2015 AG Institute in Utah: Day 18

(For background on the conference and my time in Utah see my previous updates: Day 1,  Day 3Day 7, Day 14.)

Today marks the 18th and penultimate day of the 2015 AG Institute! This means that yesterday was essentially my last free day in Utah. Having explored the campus and gone hiking I decided to explore downtown Salt Lake more. In particular, I visited Temple Square, which is home to the Mormon Temple, the Tabernacle, and a bunch of museums dedicated to various parts of Mormonism.


Everything was immaculate and beautiful. I was also taken surprised with just how big the temple is. (Apparently this is because I’ve never seen a proper cathedral because some of the conference participants from Europe were surprised with how small it is.) I was also able to go inside the Tabernacle and see a organ recitle. That was somewhat underwhelming, but then again I am not an avid enjoyer of organ music. That said it was pretty cool to at least say I’ve been there. (Pro-tip: Don’t take pictures, even with the flash off, they don’t appreciate that.)



I also spent some time exploring some of the used bookstores downtown. Sadly they didn’t have any interesting math books. (One of them literally had a shelf of random high school algebra solution books that looked to be from the last 15 years. Who in the world goes to a rare book store to buy high school algebra solution books?) That said it was nice to walk around downtown, and I ended up finding a bit of Salt Lake that didn’t feel like one big parking lot!!

Finally, I visited the olympic cauldron from the 2002 Winter Olympics. The cauldron sits next to the University of Utah’s football stadium, and it is a somewhat sad sight. There is no advertisement for it, except one column I had mistake previously mistake for the cauldron.

When you talk towards the torch you eventually find a small room attached to the football ticket office called the Olympic Visitors Center. The room is entirely empty. Even the olympic logo has been removed from the wall. It’s actually kind of sad (especially when one thinks about how much was probably spent on the olympics.)


Finally as always here are a few things I have learned about since last posting:

  • Arithmetic, Topology, and Algebra (Jesse Wolfson): Sitting in a French jail cell in 1940, Andre Weil wrote a letter to his sister, Simone Weil, in which he discussed the role of analogy in mathematics. In particular, Weil focused on the growing connections between three fields: algebra, arithmetic, and topology. This letter, which has come to be know has Weil’s Rosetta Stone, has served to inspire a lot of work in mathematics — at least indirectly — over the last 60 years. One nice way to summarize Weil’s analogy is by the following triangle: thumb_IMG_2123_1024Jesse Wolfson gave a very good talk discussing how starting on the topology corner of this triangle one can obtain arithmetic and algebro-gemoeric results. One of the interesting things Jesse Wolfson is doing is that not only is he able to obtain point counts, but he is also able to understand the Galois actions and weights!! I really like when different areas of math connect like this, and I think there is much more to said  in this realm. (For example, adding a fourth node to the triangle with motive stuff.) See his papers I & II co-authored with Benson Farb.


  • Heuristics For The Rank of An Elliptic Curve (Melanie Matchett Wood): Given an elliptic curve E defined over the rationals the rational points of E form a finitely generated abelian group i.e. E(\mathbb{Q})=\mathbb{Z}^r\oplus T 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 T 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 y^2=Ax^3+Bx+C where A, B, C\in \mathbb{Z} and if p is a prime such that p^4 \mid A then p^4 \nmid B. Let \mathcal{E} 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 E:=y^2=Ax^3+Bx+C is equal to the maximum of 4|A|^3 and $27 |B|^2$. (Don’t worry about the constants out front.) With all of this in mind we let:

    \mathcal{E}_{H}=\{E\in \mathcal{E} \; : \; \text{ht}(E)\leq H\}.

    The question the talk tried to answer, or provide a heuristic for answering is: What can be said about:

    \#\{E\in \mathcal{E}_{H} \; : \; \text{rk}E=r\}.

