementary, I found it worthwhile to work carefully through it, not only for its intrinsic interest but also as motivation for the study of intersection theory on moduli spaces in general. Once again, most of this material is from Eisenbud-Harris’s draft book 3264 and All That.

1. The normal bundle as self-intersection

Suppose {X = S} is a smooth surface, imbedded in some projective space, and consider the scheme {F_1 S} of lines in {S}.

Fix a line {L} in S. In this case, the normal sheaf {N_{S/L}} is actually a vector bundle of normal vector fields, given by the adjunction formula

\displaystyle N_{S/L} = \left(\mathcal{I}_L/\mathcal{I}_L^2\right)^{\vee} = \left(\mathcal{O}_S(-L)/\mathcal{O}_S(-2L)\right)^{\vee} = \mathcal{O}_L(L).

In particular, {N_{S/L}} is a line bundle on {L \simeq \mathbb{P}^1} and has a well-defined degree. This degree is in fact the self-intersection {L.L} of {L}, considered as a divisor on the smooth surface {S}. (more…)

Let {X \subset \mathbb{P}^r} be a subvariety (or scheme). A natural question one might ask is whether {X} contains lines, or more generally, planes {\mathbb{P}^{k} \subset X \subset \mathbb{P}^r} and, if so, what the family of such look like. For example, if {Q \subset \mathbb{P}^3} is a nonsingular quadric surface, then {Q} has two families of lines (or “rulings”) that sweep out {Q}; this corresponds to the expression

\displaystyle Q \simeq \mathbb{P}^1 \times \mathbb{P}^1,

imbedded in {\mathbb{P}^3} via the Segre embedding. For a nonsingular cubic surface in {\mathbb{P}^3}, it is a famous and classical result of Cayley and Salmon that there are twenty-seven lines. In this post and the next, I’d like to discuss this result and more generally the question of planes in hypersurfaces.

Most of this material is classical; I recently learned it from Eisenbud-Harris’s (very enjoyable) draft textbook 3264 and All That.

1. Varieties of planes

Let {X \subset \mathbb{P}^r} be a variety. There is a natural subset of the Grassmannian {\mathbb{G}(k, r)} of {k}-planes in {\mathbb{P}^r} (i.e., {k+1}-dimensional subspaces of {\mathbb{C}^{r+1}}) that parametrizes those {k}-planes which happen to be contained in {X}. This is called the Fano variety.

However, the Fano variety has a natural (and possibly nonreduced) subscheme structure that arises from its interpretation as the solution to a moduli problem, so perhaps it should be called a Fano scheme. The first observation is that the {\mathbb{G}(k, r)} itself has a moduli interpretation: it is the Hilbert scheme of {k}-dimensional subschemes of {\mathbb{P}^r} consisting of subschemes whose Hilbert polynomial is given by {n \mapsto \binom{n+k}{k}}; such a subscheme is necessarily a linear subspace.

This suggests that we should think of the Fano scheme as a Hilbert scheme.

Definition 1 The Fano scheme {F_k X} of {X} is the subscheme of {\mathrm{Hilb}_X} parametrizing subschemes {L \subset X} whose Hilbert polynomial is {n \mapsto \binom{n+k}{k}}. (more…)

Let {C \subset \mathbb{P}^2} be a smooth degree {d} curve. Then there is a dual curve

\displaystyle C \rightarrow (\mathbb{P}^2)^*,

which sends {p \in C \mapsto \mathbb{T}_p C}, to the (projectivized) tangent line at {p \in C}. Such lines live in the dual projective space {(\mathbb{P}^2)^*} of lines in {\mathbb{P}^2}. We will denote the image by {C^* \subset \mathbb{P}^2}; it is another irreducible curve, birational to {C}.

This map is naturally of interest to us, because, for example, it lets us count bitangents. A bitangent to {C} will correspond to a node of the image of the dual curve, or equivalently it will be a point in {(\mathbb{P}^2)^*} where the dual map {C \rightarrow (\mathbb{P}^2)^*} fails to be one-to-one. In fact, if {C} is general, then {C^*} will have only nodal and cuspidal singularities, and we we will be able to work out the degree of {C^*}. By the genus formula, this will determine the number of nodes in {C^*} and let us count bitangents.

