User alexander woo - MathOverflow

Answer by Alexander Woo for Smoothness properties of the Springer fiber

2012-10-29T06:10:11Z

I am pretty sure what you are aware of is all that is known (publicly at least) about Springer components. You may want to try asking Fresse or Melnikov or Perrin directly. You should not expect a general result, but they may be making further progress on special cases.

As for your secondary question, my understanding (keeping in mind that I only understand this vaguely) is that Kazhdan--Lusztig polynomials were originally defined to answer precisely this question. In particular, the coefficients of $q^{(\ell(w)-\ell(v)-1)/2}$ (the largest possible by definition) in $P_{v,w}(q)$ give these matrix coefficients. (In particular, the Kazhdan--Lusztig cell representations of the Hecke algebra are the Springer representations.) Either you consider this an explicit and practical expression, or, given the state of explicit combinatorial knowledge about Kazhdan--Lusztig polynomials, you consider this as evidence the problem is impossible.

Answer by Alexander Woo for Schubert varieties which admit small resolutions of singularities

2012-09-04T02:01:54Z

For the Hermitian symmetric $G/P$, Nicolas Perrin explicitly classified all the Schubert varieties admitting a small resolution. (More explicitly, he classifies all the minimal models and quotes a theorem that says a small resolution is a smooth minimal model.) If I remember correctly, except for some very small rank exceptions, type A, and the obvious cases (e.g. projective space), all $G/P$ have a Schubert without a small resolution. Fortunately, there are few enough minimal models for the Hermitian symmetric cases that even in the non-smooth case, their intersection cohomology have a reasonable description, leading to the (previously known) explicit formulas for KL-polynomials. (These formulas are summarized in a paper by Boe that proves the final cases and summarizes and refers to the cases done earlier.)

In type A, there is a Schubert without a small resolution for any 2-step flag variety where the two steps are not adjacent. (The example mentioned in the Zelevinsky paper (which he notes as being due to MacPherson) generalizes to all such cases.) This also means there is a Schubert without a small resolution for any 3-or-more-step flag variety.

This leaves as the only case (barring low rank exceptions) adjacent 2-step flag varieties. I have not seen this resolved in the literature. I suspect there is a Schubert variety in those cases without a small resolution provided the steps are at least two away from both ends. In fact I have an explicit expected counterexample, but I have never learned the tools necessary to prove it actually is one.

Answer by Alexander Woo for What can an algebraic geometer do outside academia?

2012-08-27T01:45:17Z

People not originally from the US (or at least not from an English speaking country) tend to have a little more trouble with this question, because they have more difficulty understanding the following:

Traditionally in the United States, higher education is not specific training for a career. Rather, it is training to be a thoughtful contributing citizen. Skills necessary for being a thoughtful contributing citizen tends to be useful in all kinds of intellectual and non-intellectual work. Companies looking for intelligent employees therefore look for intelligent, hard-working people who have done well at studying something, not necessarily anything related to what they do.

As a graduate of a liberal arts college, I saw music majors and English majors get all kinds of jobs. What were the most common?

1) Sales. Plenty of companies need people to sell something. One might sell large software packages to businesses. One might sell re-insurance to insurance companies. One might sell medical devices to doctors. Since salespeople generally are mostly paid on commission, hiring a salesperson is low risk. Traditionally, salespeople know almost nothing about what they are selling; they are just good at convincing people to buy whatever they are selling.

2) Technical sales support. Someone has to understand what they are selling. Except technical sales support people are not actually experts in what is being sold. They've just had a little more training so they can answer the 98% of questions that aren't actually that technical.

3) Software. If you've learned how to program, the vast majority of jobs in the software industry don't require that much software engineering knowledge, and you've probably learned enough. We are not talking about making a database or operating system run faster; we're talking about changing the interface of MathOverflow to move the answer box 5 pixels to the right, or just making sure this webpage is displaying the answers to this question and not some other random question.

4) Consulting. There is both software consulting (see software) and management consulting. This involves being in a group that is hired to look at a company's operations, gather qualitative and quantitative data, and make recommendations. The whole point is that you are mostly not experts and therefore (at least supposedly) have a fresh view on what they are doing and can recommend the obvious stupid things they are missing.

5) Investment banking. See Sales, except you're selling financial products to companies. Except that in this industry, sales is frequently done in teams, so your role might be closer to that of a consultant (assembling data to support the sale) than an actual salesperson.