    Note that \#\mathcal{E}_{H} is equal to H^{5/6}. 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 Bhargava et al. on heuristics for the Tate-Schaferavich group. The interesting thing is precisely what their model predicts:

    1. 50% of elliptic curves have even rank and 50% have odd rank.
    2. 50% of elliptic curves have rank 0 and 50% have rank 1.
    3. The Tate-Shaferavich group is finite 100% of the time.
    4. \#\{E \in \mathcal{E}_{H} \; : \; \text{rk}E=r\}=H^{\frac{(21-r)}{24}+o(1)}, \quad 1\leq r \leq 21.
    5. The number of elliptic curves with rank greater than or equal to 21 is finite!!!

PS: Continuing the game of fun things you find in a business school I present you with ethics contests! This raises so many questions: What exactly is an ethics contest? Does the need for an ethics contests suggest a problem with business culture / business school? How would one cheat in an ethics contest?


Fun fact, Spencer Eccles is accused of having helped relatives of IOC members get jobs in the lead up to the bidding process for the 2002 Olympics.


2015 AG Institute in Utah: Day 14

(For background on the conference and my time in Utah see my previous updates: Day 1,  Day 3, Day 7.)

Week two of three is now officially in the books, and I have exactly seven six days left in Salt Lake. The second week was a little more laid back with a bit less exploring. Instead I spent most of my free time playing basketball at the gym, talking to other participants, and studying. Oh and on Wednesday a couple other Wisconsin grad students and I found a decent bar downtown!!

2015-07-22 14.42.33

First good beer in SLC.

2015-07-22 18.02.56

I wonder how well Mormons take this…

2015-07-22 18.02.13

I like the Food Pairings…

Friday turned out to be Pioneer Day, which celebrates the arrival of the Mormons in the Salt Lake Valley. According to a local it is bigger than the 4th of July, which based on the number of things that were closed on a Friday just might be true. (As an aside can anyone else think of another local holiday — state or city/town — that is widely observed and results in things shutting down?)



The talks this week focused on derived algebraic geometry, mirror symmetric, tropical geometry and so I was pretty far outside of my comfort zone. Thus, these summaries are going to be a bit shorter than previously. However, here are a few things I found interesting/learned:

  • Uniform Bounds on Rational Points (Joseph Rabinoff): Given a curve C over a the rational numbers how many rational points does C have? Of course this question, and many similar questions, have been intensely studied by numerous people over the years. In his talk Joseph Rabinoff, speaking about joint work with Eric Katz and David Zureick-Brow, discussed recent work on this question. In particular, he presented the following really cool theorem:

    Theorem (KRZ-B): If C is a smooth curve of genus g over \mathbb{Q} satifying some additional conditions then:

    \#C(\mathbb{Q})\leq 76g^2-82g+22.

    Notice that in this theorem the bound only depends on the genus of the curve, nothing else! The precise statement of their theorem, which is vastly more general can be found in their paper. In the paper they also address a similar question called the Mumford-Manin conjecture. Once again providing uniform bounds for a class of specific curves.

  • Generalized Fields (Jacob Lurie): The idea behind recent Breakthrough Prize winner Jacob Lurie’s three plenary lectures is that algebraic topology seems to provide a way to generalize many of the algebraic structures we know and love. To illustrate this concept Jacob spent his lectures focusing on the example of generalized fields. A generalized field is an associative ring spectrum E such that every module over E^*(\{x\}) 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.

    One can also transfer other algebraic notions from algebra to algebraic topology; often to surprising results. For example, you can define the characteristic of a generalized field, and instead of there being just two flavors — zero and p — there turns out to be generalized fields of intermediate characteristics. Jacob went on to discuss how one can do other algebraic things over these fields of intermediate characteristic — representation theory & roots of unity — and how the results differ from the results we know over regular fields.

    All of this was very interesting and exciting, but did leave me somewhat unsure of whether the analogy between algebra and topology is a deep connection or simply a useful mental crutch for understanding these new ideas. (Possibly both??) That said this almost certainly stems from a lack of understanding on my part. Regardless his talks were extremely enjoyable and down to Earth. I would high recommend them if only as a glimpse to a new area of math that seems to potentially have a lot of promise. You can find his slides here, and videos of the talks should be posted shortly.


Tom Bridgeland also gave a series of excellent talks on stability conditions, but I am not sure I understood them quite well enough to write about. So if you’re interested check out his slides and video when posted.

