Max LieblichProfessor of Mathematics at the University of Washington. Partially supported by NSF CAREER Grant DMS-1056129.
http://max.lieblich.us/
Wed, 12 Jul 2017 23:46:21 +0000Wed, 12 Jul 2017 23:46:21 +0000Jekyll v3.4.5Versal torsors are awesome<p>I was just at Bar Ilan University for Louis Rowen’s
<a href="http://u.math.biu.ac.il/~vishne/Conferences/Rowen2017/index.html">great birthday conference</a>.
Included among my fellow speakers were Danny Krashen and Alexander Merkurjev.
They told me about something I should have seen long ago: versal torsors.
Among other things, versal torsors give a way to produce <em>universal bounds on symbol lengths</em> for
central division algebras with bounded index. This blew my mind. Let me show you how it works.
<!--more--></p>
<h3 id="look-at-what-happened-to-my-classifying-space">Look at what happened to my classifying space!</h3>
<p>Let <script type="math/tex">G</script> be a linear algebraic group. Suppose I want to understand <script type="math/tex">G</script>-torsors.
As a card-carrying algebraic geometer and stack aficionado, I should just “use”
the classifying stack <script type="math/tex">\operatorname{B}G</script>, right? <strong>WRONG</strong>! Here’s another thing you can do.</p>
<p>Choose a faithful representation <script type="math/tex">G\hookrightarrow\operatorname{GL}_n</script> (over whatever base we’re using) and form the quotient <script type="math/tex">B=\operatorname{GL}_n/G</script>. By the definition of quotients, there is an induced <script type="math/tex">G</script>-torsor <script type="math/tex">T\to B</script> (i.e., <script type="math/tex">T=\operatorname{GL}_n</script>!!). This is called a <em>versal torsor</em>. Why versal? For any field <script type="math/tex">K</script> over the base we get an exact sequence of pointed sets</p>
<script type="math/tex; mode=display">G(K)\to\operatorname{GL}_n(K)\to B(K)\to\operatorname{H}^1(\operatorname{Spec}(K),G)\to\operatorname{H}^1(\operatorname{Spec}(K), \operatorname{GL}_n(K)).</script>
<p>Hilbert’s Theorem 90 tells us that the last set is a singleton. We conclude the following amazing fact: for <em>any</em> field <script type="math/tex">K</script> over the base and <em>any</em> <script type="math/tex">G</script>-torsor <script type="math/tex">\Gamma</script> over <script type="math/tex">K</script>, there is a <script type="math/tex">K</script>-point of <script type="math/tex">B</script> such that the fiber of <script type="math/tex">T</script> over that point is isomorphic to <script type="math/tex">\Gamma</script>.</p>
<p>I don’t know why this freaks me out so much; perhaps it has to do with my perception that there are a lot of <script type="math/tex">G</script>-torsors over arbitrary fields, whereas this tells us that they are all points of a single variety. (Corollary: the essential dimension of <script type="math/tex">G</script> is finite. When you put it that way, it doesn’t sound so outrageous.)</p>
<h3 id="now-that-they-gave-me-this-thing-what-can-i-do-with-it">Now that they gave me this thing, what can I do with it?</h3>
<p>Here’s an amazing application of versal torsors. Let <script type="math/tex">G</script> be the group <script type="math/tex">\operatorname{PGL}_n</script> over the ring <script type="math/tex">A:=\mathbf{Z}\left[\frac{1}{n}\right][\mu_n]</script>.