Realizing not-quite-barycentric subdivision of a polytope

2012-07-19T19:48:12Z

Given a poset $S$, one can form a new poset $I(S)$ whose elements are intervals in $S$ (i.e. either $\emptyset$ or $[a,b]$ for some $a\leq b\in S$) with ordering by (set) inclusion. If $S$ is ranked, then $I(S)$ will also be ranked (by $r([a,b])=r(b)-r(a)$).

If $S$ is the face lattice of a $d$-dimensional polytope $P$, is there a canonical way to construct a $d+1$-dimensional polytope $I(P)$ with face lattice $I(S)$? Is there a name for this construction?

Notes:

1) The 2-faces will always be quadrilaterals.

2) The underlying cellular complex is not the barycentric subdivision, whose faces are the chains of $S$, not the intervals.

3) If you apply the construction to a simplex, you should get a cube (of one higher dimension).

4) Of course the best construction should preserve symmetries and intertwine the inclusion of a face $F$ into $P$ with that of $I(F)$ into $I(P)$.

5) The only polytope I actually need an answer for right now is the regular 3-dimensional cube. If this construction only works for, say, simple polytopes, I'm fine with that.

Answer by Alexander Woo for Computer package for representation theory of the symmetric group

2012-05-11T21:05:11Z

Another answer: John Stembridge's Maple code for dealing with Weyl groups may also have something along these lines.

Answer by Alexander Woo for Computer package for representation theory of the symmetric group

2012-05-11T21:04:11Z

The combinatorics package in Sage should do these things, at least if you are just interested in the decompositions into irreducibles and not actual matrices.

Answer by Alexander Woo for any software to compute multivariable resultant?

2012-03-08T03:57:02Z

Singular and Macaulay2 do it according to the documentation, though I haven't personally tried using either of them for this purpose. They are likely to be faster than Maple; I don't know about MARS.

Answer by Alexander Woo for Symmetric polynoms are Hopf algebra ? What for one needs co-product ?

2012-01-18T19:03:26Z

There is also a motivation coming from the (co)homology of Grassmannians.

(Note I'm making this answer community wiki - I would like someone to expand on this, since I don't have the time right now to look up sources to do so and don't want to risk getting something wrong by going off the top of my head.)

Answer by Alexander Woo for Schubert problems to cycle class in Grassmanian

2011-12-29T17:23:47Z

For simplicity, I am supposing your family is a pure-dimensional (though not necessarily irreducible) subvariety $V$ in the Grassmannian.

As you probably know, given a fixed $i$, the classes $[X_\lambda]$ of Schubert subvarieties $X_\lambda$, where $\lambda$ is a partition which has $i$ boxes and fits inside a $k \times n-k$ rectangle, form a basis of $H^{2i}(G_{k,n})$. (The Chow ring and cohomology ring are the same for Grassmannians over $\mathbb{C}$.) (I am assuming a particular indexing convention for Schubert varieties; with a different indexing convention, $\lambda$ should have $k(n-k)-i$ boxes.)

Under the intersection pairing $\langle \cdot, \cdot\rangle$ between $H^{2i}$ and $H^{2[k(n-k)-i]}$, the Schubert bases are dual to each other. To be precise, $\langle[X_\lambda],[X_\mu]\rangle = 1$ if $\lambda^*=\mu$ and $\langle[X_\lambda],[X_\mu]\rangle=0$ if $\lambda^*\neq\mu$ . Here $\lambda^*$ is the box-complement to $\lambda$. Take all the squares in the $k\times (n-k)$ rectangle which are not part of $\lambda$, rotate 180 degrees, and you have the partition $\lambda^*$ . In notation, $\lambda^*_i = n-k+1-\lambda_{k+1-i}$ .

This means that the class $[V]$ is given by $$[V]=\sum_{\lambda} \langle [V], [S_{\lambda^*}]\rangle [S_\lambda],$$ where the sum is over all partitions $\lambda$ fitting inside a $k\times n-k$ rectangle with $\mathrm{codim} V$ boxes.

There cannot be any easier method because it takes $d$ pieces of information to determine an element in a vector space of dimension $d$.

Jacobian Conjecture for unit triangular matrices

2011-12-14T06:27:15Z

This question is about the Jacobian conjecture for a special case. I will first explain the Jacobian conjecture (since it is something every mathematician should know about).

Let $k$ be an algebraically closed field.

Consider a map $$F: k^n \rightarrow k^n,$$ defined by $$F(x_1,\ldots,x_n)=(f_1(x_1,\ldots,x_n),\ldots,f_n(x_1,\ldots,x_n)),$$ where $f_1,\ldots,f_n$ are polynomials.