PS: The afternoon talks are hosted in the business school, and on Friday I found something called the “Leadership Lounge” which appears to be a lounge for business (grad) students complete with multiple ping pong tables and a foozball table.



I am starting to think the lives of a math grad student and a business grad student are very different… (Also what exactly makes a lounge “leadership”?)

2015 AG Institute in Utah: Day 7

(As has consistently been the case this post is one day late, and today is actually Day 8. For background on the conference and my time in Utah see: Day 1 and Day 3.)

Today marks one week into the three week mega conference in algebraic geometry! Things have begun to settle in a bit, and this weekend was a nice break from the usual five+ hours of talks a day. That said the weird thing about this conference is that since it is 3 weeks long people are always coming and going. It’s almost as if the conference is actually three miniature conferences in one.

2015-07-16 17.57.41

This weekend has also given me a bit more time to explore Salt Lake City. Here are a couple things I have noticed:

  • Everything in this city feels fair away, and not just because I am living at the furtherest edge of the city on the foothills. (It is actually about a 3 mile walk from my place to the edge of what appears to be the start of the city.) I mean that the blocks seem really long and streets very wide, and so getting from place to place takes a while. Someone told me that I am not crazy and Salt Lake does in fact have abnormally large block and wide streets, and apparently this dates back to the city’s founding and Brigham Young. Apparently Brigham Young wanted each block to be exactly 10 acres with streets being something like 100 feet wide. He thought each family should essentially have a small farm or something. While this might have been great back in the 19th century when Salt Lake was being founded it seems less ideal in the 21st century. As a plus Brigham Young did have a pretty sensical way to name streets as everything is essentially labeled as if it were on the cartesian plane.
  • In addition, to the large blocks Salt Lake also seems really spread out — did I mention I am like 3 miles to the start of down town. Luckily there is a pretty nice public transportation system featuring buses (haven’t used them) and a train (have used it). The train is incredibly nice. It is quite, smooth, and pretty clean. Moreover it covers a large amount of the city. (This is how I’ve been getting into the city.) That said it does have a few huge down sides: 1) It is like 25-3oish minutes between trains and 2) The train stops at 11pm on Friday nights!! Really 11pm on a Friday! Does Salt Lake have any nightlife? All of these things combined have meant that for the most part going out at night is a huge chore and has generally not seemed worth the effort.
2015-07-18 18.28.38

Strange things you see from the train.

  • The landscape in Salt Lake is amazing with mountains on all sides providing great views. However, the city itself — or at least the bits I’ve been to — are not that pretty. Maybe it is due to the fact that all of the roads seem to be 8 lanes wide and every business has a huge parking lot, but walking downtown is just not enjoyable.
Salt Lake City From Campus

Salt Lake City From Campus

Finally here again are some of the interesting things I learnt / heard during talks on Day 4 and 5. (Once again a quick disclaimer that there are most likely errors in what follows and those are due entirely to myself.)

  • Non-Negativity and Sums of Squares (Greg Smith): Given a real polynomial f\in \mathbb{R}[x_0,\cdots,x_n] we say that:
    • f is non-negative iff f(x)\geq0 for all $latex x\in S:=\mathbb{R}^{n+1}.
    • f is a sum of squares iff f=g_1^2+\cdots+g_s^2 for some $;atex g_i\in S$.

    Having made these definitions there are two fairly obvious, but important, observations one can make: 1) the definition of non-negative only makes sense for real polynomials and 2) a sum of squares is clearly non-negative. In light of observation 2) one might ask when the converse holds i.e. when is a non-negative polynomial necessarily a sum of squares. It turns out one of math’s most influential question asker — David Hilbert — asked essentially this question as his 17th question back in 1900. As is the case with many of Hilbert’s problems this one was solved in the form of the following theorem due to E. Artin:

    Theorem (E. Artin): A polynomial f\in S is non-negative if and only if there exists h_1,\ldots, h_s, g_1,\ldots,g_r\in S such that:

    f(h_1^2+\cdots+h_s^r)=g_1^2+\cdots+g_r^2 \quad \quad (1).