Choosing a faithful linear representation (example: the adjoint) <script type="math/tex">\operatorname{PGL}_n\to\operatorname{GL}
_N</script> and forming the versal torsor <script type="math/tex">\operatorname{GL}_N\to\operatorname{GL}_N/\operatorname{PGL}_n</script> gives me
a scheme <script type="math/tex">X</script> (the quotient) and a <script type="math/tex">\operatorname{PGL}_n</script>-torsor (let’s call it <script type="math/tex">T\to X</script>) with the amazing property that
for <em>any</em> field <script type="math/tex">K</script> in which <script type="math/tex">n</script> is invertible and containing a primitive <script type="math/tex">n</script>th root of unity
and <em>any</em> <script type="math/tex">\operatorname{PGL}_n</script>-torsor <script type="math/tex">Y</script> over <script type="math/tex">K</script>, there is a <script type="math/tex">K</script>-point
of <script type="math/tex">X</script> such that the fiber of <script type="math/tex">T</script> over that point is isomorphic to <script type="math/tex">Y</script>. (I’m just restating the above.)</p>
<p>Here’s something we can do with this. Given a central division algebra <script type="math/tex">D</script> over such a field <script type="math/tex">K</script> with index <script type="math/tex">n</script>, the
Merkurjev-Suslin theorem says that there is a sequence of symbol algebras <script type="math/tex">(a_1,b_1),\ldots,(a_m,b_m)</script> over <script type="math/tex">K</script>
such that <script type="math/tex">D</script> is Brauer-equivalent to the tensor product <script type="math/tex">E=\bigotimes_i (a_i,b_i)</script> (that is, there are matrix
algebras over <script type="math/tex">D</script> and <script type="math/tex">E</script> that are isomorphic). The question is: how many symbols do we need?</p>
<p>The surprising answer is: <em>there is a universal bound on the number of symbols we need</em>! Here’s how to prove this.
Let <script type="math/tex">\alpha</script> be the Brauer class of the versal torsor over <script type="math/tex">X</script>. Merkurjev-Suslin gives us an expression
for <script type="math/tex">\alpha</script> restricted to the generic point of <script type="math/tex">X</script> in terms of symbols. Standard spreading out arguments
make this expression valid in an open neighborhood (i.e., the symbols extend to Azumaya algebras in a neighborhood).
Take the complement of this open subset; this is some closed subset <script type="math/tex">X'\subset X</script>. Repeat this at a generic point
of <script type="math/tex">X'</script>. Since any locally-closed stratification of <script type="math/tex">X</script> is finite (<script type="math/tex">X</script> being Noetherian), this process will terminate.
Take the largest number of symbols you used and that’s your bound! Every algebra of degree <script type="math/tex">n</script> over any field factors through
<script type="math/tex">X</script>, so it will factor through a piece of the stratification, and we can use the symbol decomposition on that stratum.</p>
<p>As Harry Caray used to say, “Holy Cow!”</p>
Sun, 02 Jul 2017 00:00:00 +0000
http://max.lieblich.us/research/2017/07/02/versal-torsors-are-awesome.html
http://max.lieblich.us/research/2017/07/02/versal-torsors-are-awesome.htmlAlgebra,ModuliresearchForsyth = Chisini + Moishezon<p>Our May <a href="https://aimath.org/pastworkshops/algvision.html">AIM workhop on Algebraic
Vision</a> was an absolutely
fascinating experience. I will eventually write more about things that I
learned there, but what I will write about first is of the most interesting
from a sociological perspective. It sheds some light on the culture gap between
computer vision and pure mathematics. <!--more--></p>
<h3 id="forsyth">Forsyth</h3>
<p>In 1993, David Forsyth published a paper at the Fourth International Conference
on Computer Vision entitled <a href="http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=378177&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D378177">“Recognizing algebraic surfaces from their
outlines”</a>.