The Jacobian of $F$, which I denote $J$, is the determinant of the matrix $dF$ where the $(i,j)$-th entry of $dF$ is $\partial f_i/\partial x_j$. (The matrix $dF$ gives the induced map on the tangent bundle, or maybe it's the cotangent bundle; it doesn't matter for this question.)

Since $F$ is given by polynomials, the entries of $dF$ are polynomials and $J$ is a polynomial. Hence $F$ is a nonsingular map if and only if $J$ is a constant.

The inverse function theorem tells us that $F$ has a smooth inverse map if and only if $J$ is constant. The Jacobian conjecture says that this smooth map is in fact also given by polynomials (in the case where the original map $F$ is given by polynomials).

Question: I would like to know if the Jacobian conjecture is known (or trivial) for the special case where the matrix $dF$ is a triangular matrix with $1$'s on the diagonal. If it is not known, I would like to know if the full Jacobian conjecture is known to be equivalent to this special case.

Motivation: This is a possible strategy for proving that a particular family of maps I have constructed for a particular purpose is in fact invertible within the category of affine algebraic varieties.

EDIT: Clarifying in light of Tom's remark. The inverse function theorem just says that $F$ has a local inverse. The Jacobian conjecture is that $F$ has a global inverse which is given by polynomials.

Answer by Alexander Woo for Motivating Algebra and Analysis for Average Undergraduates

2011-11-02T01:46:22Z

If you color the Cayley table of a group $G$ according to what coset of some subgroup $H$ an element belongs to, then you get a nice pattern if and only if $H$ is normal.

Answer by Alexander Woo for Motivating Algebra and Analysis for Average Undergraduates

2011-11-01T19:35:50Z

1) For everyone except mathematicians (and, prior to 1820 or so, everyone except George Berkeley), the ultimate reason for believing calculus is that it helps engineers build bridges that don't collapse. Analysis does not justify calculus; applications do. Do some naive manipulations with series get one into trouble? Yes, but it is not necessary to develop analysis to deal with the problem; one just has to learn by experience not to do those manipulations. (This is not a new point of view. I am just trying to explain what Wittgenstein tried to explain to Turing in 1946.)

2) In the same sense that literature is an unnecessary, parasitic phenomenon upon ordinary language, "higher mathematics" is an unnecessary, parasitic phenomenon upon ordinary calculation. English majors study Shakespeare because it is a great historical achievement of our civilization and its study teaches us various useful skills. Math majors study analysis and algebra for the same reason. (EDIT: I reread this, and realized that it's possible for people to misread it. I have the greatest respect both for the study of literature and the study of higher mathematics and think both are worthwhile pursuits. I think the viewpoint that denigrates these pursuits is a bad viewpoint, but at the same time I don't think it is an irrational viewpoint.)

3) In my experience, although students may ask for motivation, what they are really looking for is something with which they are familiar to which they can compare the new stuff they are learning, so that they can build a context for the new concepts. I hope you will get some answers answering this implied question, but since I believe in brutal honesty with students, I think my above points needed mentioning.

Answer by Alexander Woo for Virtual algebraic calculation within proofs

2011-10-15T07:02:42Z

A very elementary example (simpler than the ones you've given) is the generating function for the number of partitions of $n$, denoted $p_n$: $$\sum_{n\geq 0} p_n q^n = \prod_{i\geq 0} \frac{1}{1-q^i}.$$

Answer by Alexander Woo for What motivates modern algebraic geometry for a combinatorial/constructive algebraist?

2011-10-02T04:46:18Z

Positivity of Kazhdan--Lusztig polynomials (and all the other positivity results in Kazhdan--Lusztig theory in general).

Consider the Hecke algebra $H_n(q)$. It is a particular deformation of the group algebra of the symmetric group (or some other Coxeter group). As such, it has a basis $T_w$ indexed by permutations, and multiplication is given by $$T_wT_{s_i}=T_{ws_i}$$ if $\ell(ws_i)=\ell(w)+1$, and $$T_wT_{s_i}=qT_{ws_i}+(1-q)T_w$$ if $\ell(ws_i)=\ell(w)-1$.

Define an involution on $H_n(q)$ (usually called the bar involution) by $\overline{q}=q^{-1}$ and $\overline{T_w}=(T_{w^{-1}})^{-1}$. Kazhdan and Lusztig proved that there exists a unique basis $C^\prime_w$ such that

1) $\overline{C^\prime_w}=C^\prime_w$

2) If we write $C^\prime_w=\sum_x P_{x,w}(q)T_x$, then the degree of $P_{x,w}(q)$ is bounded above by $(\ell(w)-\ell(x)-1)/2$.

3) $P_{w,w}(q)=1$.