The purpose of this post is to describe this, and to discuss this map from the point of view of jet bundles, discussed in the previous post. (more…)

Let {C \subset \mathbb{P}^2} be a smooth plane quartic, so that {C} is a nonhyperelliptic genus 3 curve imbedded canonically. In the previous post, we saw that bitangent lines to {C} were in natural bijection with effective theta characteristics on {C}, or equivalently spin structures (or framings) of the underlying smooth manifold.

It is a classical fact that there are {28} bitangents on a smooth plane quartic. In other words, of the {64} theta characteristics, exactly {28} of them are effective. A bitangent here will mean a line {L \subset \mathbb{P}^2} such that the intersection {L \cap C} is a divisor of the form {2(p + q)} for {p, q \in C} points, not necessarily distinct. So a line intersecting {C} in a single point (with contact necessarily to order four) is counted as a bitangent line. In this post, I’d like to discuss a proof of a closely related claim, that there are {24} flex lines. This is a special case of the Plücker formulas, and this post will describe a couple of the relevant ideas.  (more…)

Let {C} be an algebraic curve over {\mathbb{C}}. A theta characteristic on {C} is a (holomorphic or algebraic) square root of the canonical line bundle {K_C}, i.e. a line bundle {L \in \mathrm{Pic}(C)} such that

\displaystyle L^{\otimes 2} \simeq K_C.

Since the degree of {K_C} is even, such theta characteristics exist, and in fact form a torsor over the 2-torsion in the Jacobian {J(C) = \mathrm{Pic}^0(C)}, which is isomorphic to {H^1(C; \mathbb{Z}/2\mathbb{Z}) \simeq (\mathbb{Z}/2\mathbb{Z})^{2g}}.

One piece of geometric motivation for theta characteristics comes from the following observation: theta characteristics form an algebro-geometric approach to framings. By a theorem of Atiyah, holomorphic square roots of the canonical bundle on a compact complex manifold are equivalent to spin structures. In complex dimension one, a choice of a spin structure is equivalent to a framing of {M}. On a framed manifolds, there is a canonical choice of quadratic refinement on the middle-dimensional mod {2} homology (with its intersection pairing), which gives an important invariant of the framed manifold known as the Kervaire invariant. (See for instance this post on the paper of Kervaire that introduced it.)

It turns out that the mod {2} function {L \mapsto \dim H^0(C, L)} on the theta characteristics is precisely this invariant. In other words, theta characteristics give a purely algebraic (valid in all characteristics, at least {\neq 2}) approach to the Kervaire invariant, for surfaces!

Most of the material in this post is from two papers: Atiyah’s Riemann surfaces and spin structures and Mumford’s Theta characteristics of an algebraic curve. (more…)

I’ve been trying to learn a little about algebraic curves lately, and genus two is a nice starting point where the general features don’t get too unmanageable, but plenty of interesting phenomena still arise.

0. Introduction

Every genus two curve {C} is hyperelliptic in a natural manner. As with any curve, the canonical line bundle {K_C} is generated by global sections. Since there are two linearly independent holomorphic differentials on {C}, one gets a map

\displaystyle \phi: C \rightarrow \mathbb{P}^1.

Since {K_C} has degree two, the map {\phi} is a two-fold cover: that is, {C} is a hyperelliptic curve. In particular, as with any two-fold cover, there is a canonical involution {\iota} of the cover {\phi: C \rightarrow \mathbb{P}^1}, the hyperelliptic involution. That is, every genus two curve has a nontrivial automorphism group. This is in contrast to the situation for higher genus: the general genus {g \geq 3} curve has no automorphisms.

A count using Riemann-Hurwitz shows that the canonical map {\phi: C \rightarrow \mathbb{P}^1} must be branched at precisely six points, which we can assume are {x_1, \dots, x_6 \in \mathbb{C}}. There is no further monodromy data to give for the cover {C \rightarrow \mathbb{P}^1}, since it is a two-fold cover; it follows that {C} is exhibited as the Riemann surface associated to the equation

\displaystyle y^2 = \prod_{i=1}^6 (x - x_i).

More precisely, the curve {C} is cut out in weighted projective space {\mathbb{P}(3, 1, 1)} by the homogenized form of the above equation,

\displaystyle Y^2 = \prod_{i = 1}^6 ( X - x_i Z).