The key result is the following. (I have copied the statement of the theorem
verbatim.)</p>
<p><strong>Theorem</strong>. <em>The equation of its outline in a perspective image completely
determines the projective geometry of an algebraic surface of degree 2 or
greater, for a generic view of a generic algebraic surface.</em></p>
<p>What does the statement mean? Let me define various terms that come up.</p>
<ul>
<li><em>Algebraic surface</em> is easy: this means projective algebraic surface
<script type="math/tex">X\subset\mathbf{P}^3.</script></li>
<li><em>View</em> means: linear projection <script type="math/tex">\mathbf{P}^3\dashrightarrow\mathbf{P}^2.</script></li>
<li><em>Perspective image</em> means: image in a view (as defined above).</li>
<li><em>Generic</em> is undefined in the paper, but we know what it means: true on a
Zariski-open locus.</li>
<li><em>Outline</em> is the most interesting one. Here is how to define it: we assume
our linear projection <script type="math/tex">\mathbf{P}^3\dashrightarrow\mathbf{P}^2</script> has base
locus at a point not on <script type="math/tex">X</script>. We thus get an induced finite morphism
<script type="math/tex">X\to\mathbf{P}^2.</script> This morphism ramifies along a curve <script type="math/tex">D\subset X</script>, and
the outline is the image of <script type="math/tex">D</script> in <script type="math/tex">\mathbf{P^2}</script>.</li>
<li><em>Projective geometry</em> of <script type="math/tex">X</script> means “<script type="math/tex">X</script> up to automorphism of
<script type="math/tex">\mathbf{P}^3</script>”.</li>
</ul>
<p>We can restate the theorem like this.</p>
<p><strong>Theorem</strong> (restated). <em>Suppose <script type="math/tex">X\subset\mathbf{P}^3</script> is a generic smooth
projective surface of degree at least <script type="math/tex">2</script> and
<script type="math/tex">f:\mathbf{P}^3\dashrightarrow\mathbf{P}^2</script> is a generic linear projection
that is regular along <script type="math/tex">X</script>. Let <script type="math/tex">D\subset X</script> be the ramification curve of
the restriction <script type="math/tex">f_X</script>. Then <script type="math/tex">f(D)</script> uniquely determines <script type="math/tex">X</script>, up to
automorphism of <script type="math/tex">\mathbf{P}^3</script>.</em></p>
<p>(This is close enough to a rigorous statement that it would pass muster to
almost any algebraic geometer, even though some <a href="http://math.stanford.edu/~conrad/">number
theorists</a> might not like the lax use of
genericity. In fact, I don’t like it either this way. We should be careful and
say that we are looking at a generic pair <script type="math/tex">(X,f)</script>.)</p>
<h3 id="chisini">Chisini</h3>
<p>When stated this way, it looks awfully similar to the following, which I will
state as a conjecture.</p>
<p><strong>Conjecture</strong>. <em>A generic finite morphism <script type="math/tex">f:X\to\mathbf{P}^2</script> is uniquely
determined by its branch divisor.</em></p>
<p>This conjecture is simpler – it removes linear projections from the picture –
and harder – it removes linear projections from the picture! But now it has
transformed into a standard problem in algebraic geometry, called
<a href="http://www-history.mcs.st-and.ac.uk/Biographies/Chisini.html">Chisini</a>’s
conjecture, first stated in 1944. In fact, Chisini also assumed for his
conjecture that the degree of <script type="math/tex">f</script> is at least <script type="math/tex">5</script>. And the word “generic”
allows us to assume that all ramification of <script type="math/tex">f</script> is simple (i.e., is locally
around source and target given analytically as <script type="math/tex">x\mapsto x^2</script>), that the
ramification curve <script type="math/tex">R</script> in <script type="math/tex">X</script> is smooth, that its image <script type="math/tex">B</script> in
<script type="math/tex">\mathbf{P}^2</script> has only nodes and cusps, and that the induced map <script type="math/tex">R\to B</script>
is birational.</p>
<p>Chisini himself stated a solution to some cases of this conjecture in [1], but
his proof was incorrect. (This is briefly explained by Antonio Lanteri in his
MathSciNet review of Catanese’s paper [2]. The error seems to be related to the
degeneration types of the curves that appear.)</p>
<p>As an algebraic geometer, the immediately intuitive way to approach a problem
like this is to study the fundamental group of the complement of the branch
divisor in <script type="math/tex">\mathbf{P}^2</script>. It looks like one is trying to prove that if the