The polynomials $P_{x,w}(q)$ (and in particular their coefficient in the maximum degree they are allowed) turn out to give a very nice combinatorial way to construct representations of $S_n$ (or the Coxeter group in question), and a similar theory also constructs representations of finite groups of Lie type.

Now, the only way to prove that $P_{x,w}(q)$ have positive integer coefficients so far is to show that they are the Poincare polynomials for local intersection cohomology on Schubert varieties. Even better, one should interpret the Hecke algebra as a kind of Grothendieck group on the category of perverse sheaves on the flag variety. Springer in the early 1980s used this interpretation to show that, if one takes a product $C_vC_w$ and expands this product in the $C$ basis, the coefficients are all polynomials with positive integer coefficients. (The $C^\prime$ basis is a variant of the $C$ basis that is a little easier to write.)

(The best references I know are Humphrey's book on reflection groups and Coxeter groups and Bjorner and Brenti's book on Combinatorics of Coxeter groups, both of which have a chapter devoted to this subject.)

Answer by Alexander Woo for Giambelli and Porteous Formula

2011-09-28T18:02:50Z

William Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math J. 65 (1992) 381--420

Parabolic convolution of perverse sheaves in terms of the Hecke algebra

2011-08-30T00:21:40Z

It is "well-known" that the Hecke algebra $\mathcal{H}$ can be thought of as the Grothendieck group for the category of perverse sheaves on $G/B$, where the product in $\mathcal{H}$ corresponds to convolution of sheaves by the Borel subgroup. This means, given perverse sheaves $X$ and $Y$ on $G/B$ and their classes $[X]$ and $[Y]$ in the Grothendieck group considered as elements of $\mathcal{H}$, the element $[X][Y]$ is (the class of) the sheaf

$$q_B(\pi^*(X)\boxtimes Y),$$

where $\pi$ is the quotient map $G\rightarrow G/B$ and $q_B$ is the operation that takes a sheaf on $G\times G/B$ which is equivariant under the action of $B$ by $b\cdot (g, hB) = (gb^{-1}, bhB)$ and quotients out by this action, mapping to a sheaf on $G/B$.

Now I want to think about convolution using a parabolic subgroup larger than the Borel. Suppose $P$ is a parabolic, $X$ is a perverse sheaf on $G/B$ which is pulled back from $G/P$, and $Y$ is $P$-equivariant.

What is the class of

$$q_P(\pi^*(X)\boxtimes Y),$$

where $q_P$ now quotients by the action of $P$ by $p\cdot (g, hB) = (gp^{-1}, pHB)$ in terms of the Hecke algebra?

Motivation: Actually, I only care about the case where $X$ and $Y$ are IC sheaves of Schubert varieties, so $[X]$ and $[Y]$ are Kazhdan--Lusztig basis elements. In this case, by the Decomposition Theorem the product will give a positive, bar--invariant sum of Kazhdan--Lusztig basis elements, so if you find good choices of $X$ and $Y$ and understand this product, you get a good inductive method for calculating Kazhdan--Lusztig polynomials. In some cases I am interested in, I have calculated the product geometrically by localization, but sticking to algebra would make things cleaner and likely easier to write up in general. There are cases in which this has been done, most notably Polo's paper showing that any polynomial with positive integer coefficients and constant term 1 is a Kazhdan--Lusztig polynomial; an answer to this question should allow one to dispense with geometry in his paper and formulate his calculations entirely within the Hecke algebra (given a theorem that says the algebraic analogue of parabolic convolution produces positive sums of Kazhdan--Lusztig elements).

Although everything makes sense for an arbitrary Kac-Moody group, I am happy with an answer for the finite dimensional case, or even with answers for type A.

Pre-emptive requests: (1) I expect parabolic Kazhdan--Lusztig elements will come up. As there are several variants, please tell us which one you mean, with reference to a paper using your notation. (2) As you may have noted from my vague description of $q_B$ and any other mistakes I might have made above, I don't really understand perverse sheaves and intersection cohomology. Please feel free to correct me, and please give me an answer I can understand.

Answer by Alexander Woo for A meta-mathematical question related to Hilbert tenth problem

2011-09-26T05:23:16Z

I would take the sentence to mean something like:

Given any model of PA in any axiomatic system, there is no Turing machine that ... according to whether the sentence is true or false for that model.

Answer by Alexander Woo for How important is it for one on the job market to have thought about suitable REU projects?