    While this theorem nicely answers the question at hand, it does so in an entirely non-constructive way — the proof apparently uses the axiom of choice — and so one could wonder if there is an effective version. That is if f is a non-negative polynomial then is there an upper bound on how large the degree of the h_i we might need to check for the above theorem. It turns out a 2014 paper of Lombardi, Perrucci, and Roy provides such a bound:

    Theorem (L,P,R): If f\in S is non-negative polynomial such that \deg(f)=2j then there exists h_1,\ldots, h_s, g_1,\ldots,g_r\in S satisfying (1) such that:

    \deg(h_i)\leq 2^{2^{2^{j^{4^{n+1}}}}}.

    Ok so that bound while theoretical effective is in practice no where near useful. Moreover it would seem mildly ridiculous if that bound was actually sharp. In his talk Greg Smith discussed how this result can be improved.

    The general idea behind Smith and collaborators’ — whose names I cannot find in my notes — work is to utilize the power of algebraic geometry. Most of the work done on this question to this part goes via the use of semi-algebraic sets and not varieties. However, in many ways varieties are much better understood, and more intrinsic, than semi-algebraic sets. Thus, the idea is try and rephrase this question in terms of varieties and then leverage the machinery that is algebraic geometry.

    In order to do this Smith et al. first have to rephrase the problem in terms of algebraic geometry. From now on X\subset \mathbb{P}^n_{\mathbb{C}} is will be a real projective sub-variety. By real sub-variety we mean that X is defined by real polynomials. Moreover we require that X(\mathbb{R}) be Zariski dense in X. While this last condition might seem restrictive it is actually equivalent to X having a smooth real point. Finally let R be the coordinate ring associated to X. With this set up we say f\in R_{2j} is:

    • non-negative iff its evaluation at each point in X(\mathbb{R}) is greater than zero.
    • a sum of squares iff there exist g_1,\ldots, g_r\in R_{j} such that f=g_1^2+\cdots g_2^2.

    As a quick reality check you should think why the definition of non-negativity makes sense. Remember f is not necessarily a function and we are working with projective points. In this set up they are able to obtain the following theorem:

    Theorem (Smith, et al.): Let X\subset \mathbb{P}^n be a curve of degree d and arithmetic genus p_a. 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:

    k\leq \max\{d-n+1, \text{ceil}(2p_ad^{-1})\}.

    Notice that this bound is way way more reasonable looking that the bound found by Lombardi, Perrucci, and Roy. Moreover Smith, et al. have been able to show that their bound is sharp!! Also notice that the bound is entirely independent of f and depends solely on the geometry of X!!! Sadly they are not able to extend this result to higher dimensions. However, they do know any optimal bound in higher dimensions cannot be independent of f.

    The idea behind the proof seemed a bit complicated, and I don’t want to go into the sketch presented in the talk. However, I will say this it utilizes everyone’s second favorite type of cone. (For those who didn’t know what that I was assured it was spectrahedral cones.

  • Geometry and Syzygies of Curves (Robert Lazarsfeld): Robert Lazarsfeld also gave an amazing talk highlighting the connected between the geometry of a curve and its syzygies. The general set up is very similar to that of Gavril Farkas’s talk (see Day 3). Let C be a smooth curve of genus g\geq2 and L a line bundle on C such that \deg(L)=d>>0. Now |L| defines an embedding of C into \mathbb{P}^{d-g}, and we will let S=\text{Sym} H^0(L) be the coordinate ring associated to C. Given another line bundle B on C we define R(B, L) to the S-module:

    R(B, L)=\bigoplus_{m\geq0} H^0(C, B+mL).

    If we let $B=K_c$ be the canonical canonical class it turns out R(K_C, L) is finitely generated as an S-module by H^0(C, K_C). In fact, the multiplication map \mu: H^0(C, K_c)\otimes H^0(C, L)\rightarrow H^0(C, K_C+L) is surjective. Taking the kernel of this map we get the following complex:

    S(-1)^N\rightarrow S^g\rightarrow R(K_C, L) \rightarrow 0.