1. Moduli of genus two curves

It follows that genus two curves can be classified, or at least parametrized. That is, an isomorphism class of a genus two curve is precisely given by six distinct (unordered) points on {\mathbb{P}^1}, modulo automorphisms of {\mathbb{P}^1}. In other words, one takes an open subset {U \subset (\mathbb{P}^1)^6/\Sigma_6 \simeq \mathbb{P}^6}, and quotients by the action of {PGL_2(\mathbb{C})}. In fact, this is a description of the coarse moduli space of genus two curves: that is, it is a variety {M_2} whose complex points parametrize precisely genus two curves, and which is “topologized” such that any family of genus two curves over a base {B} gives a map {B \rightarrow M_2}. Moreover, {M_2} is initial with respect to this property.

It can sometimes simplify things to assume that three of the branch points in {\mathbb{P}^1} are given by {\left\{0, 1, \infty\right\}}, which rigidifies most of the action of {PGL_2(\mathbb{C})}; then one simply has to choose three (unordered) distinct points on {\mathbb{P}^1 \setminus \left\{0, 1, \infty\right\}} modulo action of the group {S_3 \subset PGL_2(\mathbb{C})} consisting of automorphisms of {\mathbb{P}^1} that preserve {\left\{0, 1, \infty\right\}}. In other words,

\displaystyle M_2 = \left( \mathrm{Sym}^3 \mathbb{P}^1 \setminus \left\{0, 1, \infty\right\} \setminus \left\{\mathrm{diagonals}\right\}\right)/S_3.

Observe that the moduli space is three-dimensional, as predicted by a deformation theoretic calculation that identifies the tangent space to the moduli space (or rather, the moduli stack) at a curve {C} with {H^1(T_C)}.

A striking feature here is that the moduli space {M_2} is unirational: that is, it admits a dominant rational map from a projective space. In fact, one even has a little more: one has a family of genus curves over an open subset in projective space (given by the family {y^2 = \prod (x - x_i)} as the {\left\{x_i\right\}} as vary) such that every genus two curve occurs in the family (albeit more than once).

The simplicity of {M_2}, and in particular the parametrization of genus two curves by points in a projective space, is a low genus phenomenon, although similar “classifications” can be made in a few higher genera. (For example, a general genus four curve is an intersection of a quadric and cubic in {\mathbb{P}^3}, and one can thus parametrize most genus four curves by a rational variety.) As {g \rightarrow \infty}, the variety {M_g} parametrizing genus {g} curves is known to be of general type, by a theorem of Harris and Mumford. (more…)

I’ve just uploaded to arXiv my paper “The homology of {\mathrm{tmf}},” which is an outgrowth of a project I was working on last summer. The main result of the paper is a description, well-known in the field but never written down in detail, of the mod {2} cohomology of the spectrum {\mathrm{tmf}} of (connective) topological modular forms, as a module over the Steenrod algebra: one has

\displaystyle H^*(\mathrm{tmf}; \mathbb{Z}/2) \simeq \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2,

where {\mathcal{A}} is the Steenrod algebra and {\mathcal{A}(2) \subset \mathcal{A}} is the 64-dimensional subalgebra generated by {\mathrm{Sq}^1, \mathrm{Sq}^2,} and { \mathrm{Sq}^4}. This computation means that the Adams spectral sequence can be used to compute the homotopy groups of {\mathrm{tmf}}; one has a spectral sequence

\displaystyle \mathrm{Ext}^{s,t}( \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2, \mathbb{Z}/2) \simeq \mathrm{Ext}^{s,t}_{\mathcal{A}(2)}(\mathbb{Z}/2, \mathbb{Z}/2) \implies \pi_{t-s} \mathrm{tmf} \otimes \widehat{\mathbb{Z}_2}.

Since {\mathcal{A}(2) \subset \mathcal{A}} is finite-dimensional, the entire {E_2} page of the ASS can be computed, although the result is quite complicated. Christian Nassau has developed software to do these calculations, and a picture of the {E_2} page for {\mathrm{tmf}} is in the notes from André Henriques‘s 2007 talk at the Talbot workshop. (Of course, the determination of the differentials remains.)