2011-09-12T04:59:14Z

1) If you are genuinely seriously interested in mentoring undergraduates in REU projects, then you would naturally be fantasizing about projects you might do. If you aren't all that interested in it, you can't fake it just by thinking up some projects. Note that there is some middle ground between 'genuinely seriously interested' and 'not all that interested'. Thinking up projects does show that you are interested in it. The ones you think up don't need to be great; they just need to be not delusional. Keep in mind that weaker schools will only have someone who is potential grad student material only once every several years, so those schools, if they are interested in undergraduate research, will want projects that someone who is not grad student material can tackle. (As a rough and not particularly accurate guide, by 'weaker school' I mean median SAT of incoming freshman below 1150 or so for a small liberal arts college; larger universities may have some stronger students even if the average is somewhat lower.) People in applied and computational areas have a huge advantage here because they can propose research that does not involve proving any theorems.

The truth is that you'll probably have a failure or two before you have a success, and it's not a big deal unless it is a total failure where the student gets nothing out of it. So don't worry too much about the specifics of the project; the point of having one is to show you are serious and not completely delusional (given the school and its students) about it.

2) Unless the advertisement does not ask for a research statement or asks for undergraduate research to be addressed elsewhere, it goes in a section of your research statement.

Keep in mind that many schools that care seriously about undergraduate research will care only a little about your research except inasmuch as it generates projects for undergraduates. When applying for positions at such schools, you may need to reshape your research statement to say only generalities about your own research (since no one there will understand most of it anyway) and focus on the accessible parts and the undergraduate research. It may also be the case that such a school will not ask for or read a research statement, in which case you need to be prepared to put the material as a section in your teaching statement.

3) It absolutely does NOT have to be in your primary research area, as long as it is an area where you have enough of an idea what is going on to actually guide a project. (In other words, you should know enough that having the project turn out to be a fairly well-known solved problem is not at all a risk.) In fact, many professors at undergraduate institutions have gradually migrated out of their dissertation areas to research areas that are more friendly to their students.

Answer by Alexander Woo for Smoothness of hypersurfaces in Grassmannians

2011-09-01T01:46:29Z

For Grassmannians and more generally for homogeneous spaces, it is sometimes far easier to check if a subvariety is smooth by checking on each chart since all the charts are affine space (of much smaller dimension).

Take local equations defining your variety on each chart and use the Jacobian criterion on each of those sets of equations.

To get local equations from global equations (for the Grassmannian), recall that a chart is where a given particular Pluecker coordinate $p_I$ does not vanish, and the chart itself is affine space with coordinates $p_J/p_I$ where $J$ differs from $I$ by one element, and other Pluecker coordinates can be written as determinants of these. (I can give more details, but I won't figure them out if no one is interested.)

Sometimes for general reasons you will know that if the variety is singular there must be a singular point on a specific chart, which will make your life even easier.

Answer by Alexander Woo for How misleading is it to regard $\frac{dy}{dx}$ as a fraction?

2011-08-24T03:38:08Z

I find $dy/dx$ misleading because it treats $x$ and $y$ as similar objects.

When you use this notation, you lose the important point that $y$ is a function of $x$; instead you end up looking at $x$ and $y$ as related quantities.

I think it is important for calculus students to get the idea that differentiation is an operation that takes one function and produces a new function. In that way, it is fundamentally different from addition (or unary negation) of numbers (which is not the same thing as addition of functions).

Note that I am a lot more interested in (theoretical) computer science than (any form of) physics - this may bias my point of view.

What does the space induced by this unusual metric(?) on R/Z look like?

2011-05-11T05:44:11Z

The motivation for this question comes from music theory. Dmitri Tymoczko models "good" voice leading as minimizing distance between pitches in successive chords. While this theory works well for upper voices, it does not work so well for the bass, which tends to move by 4ths and 5ths quite frequently. (Tymoczko explicitly excludes the bass from his model.)

Taking log (all logs are base 2 in this question) of frequency and taking pitches which differ by an octave as equivalent, we get $\mathbb{R}/\mathbb{Z}$. For the upper voices, we want the standard metric on this.

For the bass, we want moving by a fifth or a fourth - meaning by $\pm \log (3/2)$ to be small. So we want $d(x,x\pm\log(3/2))=k_1$, where $k_1$ is probably somewhere around $0.05$. To make this a metric space, let's declare that $d(x,y)$ should be the minimum of $|x-y|$, $k_1+||x-y|-\log(3/2)|$, and $k_1+||x-y|-\log(4/3)|$.

We probably want moving by a major third - meaning by $\pm\log(5/4)$ to also be small, but not as small. I suspect $2k_1$ would make the most mathematical sense, but any constant of roughly that magnitude is fine. Ditto for minor thirds - this would be movement by $\pm\log(6/5)$, with a slightly larger constant.