monodromy action of cycles around the divisor are a given set of
transpositions, then the monodromy representation is uniquely determined. This
is pretty close to wanting to know a presentation of the fundamental group. For
example, is it abelian with a specific set of generators?</p>
<p>Indeed, Zariski conjectured that if a plane curve has only nodes, then its
complement has abelian fundamental group. This was proven by Fulton (in the
sense of profinite fundamental groups, although this was improved by Deligne in
the complex case) [3] in 1980. (Note: knowing that such curves generally
degenerate to unions of lines plays an important role, echoing the issues with
Chisini’s original incorrect proof.)</p>
<p>So we’re done, right? Unfortunately, no: the branch curve of a projection need
not be nodal. In fact, a simple dimension count argument shows that it <em>can’t</em>
be, in general.</p>
<h3 id="moishezon">Moishezon</h3>
<p>As it turns out, a number of authors have studied Chisini’s conjecture over the
years, including Catanese, Kulikov, Nemirovskii, and Moishezon. Moreover,
Moishezon solved the precise case encountered by Forsyth in 1981. The hard part
of Chisini’s problem is that it is stated very generally, with no assumptions
on the structure of the morphism. Things are much easier for linear
projections, although Kulikov still has to work hard for high dimensional
projections, as opposed to projections from 3-space.</p>
<p>Moishezon’s proof is embedded in a larger work that studies problems related to
fundamental groups of curve complements [4]. So my mathy
gut instinct was “right” in a certain vague sense, but it is hard to implement.</p>
<h3 id="what-did-forsyth-do">What did Forsyth do?</h3>
<p>Forsyth’s argument is much easier. It relies heavily on input from Debarre, who
was at Iowa at the same time as Forsyth (the early 1990s), consisting of a few
simple cohomology calculations and some clever observations about generators of
ideals in various degrees (related to the degree of the surface). At some
point, I intended to sketch it here, but that is stopping me from actually
finishing this. Instead, I encourage you to read the paper, not only to see how
the argument works, but also to get a taste of the comparison between the cultures
of algebraic geometry and computer vision.</p>
<h3 id="what-happened">What happened?</h3>
<p>The main point I want to make here is that Forsyth, a computer vision
researcher, seems to have rediscovered the results of Chisini, etc., without
being aware of the algebro-geometric literature. There’s one small wrinkle:
Forsyth got help from Debarre, a well-known algebraic geometer. Chisini’s
conjecture and the work of Catanese and Moishezon seem to have been obscure
enough within the subject that Debarre didn’t immediately point Forsyth toward
those results. Or perhaps Debarre realized that to answer Forsyth’s immediate
question, much simpler techniques would suffice. Or maybe Forsyth went to
Debarre with a specific bit of the question, and Debarre didn’t know about the
bigger context. Enough speculating! Regardless of how this came to be, Forsyth
and Debarre cooked up a very nice and simple proof in the case of linear
projections.</p>
<p>There is a lot that computer vision researchers and algebraic geometers have to
learn from one another. But that is the subject of another post.</p>
<h3 id="references">References</h3>
<p>[1] Chisini, “Sulla identita birazionale delle funzioni algebriche di due variabili
dotate di una medesima curva di diramazione.” <em>Ist. Lombardo Sci. Lett. Cl.
Sci. Mat. Nat. Rend.</em> (3) <strong>8(77)</strong>, (1944). 339–356.</p>
<p>[2] Catanese, “On a problem of Chisini.” Duke Math. J. 53 (1986), no. 1, 33–42.</p>
<p>[3] Fulton, “On the fundamental group of the complement of a node curve.” Ann. of Math.
(2) 111 (1980), no. 2, 407–409.</p>
<p>[4] Moishezon, “Stable branch curves and braid monodromies.” Algebraic geometry (Chicago, Ill., 1980), pp. 107–192,
Lecture Notes in Math., 862, Springer, Berlin-New York, 1981.</p>
Wed, 07 Dec 2016 00:00:00 +0000
http://max.lieblich.us/research/2016/12/07/forsyth.html
http://max.lieblich.us/research/2016/12/07/forsyth.htmlAlgebraicvisionresearchA stronger derived Torelli theorem for K3 surfaces<p><a href="http://math.berkeley.edu/~molsson">Martin Olsson</a> and I have a <a href="http://arxiv.org/abs/1512.06451">new paper</a> about
derived Torelli theorems for K3 surfaces. It is a piece of our gradual attempt to
recast Torelli theorems in terms of a combination of the derived category and the Chow theory.