    Thus, the question becomes: When is the above complex exact? Put differently when is the above a linear presentation for R(K_C, L)? It turns out that Mark Green gave an answer to this question:

    Theorem (Green): If C is a smooth projective curve of genus g and L is a line bundle on C such that \deg(L)>>0 then:

    S(-1)^N\rightarrow S^g\rightarrow R(K_C, L) \rightarrow 0.

    is exact if and only if C is not hyper-elliptic.

    The question Lazarsfeld went on to discuss when does this pattern continue? That is if E_{-}\rightarrow R(B, L)\rightarrow 0 is a free resolution of R(B, L) when can we guarantee that E_{p}=S(-p)^{n_p} for some range of $p$? With this question in mind we say that B satisfies M_l iff E_p=S(-p)^{n_p} for all 0\leq p \leq l.

    With all of this set up the main conjecture that Lazarsfeld discussed was the following:

    Conjecture: If C is a smooth projective curve of genus g and L is a line bundle on C such that \deg(L)>>0 then K_C fails M_l if and only if \text{gon}(C)\leq l+1.

    Recall that a curve C has gonality at least l if there is a degree l+1 map from C to \mathbb{P}^1. 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.

    This conjecture turns out to be true!! In fact a stronger conjecture relating the vanishing of syzygies to the ampleness of the line bundle of interest is true!! However, this post is already getting long so I will spare righting about that. As Lazarsfeld explains it the proofs of these conjectures were hanging in plain sight, however, they only were noticed following Voisin’s work on Green’s Conjecture.

    Overall this talk was awesome, and I am really hoping to learn more about this area at some point in the near future. In fact I found it so exciting I purchased this book on Koszul cohomology in algebraic geometry. As a final note one subtle thing regarding all of these theorems is that the degree of the embedding line bundle needs to sufficiently positive. However, from the talk it wasn’t clear what sufficiently positive means. Is there an explicit bound above, which these theorems work, or is it just an eventually type statement?

Well that’s all for now. Stay tune for another update either Wednesday or Thursday!!

2015 AG Institute in Utah: Day 3

Day four of the 2015 AG Institute has just ended!! As with the intro post for this conference I am a bit delayed getting this post up, and so this post will only cover things through Day 3 (Wed, July 16)  of the conference. Day 4 and beyond will be covered in a future post.

Day 3 was sort of a rest day and so there were no lectures in the afternoon. I spent much of this time starting to explore Salt Lake City. First I tried to hike up a small hill located behind the dorms. (It looks like it would give  a great view of the city.) However, the trail I wanted to hike was closed :/

It would have been so much better from the top.

Salt Lake from head of the trail


So instead of hiking I ended up exploring campus and a bit of the city. Didn’t end up seeing to much, but I did see Ute Stadium and what I think is the 2002 Olympic torch!

Where the reign of Harbaugh will begin!

Where the reign of Harbaugh will begin!

The University of Utah campus is completely dry (see the post-script for more about this) and where we are staying is fairly far away so the first few days of the conference saw little of the usual night time conference socialization. However, someone discovered the Marriott near campus has a bar, and so there was an invasion of mathematicians.

Finally, a quite rundown of a few of the cooler things I have heard/learned so far — this is in no way an exhaustive list. Disclaimer: All of this is based on the talks I’ve seen this week and I do not understand much of it. Any errors are due to me and not the speakers :

  • Green-Lazarsfeld Secant Conjecture (Gavril Farkas): The Green-Lazarsfeld Secant (GLS) conjecture is a generalization of Green’s conjecture concerning the syzygies of a curve. The set up for the conjecture is as follows: we have a smooth projective curve C of genus g and a very ample line bundle L on C. Now |L| determines an embedding of C into \mathbb{P}^n where n= \text{deg}(L)-g. We let S_C be the homogenous coordinate ring of C with respect to this embedding and S the homogenous coordinate ring of the ambient space i.e. the polynomial ring. (Remember S_C is really really dependent on L.) Now one way to study S, and so study C, is by looking at its minimal free resolution, and the associated Betti numbers. (If F_{\cdot}\rightarrow S\rightarrow 0 is a free resolution then each $F_i$ is free and so is of the form S(-j)^{b_{ij}} were S(-j) is S with its grading twisted down by j. Recall this is a resolution as graded modules so we need our maps to respect the grading hence the twisting. These b_{ij} are called the Betti numbers of S_X.) The general idea behind the (GLS) conjecture is that these the level of ampleness of L restrain these $b_{ij}$. Explicitly (as presented here) the GLS conjecture says:

    Conjecture (GLS): Let C is a smooth projective curve of genus g and L be a globally generated line bundle of degree d on C such that
    d\geq 2g+p+1-2h^1(C, L)-\text{Cliff}(C)
    then L fails N_p if and only if L is not (p+1)-very ample.

    Certainly this conjecture is a mouthful, but my understanding is that it basically boils down to the idea that if L has a very very large degree then ampleness of L determines when the Betti numbers start to vanish. Note the conjecture gets its name because the condition that L is not (p+1)-very ample is equivalent to saying the embedding of C induced by |L| has a (p+2)-secant $\latex p$-plane. (Not sure exactly why…) Regardless the big point of the talk was the following theorem:

    Theorem (Farkas-Kemeny): The GLS conjecture holds for a general curve and general line bundle.

    Note the GLS conjecture is a generalization of Green’s conjecture for canonically embedded curves, which Claire Voisin proved was true for general curves about 8 years ago. My understanding of the proof is that satisfying the GLS conjecture is an open condition (not sure why…) in a moduli space (not sure precisely which…) hence to prove it for a general curve it suffices to cook up examples of the theorem. Farkas and his co-author Michael Kemeny have recently managed to do this!! I did not follow there exact constructs, but somehow utilize K3 surfaces — I think Voisin’s work also goes via a similar route. This is a talk I really want to understand better, and would like to know more about the general area.

  • Groups of Birational Transforms (Serge Cantat): Serge Cantat has given a series of great talks discussing the complexity of the group \text{Bir}(V) of birational transformations of a variety V. The general theme of this has been that as \text{dim} V gets larger the complexity of \text{Bir}(V) increases. Cantat has presented many different incarnations of the this general idea, but one example I find particularly interesting is the following theorem:

    Theorem (Cantat-Xie): If \Gamma\subset \text{SL}_n(\mathbb{Z}) is a finite index subgroup with n\geq3 and V is a complex projective variety (irreducible) then if \Gamma embeds into \text{Bir}(V) then $\text{dim}(V)\geq n-1$. Moreover in the case of equality V is rational and \Gamma acts via conjugation by regular automorphisms.

    This seemed quite surprising to me, and the proof uses p-adic dynamics! (In particular, that p-adic dynamics is much more well-behaved as compared to complex dynamics.) Notice from this we can also get the following corollary regarding the Cremona group. Recall that the nth Cremona group over k is \text{Bir}(\mathbb{P}^n_k). (Alternatively it is the group of automorphisms of k(x_1,\ldots,x_n).

    Corollary: \text{Cr}_n(k) is isomorphic to \text{Cr}_m(k) if and only if n=m. (k=\mathbb{C}=\mathbb{Q}=\cdots)

    I’ve never really thought about the group of birational transformations, and I’ve found it really interesting — and to me surprising, although maybe I shouldn’t have — that such groups can be used to be prove facts about the geometry of a variety e.g. the moreover bit of the above theorem. (The slides for his talks are available here and videos should be posted shortly after the conference.)