If we do this, we might as well make all movements by $\pm\log(p/q)$ small if $q$ is small.

CLARIFICATION: I also want the standard metric to be one of the options for getting from $x$ to $y$. So the distance between $0$ and $\sqrt{1/500}$ should be $\sqrt{1/500}$, while the distance between $0$ and $7/12$ should be $k_1+|\log(3/2)-7/12|$. (Musically, $|\log(3/2)-7/12|$ is how far off an even-tempered 5th is from Pythagorean tuning.)

Question 1: Can one actually define something along these lines that satisfies the triangle inequality? (I don't think I actually have; I probably need to take the minimum (or infimum) of some infinite sequence, but am not entirely sure that works.)

Question 2: Assuming the answer to (1) is yes, what does this metric space look like? Can someone help me with a picture that seems less exotic, perhaps comparing it to the Hawaiian Earring or something of that sort? In particular, what might the fundamental group look like?

My background: I'm a combinatorialist and algebraic geometer who happens to be the one least unqualified here to be supervising an undergraduate independent study on mathematics in music theory. I did the standard first year graduate courses in point set topology and algebraic topology, but that was almost a dozen years ago.

Answer by Alexander Woo for Picard group of Schubert varieties

2011-04-14T16:25:21Z

Alex Yong and I work this out in our paper for the case of the Borel subgroup, but I'm pretty sure it's the same for every parabolic.

When is a Schubert variety Gorenstein?, Advances in Math. 207 (2006), 205-220.

Please note our conventions in that paper are backwards from yours in that our Schubert varieties are $\overline{B_-wB/B}$.

We don't say explicitly what happens for groups other than GL_n, but you can do the same thing using the appropriate Monk-Chevalley formula for the group in question.

EDIT: More details upon glancing at my own paper... Mathieu (reference in comments) shows that every line bundle on a Schubert variety is the restriction of a line bundle on the homogeneous space. (Actually, iirc the proof in the finite dimensional case predates Mathieu.) This means the Picard group of the Schubert variety will be the same as that of the homogeneous space unless some nontrivial line bundle restricts to a trivial one. For $G/P$ where $P$ is a maximal parabolic, this only happens if your Schubert variety is a point.

What we do is actually work out the Picard group as a subgroup of the class group.

Answer by Alexander Woo for Teaching undergraduate students to write proofs

2011-01-13T05:21:39Z

I am teaching such a course - in a 4 week intensive one course at a time format - right now. This is a reminder to myself to say something intelligent about it in February, and a placeholder for my future answer. (If you see this in mid-February, a reminder to actually put up an answer will be much appreciated. I am posting under my real name and can be Googled.)

One of the things I am thinking about now, 2 weeks in:

It has surprised me the extent that what one might simply call cognitive deficits are an obstacle. Some of my students have trouble consistently being able to keep three ideas with their precise statements in their head at the same time. (I mean to say that if they make a special effort for one or two statements, they can, but it is a struggle for them to do this routinely over even a short proof.) This is a serious problem because when a step involves going from a statement with two quantifiers to another statement, the first statement with the quantifiers has already fills up their head and there is no room for the next statement.

What I am suggesting to my students, with absolute seriousness, is to do a Sudoku puzzle or two every day, preferably with a pen (to force themselves to think through inferences rather than going by trial and error). The inferences involved in doing Sudoku might be quite simple to most of us, but you do have to keep a few facts in your head simultaneously to make the inference, and I am hoping the practice does improve their working memory.

Resources for teaching arithmetic to calculus students

2010-11-05T18:59:16Z

Every time we teach calculus we discover that a significant portion of our students never understood arithmetic. I don't mean that they can't multiply numbers, but rather that they don't know intuitively that a car going 15 miles an hour goes 1 mile in 60/15=4 minutes (i.e. that division is the arithmetic operation corresponding to this problem).

It would be entirely inappropriate to teach them as if they were 10 years old, even if we had 3 months to teach them arithmetic.

Usually these are fairly intelligent individuals considering that they managed to get through high school mathematics well enough to get into a good college or university despite this handicap, and this deficiency in their background is not their fault in any way. They are likely to be able to pick up arithmetic quite quickly, and figure out from that why they have been a little befuddled through all of high school math.

I would hope this problem has been studied and ways to help these students have been proposed.

I am looking for references either to resources for these students or resources for instructors trying to help these students in the context of a calculus (or precalculus) class.

Answer by Alexander Woo for Reference for: the Bruhat-minimal permutations not less than a fixed permutation pi?