The approach to the calculation of {H^*(\mathrm{tmf}; \mathbb{Z}/2)} in this paper is based on a certain eight-cell (2-local) complex {DA(1)}, with the property that

\displaystyle \mathrm{tmf} \wedge DA(1) \simeq BP\left \langle 2\right\rangle,

where {BP\left \langle 2\right\rangle = BP/(v_3, v_4, \dots, )} is a quotient of the classical Brown-Peterson spectrum by a regular sequence. The usefulness of this equivalence, a folk theorem that is proved in the paper, is that the spectrum {BP\left \langle 2\right\rangle} is a complex-orientable ring spectrum, so that computations with it (instead of {\mathrm{tmf}}) become much simpler. In particular, one can compute the cohomology of {BP\left \langle 2\right\rangle} (e.g., from the cohomology of {BP}), and one finds that it is cyclic over the Steenrod algebra. One can then try to “descend” to the cohomology of {\mathrm{tmf}}. This “descent” procedure is made much simpler by a battery of techniques from Hopf algebra theory: the cohomologies in question are graded, connected Hopf algebras. (more…)

The topic of topological modular forms is a very broad one, and a single blog post cannot do justice to the whole theory. In this section, I’ll try to answer the question as follows: {\mathrm{tmf}} is a higher analog of {KO}theory (or rather, connective {KO}-theory).

1. What is {\mathrm{tmf}}?

The spectrum of (real) {KO}-theory is usually thought of geometrically, but it’s also possible to give a purely homotopy-theoretic description. First, one has complex {K}-theory. As a ring spectrum, {K} is complex orientable, and it corresponds to the formal group {\hat{\mathbb{G}_m}}: the formal multiplicative group. Along with {\hat{\mathbb{G}_a}}, the formal multiplicative group {\hat{\mathbb{G}_m}} is one of the few “tautological” formal groups, and it is not surprising that {K}-theory has a “tautological” formal group because the Chern classes of a line bundle {\mathcal{L}} (over a topological space {X}) in {K}-theory are defined by

\displaystyle c_1( \mathcal{L}) = [\mathcal{L}] - [\mathbf{1}];

that is, one uses the class of the line bundle {\mathcal{L}} itself in {K^0(X)} (modulo a normalization) to define.

The formal multiplicative group has the property that it is Landweber-exact: that is, the map classifying {\hat{\mathbb{G}_m}},

\displaystyle \mathrm{Spec} \mathbb{Z} \rightarrow M_{FG},

from {\mathrm{Spec} \mathbb{Z}} to the moduli stack of formal groups {M_{FG}}, is a flat morphism. (more…)

Let {C \subset \mathbb{P}^r} be a (smooth) curve in projective space of some degree {d}. We will assume that {C} is nondegenerate: that is, that {C} is not contained in a hyperplane. In other words, one has an abstract algebraic curve {C}, and the data of a line bundle {\mathcal{L} = \mathcal{O}_C(1)} of degree {d} on {C}, and a subspace {V \subset H^0( \mathcal{L})} of dimension {r+1} such that the sections in {V} have no common zeros in {C}.

In this post, I’d like to discuss a useful condition on such an imbedding, and some of the geometry that it leads to. Most of this material is, once again, from ACGH’s book Geometry of algebraic curves. 

1. Projective normality

In general, there are two natural commutative graded rings one can associate to this data. First, one has the homogeneous coordinate ring of {C} inside {\mathbb{P}^r}. The curve {C \subset \mathbb{P}^r} is defined by a homogeneous ideal {I \subset k[x_0, \dots, x_r]} (consisting of all homogeneous polynomials whose vanishing locus contains {C}). The homogeneous coordinate ring of {C} is defined via

\displaystyle S = k[x_0, \dots, x_r]/I;

it is an integral domain. Equivalently, it can be defined as the image of {k[x_0, \dots, x_r] = \bigoplus_{n = 0}^\infty H^0( \mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(n))} in {\bigoplus_{n = 0}^\infty H^0( C, \mathcal{O}_C(n))}. But that in turn suggests another natural ring associated to {C}, which only depends on the line bundle {\mathcal{L}} and not the projective imbedding: that is the ring

\displaystyle \widetilde{S} = \bigoplus_{n = 0}^\infty H^0( C, \mathcal{O}_C(n)),

where the multiplication comes from the natural maps {H^0(\mathcal{M}) \otimes H^0(\mathcal{N}) \rightarrow H^0( \mathcal{M} \otimes \mathcal{N})} for line bundles {\mathcal{M}, \mathcal{N}} on {C}. One has a natural map

\displaystyle S \hookrightarrow \widetilde{S},

which is injective by construction. Moreover, since higher cohomology always vanishes after enough twisting, the map {S \rightarrow \widetilde{S}} is surjective in all large dimensions.

Definition 1 The curve {C \subset \mathbb{P}^r} is said to be projectively normal if the map {S \hookrightarrow \widetilde{S}} is an isomorphism. (more…)