While we <a href="http://arxiv.org/abs/1112.5114">previously</a> showed a very coarse kind of Torelli statement – a certain
kind of derived equivalence that respects a filtration on Chow theory implies
the existence of an isomorphism – we are now able to show that there is an
isomorphism that acts on the cohomology in the same way as the derived
equivalence. (There is a slight subtlety involving the
filtration-perserving condition, but this is precisely analogous to the condition in the
Torelli theorem about preserving ample classes.) Our original proof ultimately
used <em>analysis</em>, but the new one is <em>purely algebraic</em>. <!--more--></p>
<h3 id="the-basic-idea">The basic idea</h3>
<p>The Hodge structure of a variety <script type="math/tex">X</script> is a semi-linear object
that mixes linear algebra together with various filtrations. Here’s another
such gadget: the derived category <script type="math/tex">D(X)</script>, together with data about the support of
perfect complexes. This can be interpreted in various ways. Here are two
examples.</p>
<ol>
<li>One could record the support of complexes <script type="math/tex">P\in D(X)</script>, which means for
example that a complex with adjacent cohomologies that have large support
will itself also have large support, even if the rest of the cohomologies are
supported on small loci. So here the package would be <script type="math/tex">D(X)</script> together with
the collection of subcategories <script type="math/tex">D_i(X)</script> parametrizing perfect complexes
with support of dimension at most <script type="math/tex">i</script>.</li>
<li>One could record the image of <script type="math/tex">P</script> in the Chow theory <script type="math/tex">A^\ast(X)</script> via the
Chern character, in which case the support data attached to <script type="math/tex">P</script> allows things like enormous adjacent support to cancel.
Here the semi-linear package is <script type="math/tex">D(X)</script> together with the map <script type="math/tex">D(X)\to
A^\ast(X)</script> together with the codimension filtration on <script type="math/tex">A^\ast(X)</script>.</li>
</ol>
<p>The idea is that if we start with <script type="math/tex">X</script> and <script type="math/tex">Y</script>, and we have an equivalence
<script type="math/tex">\Phi:D(X)\to D(Y)</script> that preserves filtrations (in one of the sense above or another reasonable one),
then <script type="math/tex">X</script> and <script type="math/tex">Y</script> should be
isomorphic. A stronger hope is that at least some of these
filtration-preserving equivalences should actually come from isomorphisms
themselves, at least after passing to some cohomology theory. That is to say,
equivalences are given by kernels on <script type="math/tex">X\times Y</script>, so they induce
correspondences between all kinds of cohomology theories (etale, crystalline,
etc.), and we can at least ask for an isomorphism that induces the same
cohomological correspondences.</p>
<h3 id="the-original-proof-for-k3-surfaces">The original proof for K3 surfaces</h3>
<p>In our original paper, we carefully lifted the filtered equivalence <script type="math/tex">\Phi</script> to
characteristic <script type="math/tex">0</script> and then used the filtration-preserving condition to
reduce to the classical complex Torelli theorem. In the process, we lost some
control over how the equivalence was acting on various cohomology theories.
(Now that I think about it again, I think we just didn’t try. I would guess that a
relatively minor adjustment to our methods should yield the general result
of our new paper, but it would still involve lifting to characteristic <script type="math/tex">0</script>
and invoking the classical Torelli theorem.)</p>
<p>This was really fun, but it also meant that in trying to lay foundations for a
Torelli-type theory in positive characteristic, we were in the uncomfortable
position of invoking the analytic theory over the complex numbers. This limits
the amount of insight into the true meaning of the result that comes from the proof.
(I.e., maybe this derived category plus filtered thingamajig idea is really a
red herring, and all we did was record a new consequence of the analytic
theory.)</p>
<h3 id="the-new-proof-for-k3-surfaces">The new proof for K3 surfaces</h3>
<p>In the new paper, we follow a suggestion of <a href="http://www.math.u-psud.fr/~fcharles/">F.