2010-10-11T17:40:45Z

Vic Reiner, Alex Yong, and I spell this out in Sections 4.1 and 4.2 of our paper on the cohomology rings of Schubert varieties: http://arxiv.org/abs/0809.2981

This is not really original to us: for type A we refer back to Lascoux and Schutzenberger's paper Trellis et bases des groupes de Coxeter, Elect J. Combin. 3, no. 2 R27, though perhaps they don't state things exactly in this form.

Note for Allen: The rest of our paper might not be as irrelevant to you as it might seem at first glance. Jim Carrell started a line of work back in the 80s relating cohomology rings of Schubert varieties to their local equations at the identity. Interestingly, their strongest results are only for type A.

Answer by Alexander Woo for Why do we teach calculus students the derivative as a limit?

2010-09-28T21:54:31Z

The answer I give my students is that mathematicians want to know what a word (in this case 'derivative') means in all cases, and the definition of the derivative is a communal agreement about what to say in strange cases such as the absolute value function. (Well, since I banish symbolic stuff from the first two weeks, I say 'function whose graph has a sharp corner like this one (draws on board)'.)

If students press further, I point out that in a literature class they are expected to learn the communal agreement on the difference between a 'simile' and a 'metaphor'. It helps that I am at a liberal arts institution and not a technical one.

Let me also use this opportunity to share a pedagogical trick:

I find it helpful (third time I've tried it) to break up the definition of $f^\prime(2)$ into two parts:

1) Define a new function $E_2$ by the formula $$E_2(x)=\frac{f(x)-f(2)}{x-2}.$$ 2) Take the limit of $E_2$ at 2.

To pull this off, you do need to take the function $E_2$ somewhat seriously; graph it, write formulas for it, et c.

Rationale:

1) It always helps to break up complicated definitions into smaller pieces.

2) It emphasizes that you take limits of functions (in the sense of machines that accept a single number as input and gives a single number of output) rather than of symbolic expressions.

3) Students get to really understand why a discontinuous function or something like the absolute value function is not differentiable (at the relevant point).

Answer by Alexander Woo for Explicit equations for Schubert varieties

2010-08-27T21:04:23Z

Answer by Alexander Woo for Why were matrix determinants once such a big deal?

2010-08-19T02:57:47Z

Warning: I am basically just speculating, and not commenting with actual knowledge of the history.

I suspect that a lot of the nineteenth century work on determinants was motivated by invariant theory. Before Hilbert proved abstractly the finite generation of invariants, there was a small industry trying to explicitly compute invariants (for example the projective invariants of the action of $GL_2$ acting on binary forms) with the aim of giving a constructive proof that they were finitely generated.

These invariants often have a basis consisting of various sorts of determinantal expressions, and if you want to prove finite generation, you have to construct ways of taking certain determinantal expressions and writing them in terms of other determinantal expressions.

Within a decade or so of Hilbert's paper, people generally lost interest in constructive invariant theory. (After all, the abstract methods answered the most interesting questions and seemed much more likely than constructive methods to answer the most interesting remaining questions.)

I have wondered whether all the work on syzygies of determinantal varieties actually reduces to identities which were well known (to the right people) in the 19th century.

Irreducible components of quotients of Cohen-Macaulay rings of the "correct" dimension

2010-07-31T23:13:46Z

Suppose $R$ is a Cohen-Macaulay ring. It is well known that if $I$ is an ideal of $R$ generated by $n$ elements, and $I$ has codimension $n$, then $R/I$ is also Cohen-Macaulay.

Now suppose that $I$ does not have codimension $n$, but (the scheme defined by) $R/I$ has several irreducible components, one of which has codimension $n$. Is (the coordinate ring of) that component necessarily Cohen-Macaulay?

Because being Cohen-Macaulay is a local condition, it is clear that the component is generically (in fact everywhere it does not intersect the other components) Cohen-Macaulay, but there is no reason obvious to me why this would extend to the whole component.

Answer by Alexander Woo for How can I get a small resolution for the binomial fourfold $x_1 x_2 x_3- y_1 y_2=0$ in $\mathbb{C}^5$?

2010-07-13T19:19:44Z

Warning: This seems to be a really bad way of answering this question (but it at least tells you there is one).

The intersection of the opposite Schubert cell $X_\circ^{13425}$ with the Schubert variety $X_{34512}$ is defined by that equation in the appropriate coordinates. This tells you that locally around the Schubert point $e_{13425}$, the Schubert variety is isomorphic to the product of that fourfold with $\mathbb{C}^{4}$.

Since the permutation $34512$ is 321 and hexagon avoiding, the Bott--Samelson resolution is small for that Schubert variety.

As you have a binomial, I would expect there to be some toric answer that is at least a little more general than this one.