  • YTD Conjecture (Simon Donaldson): Finally Field’s medalist and Breakthrough prize winner Simon Donaldson has been giving a series of lectures on the YTD conjecture. These conjectures concern the existence of certain types of metrics on Kahler manifolds and connect algebraic geometry with symplectic and remanian geometry and analysis. Kahler geometry is not my cup of tea — I actually barely know the definition of a Kahler manifold, however, these lectures have served to show a surprising connection between metrics and geometric invariant theory (GIT).Let G be a compact group and G^c be the complexification of G so that G^c is a reductive complex Lie group. (Just think of G^c as being a nice group i.e. \text{GL}_n(\mathbb{C}) or \text{SL}_n(\mathbb{C})). Suppose V is a complex vector space on which G^c acts linearly i.e. a complex representation of G^x. Hence $G^c$ acts on \mathbb{P}(V). Now the motivating question of GIT is: How can we parameterize the orbits of G^c on \mathbb{P}V? Put different how can we construct the moduli space of G^c orbits?One approach to this is to let R be the ring of invariants of polynomials on \mathbb{P}V. In nice cases R will be finitely generated and so \text{Proj}(R) is a somewhat reasonable thing to consider. Moreover \text{Proj}(R) is close to being the moduli we want, but not quite :/. In order to understand what \text{Proj}(R) actually parameterizes we introduce two different classes of points of \mathbb{P}(V):
    1. A point v\in \mathbb{P}(V) is stable iff \text{Orb}_{G^c}(v) is closed.
    2. A point v\in \mathbb{P}(V) is semi-stable iff 0\not\in \overline{\text{Orb}_{G^c}(v)}.

    Now \text{Proj}(R) parameterizes semi-stable points. A theorem of Mumford called the Hilbert-Mumford criteria provides a way to distinguish between stable and semi-stable points in terms of certain numerical invariants related to one parameter subgroups. Not wanting to go into the details of this here — also because I am shaky on the details of this — the important thing to take away is there is an algebraic criteria for these notions.

    That said it turns out these things can also be detected metrically!! In particular, if N is a Hermitian metric on \mathbb{P}(V) invariant under G^c then we can re-phrase the definition of stability as follows:

      1. A point v\in \mathbb{P}(V) is stable iff \text{Orb}_{G^c}(v) contains a point minimizing N.
      2. A point v\in \mathbb{P}(V) is semi-stable iff N is bound from below away from zero on \text{Orb}_{G^c}(v).

    This itself seems really cool that somehow a metric can detect something that I thought about as being an algebraic property. That said maybe this is not to surprising since the actually definition of stability and semi-stability are fairly topological in nature. (Thoughts?) However, it gets even better than this. For each point v\in V we may define a function F_v(g):G^c\rightarrow\mathbb{R} by F_v(g)=\text{log}(N(g\cdot V)). This turns out to be invariant under $G$ and so descends to a function F_v(g):G^c/G\rightarrow\mathbb{R}. Finally we have that

    1. A point v\in \mathbb{P}(V) is stable iff F_v(g) attains a minimum.
    2. A point v\in \mathbb{P}(V) is semi-stable iff F_v(g) is bounded below.

    This seems even more out there than the first re-definition of stability/semi-stablity. I tend to view the existence of minima as a fairly analytic property of a function so I found/find this really really surprising. (Of course I also have no idea how this is proved so that also adds to the intrigue.)

    Finally I said that Donaldson was speaking about when Khaler manifolds had certain types of metrics, but so fair all I have talked about is the relationships between classical GIT and metrics. It turns out that this is sort of the motivation for how one goes about proving the existence of these metrics!!! Donaldson’s slides are available here, and video from the lectures should be posted at the end of the conference.


PS: Apparently the University of Utah heard about how crazy math conferences can be and wished to be proactive:

I guess the blowing up will be restricted to the lectures.

Included in the welcome packet.


So its been a hectic last few months with grad school visits, finals, graduation… but more about that in a future post. Right now I wanted to mention that Xander and I are currently hosting the inaugural Michigan Math Undergrad Summer Seminar Lunch (MUSSL) — not the best acronym, but not the worst — which is a series of weekly mini courses given by,  hosted by, and aimed at undergrads. We are planning on having five series of talks over six weeks. Right now the schedule is:

Currently we are in the fourth week, and so far I think things have gone fairly well. Its been a great learning experience both in learning some cool math and in planning/hosting a series of talks. Maybe in the future I will write up some thoughts on this.

During the first two weeks we taped the lectures, and these are available at the links above. (Sorry in advance for the poor quality we are working on fixing that in the future.) In the near future we hope to also post lecture notes from these talks. Stay tuned here or here for future updates!


PS: On an unrelated cats are good at Jenga? Who knew?