Charles</a> to use degeneration to the
supersingular case in place of deforming to the complex case. In fact, we are
able to completely eliminate the use of Hodge theory in the situations of
interest, giving us a purely algebraic proof. (Important note: the
supersingular case relies heavily on the deep work of
<a href="https://math.berkeley.edu/~ogus/">Ogus</a>, so we don’t get a <em>transparent</em>
proof!)</p>
<p>The delicate points here involve tracing the action on cohomology as we
slide derived equivalences around over the moduli space of polarized K3 surfaces in
positive characteristic. This also involves knowing things about the geometry
of the moduli space of K3 surfaces; if one is not careful, one falls into an
analytic trap, but there are algebraic proofs of the relevant facts, even if
some of them happen in characteristic 0.</p>
<p>As an outcome, one gets a purely algebraic proof that any Fourier-Mukai partner
of a K3 surface is a moduli space of stable sheaves. Look ma, no Hodge theory! <strong>Update</strong>: I now realize that Charles relies on analytic results for his birational boundedness statement as part of his second proof of the Tate conjecture for K3s (via a Zarhin-type trick), so there is still a morsel of analysis involved, since we need the Tate conjecture to get the geometry of the moduli space of K3s right. Interesting!</p>
<h3 id="the-strongest-possible-filtration-condition">The strongest possible filtration condition</h3>
<p>Perhaps it is worth pointing out what happens if one imposes the strongest
possible filtration-preservation condition (number 1 in the list above). This
condition is very strong. In this case, it
seems pretty likely that the equivalence itself is given by composing an isomorphism
<script type="math/tex">X\to Y</script>, a twist by an invertible sheaf on <script type="math/tex">Y</script>, and a shift in the
derived category.</p>
<p>Among other things, my student <a href="http://www.math.washington.edu/~braggdan/">Dan
Bragg</a> has been thinking a bit about
this. I think Dan has worked out all of the details for arbitrary smooth <script type="math/tex">X</script> and <script type="math/tex">Y</script>, but I
don’t want to put words in his mouth (so I’ll just say “I think”).</p>
<h3 id="what-happens-next">What happens next?</h3>
<p>Martin, Dan, and I have been thinking about the situation for hypersurfaces
(including what happens to Donagi’s result in positive characteristic).
But really the interesting question is whether this holds for all (smooth
projective) varieties <script type="math/tex">X</script> or not, when one is not using the strongest
possible filtration condition. Here is a list of vague questions.</p>
<ol>
<li>What is the closest one can
get to condition 2 above and still have universal truth?</li>
<li>What classes of
varieties will be Torellish with respect to condition 2, or with respect to
some other similar condition?</li>
<li>Is there a hierarchy of conditions with different
Torellish classes of varieties?</li>
</ol>
<p>By “Torellish” I mean exactly what you think I mean.</p>
Sun, 20 Dec 2015 00:00:00 +0000
http://max.lieblich.us/research/2015/12/20/derived-Torelli.html
http://max.lieblich.us/research/2015/12/20/derived-Torelli.htmlK3,TorelliresearchWhat does grading do?<p>I’m thinking about grading again, as I have to do every quarter. My grading
scale is usually not set in stone – I say that exams will have weights that
float in some range – and I usually try to look at the data and tweak the
weights so that things are fair.</p>
<p>None of this has ever sat right with me. Every professor and lecturer sets his
or her own grading scale and we make no effort to harmonize them. Are we
hurting the students? What if Professor <em>X</em> and Lecturer <em>Y</em> use scale <strong>s</strong>, but I’ve
been using scale <strong>t</strong> and this makes my best students fail while propping up my
weaker students? (That seems like an absurd edge case, but how can be we sure
we aren’t doing something like this?)</p>
<p>I decided to investigate. <!--more--></p>
<h3 id="the-experiment">The experiment</h3>
<p>Here’s what I did. I happen to have per-question data for each of my
calculus classes going back to 2013. That is, I know how every student did on
every midterm and final question. To study the effect of grading scale on
grade, I did the following. (Note: I’m not a statistician, so I have no idea if
there is a smarter way of doing this, or if there’s a theoretical explanation
for some of what I observed. Please let me know of there is!)</p>
<ol>
<li>I chose a random weight for the final exam somewhere
between 0.3 and 0.6, and I split the rest evenly between the midterms.</li>
<li>Given these weights, I computed the class scores and looked at the rank
order of the students.</li>
<li>I repeated this process a bunch of times.</li>
<li>For each student, I then have a list of ranks (one for each random grading
run). I computed the standard deviation of this list for each student.</li>
<li>I divided these standard deviations by the number of students.</li>
</ol>
<p>This gives a score for each student under a random grading scale that measure
what percentage deviation in his or her rank one expects overall.</p>
<p>One could wonder what happens if this is done to random data (i.e., randomly
assigned question scores). I computed these as well. One might also wonder
if the students’ autocorrelations are responsible for behavior. So I also
computed the result for “random data with the given covariance matrix” (i.e.,
take random data and then use the Cholesky decomposition of the desired
covariance matrix to make linear combinations of the random data with the
right covariance).</p>
<h3 id="some-results">Some results</h3>
<p>Here are a couple of images that come from numerical
simulations of grading scales run against actual student data, as described
above. These were each generated by 10000 runs. (Like I said, this is a Monte
Carlo approach to something that might just have a simple analytic solution.