Comment by Alexander Woo

2013-05-12T06:21:58Z

I would love to be contradicted, but my educated guess is that this is not known.

Comment by Alexander Woo

2013-04-11T18:36:23Z

A silly solution: Pick one element x. Declare every finite set NOT containing x to be open. Declare the COMPLEMENT of every finite set NOT containing x to be open. This is a compact Hausdorff topology.

Comment by Alexander Woo

2013-04-10T20:45:51Z

You have the right reference. Lascoux's paper isn't a complete solution; he has to assume some constants he constructs are nonzero, but does not prove that the constants actually are nonzero. (This is acknowledged in the paper; it was the best that could be done at the time.) The paper that finishes off this problem in characteristic zero is by Pragacz and Weyman in the late 80s; I don't have the reference offhand but you'll find it in Weyman's book.

Comment by Alexander Woo

2013-03-31T05:21:12Z

their research statement, which helped since we were hiring specifically in applied algebra. Some applicants also used part of their cover letter to convince us they were interested teaching our population of students; they couldn't do so in their teaching statement since that was used for a much broader range of types of schools.

Comment by Alexander Woo

2013-03-31T05:17:37Z

I strongly disagree with what you have written about the cover letter. Beyond having a reasonable number of publications, what I looked for in our search this year was an indication that the applicant fit our position, and the easiest place to see this was the one paragraph synopses of research and teaching in the cover letter. (The research and teaching statements are too long for a first screening.) Customization of cover letters clearly made a difference. Some relatively pure algebraists made a case for their involvement with applications in their cover letter that they did not in...

Comment by Alexander Woo

2013-03-28T19:37:34Z

There are papers on "permutation arrays" by Eriksson and Linusson which might be useful; there is a paper by Billey and Vakil using this to solve Schubert problems.

Comment by Alexander Woo

2013-02-13T22:33:43Z

To be more specific, this is Equation 2.2 in Section 2 of Chapter I, found on p. 19 of the 2nd edition.

Comment by Alexander Woo

2013-02-13T00:42:14Z

What's the complete homogeneous symmetric function analogue? I might recognize that more easily. I get the feeling this is standard or at least a routine application of something standard in symmetric function theory. The right hand sides look like the squares of Cauchy kernels.

Comment by Alexander Woo

2013-02-05T05:26:34Z

Your 'original' state of mind is nonsense and madness. The use of language is a social activity. Any communication of mathematics requires language, and not only any language, but mathematical language. Therefore, any communication of mathematics is social and requires inhabiting a context of other mathematics. And if you are thinking of 'mathematics before it is communicated' I'm afraid that Wittgenstein eviscerated that notion (and all other similar notions) with his argument about the beetle in a box (Philosophical Investigations I.293).

Comment by Alexander Woo

2013-01-12T02:03:18Z

Send me an e-mail if you want suggestions of whom to ask (but I don't want to publicly put people on the spot).

Comment by Alexander Woo

2012-12-19T00:48:34Z

Michael: Many people who might have an answer to your question (myself included) don't remember the details of the increasing-Bruhat-path interpretation of $R$-polynomials and are too lazy to reach for their copy of Bjorner--Brenti to find it. (In my case, it's not just laziness; I'm about 1000 miles away from my office at the moment.) Providing some partial explanation might help you get an answer.

Comment by Alexander Woo

2012-12-18T01:45:51Z

You ask for an explicit construction of these automorphisms. Unfortunately, there are a number of different ways to understand Spin(8) explicitly, and without knowing how you understand Spin(8) explicitly, we can't really figure out how to answer your question in a way that's useful to you.

Comment by Alexander Woo

2012-12-14T05:20:51Z

Yes this is true. Unfortunately I can't remember a proof or a good reference at the moment. (I believe the fact because the covariant ring is the cohomology ring of the flag variety which has a basis of Schubert classes.)

Comment by Alexander Woo

2012-11-22T17:53:31Z

Any chance you could rewrite your question in math notation rather than physics notation, or explain the physics notation for the benefit of us mathematicians who don't understand it?

Comment by Alexander Woo

2012-11-17T16:18:02Z

The philosophical landscape on the relationship between mathematics and physics changed greatly over the 20th century. The generation of Hilbert and Riemann thought that reality IS mathematical, whether they interpreted the word 'reality' in a realist or Kantian idealist sense. Nowadays, most philosophers, and even most mathematicians and physicists, only accept the weaker claim that reality can be described mathematically, or perhaps that reality is best described mathematically. In our context, questions like Riemann's or Hilbert's 6th problem (axiomatize physics) seem quite nonsensical.