Even worse: my R code takes a painful amount of time to run. Does anyone want to weigh in? Send me email! I’ll update the post if someone
explains how this actually works.)</p>
<p>First, a plot showing the expected deviation percentage for real data (from
Autumn of 2014) vs random data, ordered from largest deviation to smallest.
The red curve is from the true data; the blue is from random data.</p>
<p><img src="/TrueVsRandom.png" alt="True v Random" /></p>
<p>Next, a plot showing the same things but comparing real data against minimally
correlated random data.</p>
<p><img src="/TrueVsCorrelated.png" alt="True v Correlated" /></p>
<p>We can also visualize a scatterplot of mean rank (over all runs) versus
normalized standard deviation of ranks (over all runs). First, true versus
random data:</p>
<p><img src="/TrueVsRandomScatter.png" alt="True v Random Scatter" /></p>
<p>Finally, true versus minimally correlated data:</p>
<p><img src="/TrueVsCorrelatedScatter.png" alt="True v Correlated Scatter" /></p>
<h3 id="relevant-observations">Relevant observations</h3>
<p>Several things jump out at us.</p>
<ol>
<li>Any reasonable grading scale (weighting the final somewhere between
0.3 and 0.6) has a reasonably high chance of placing a student within about 5% of
his or her “true” rank in the class.</li>
<li>The fluctuations are greatest near the middle of the curve.</li>
<li>This has good and bad consequences: if you are doing very well or very
poorly, there will be almost no effect, but if you are in the middle, a
small rank change is associated with a larger potential value change.</li>
<li>Linear combinations of uniformly sampled data with the given correlation
matrix <em>almost</em> correctly model true student data. What is going on?</li>
</ol>
<p>Food for thought!</p>
Thu, 15 Oct 2015 10:12:00 +0000
http://max.lieblich.us/teaching/2015/10/15/what-does-grading-do.html
http://max.lieblich.us/teaching/2015/10/15/what-does-grading-do.htmlteaching,grading,analysisteachingCalculus has begun again!<p>My online section of Math 126 (our third quarter of calculus) has begun again!
If you are taking it, you will find our lecture site
<a href="http://max.lieblich.us/math126">here</a>. There is a backing Canvas site
(available to UW students only). A few things will be different this quarter:<!--more--></p>
<ol>
<li>We will be using Canvas, Piazza, and GitHub together. Canvas will house
organizational materials, Piazza will host the discussions, and GitHub will
host the lecture content.</li>
<li>I will be making supplementary video segments to explain some of the
problems that come up in the lectures.</li>
<li>We’re going to be testing a new and improved homework practice system. More
info about this soon!</li>
</ol>
Wed, 30 Sep 2015 10:12:00 +0000
http://max.lieblich.us/teaching/2015/09/30/calculus-time.html
http://max.lieblich.us/teaching/2015/09/30/calculus-time.htmlteachingteachingWebsite refresh<p>This is my new website, built on <a href="http://jekyllrb.com">Jekyll</a> (which is a
static site and blog generator that is markdown-friendly, among other things).
This version of the site should stay more up-to-date than my last site. It will
also have a mildly blog-like aspect, but I’m not sure how often I’ll actually
write anything.</p>
Mon, 07 Sep 2015 06:56:00 +0000
http://max.lieblich.us/news/2015/09/07/welcome.html
http://max.lieblich.us/news/2015/09/07/welcome.htmlnewsnews