bio  website  math.lsa.umich.edu/~speyer 

location  Ann Arbor  
age  35  
visits  member for  5 years, 10 months 
seen  1 hour ago  
stats  profile views  34,388 
Associate Professor of Mathematics at the University of Michigan. My research interests are in combinatorial algebraic geometry, particularly Schubert calculus, matroids and cluster algebras. I also enjoy thinking about number theory and computational mathematics.
1d

awarded  Nice Answer 
1d

awarded  Nice Answer 
Jul 30 
comment 
Real varieties with enough algebraic loops
This is really nice! I'm glad I finally noticed it. 
Jul 29 
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Genus of a plane curve of the form $\prod_{i=1}^n (a_iX+b_iY+Z) = Z^n$
What the heck, abx? MO is a site for mathematicians, not algebraic geometers. There is plenty of room in math for people not to know this, plus as Peter Mueller's answer shows, there is something to be careful about even here. 
Jul 28 
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Number of bases of a matroid
It really is $a_i$, $b_i \geq 1$. But choosing all $1$'s isn't optimal. But choosing all $1$'s isn't optimal. If $k=nk$, then my choice of $(k1,1)+(1,k1)$ giving $k^2$ is much better than your choice of $(1,1)+(1,1) + \cdots + (1,1)$ giving $2^k$. 
Jul 27 
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Number of bases of a matroid
@TimothyChow Thanks! And that means that the case of no loops and no coloops reduces to a hopefully easy optimization problem: Minimize $\prod (a_i b_i+1)$, subject to $\sum a_i = k$, $\sum b_i = nk$, and $a_i$, $b_i \geq 1$. My claim is that the optimum is $((a_1, b_1), (a_2, b_2)) = ((k1,1), (1,nk1))$. 
Jul 27 
revised 
Number of bases of a matroid
added 297 characters in body 
Jul 27 
answered  Number of bases of a matroid 
Jul 27 
comment 
Number of bases of a matroid
It would be more natural to also impose that there are no coloops. (I.e. all singletons are independent in the dual matroid.) 
Jul 27 
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Determining whether $k(x + x^{1})$ is postcritically finite for $0 < k < 1$
Hensel's lemma is enough in this case: Make the change of variables $t=2u$ and note that the root $u=1$ of $1  u + 12 u^2 + 16 u^3 + 64 u^4$ lifts to $\mathbb{Q}_2$ by Hensel. But you should learn about Newton polytopes! They make spotting this sort of thing obvious, and they are a lot of fun. 
Jul 27 
answered  Determining whether $k(x + x^{1})$ is postcritically finite for $0 < k < 1$ 
Jul 27 
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Determining whether $k(x + x^{1})$ is postcritically finite for $0 < k < 1$
I also missed it. Of course, there are no nonarchimedean norms for which $1/2<1$. 
Jul 27 
revised 
A question on representation of graphs
added 3542 characters in body 
Jul 26 
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Determining whether $k(x + x^{1})$ is postcritically finite for $0 < k < 1$
If $k=1/2$, then $\pm 1$ are fixed points. I suspect a typo here? 
Jul 26 
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An algebraic strengthening of the Saturation Conjecture
A note on terminology (and then I will think about this interesting question). I usually understood the Hall algebra to refer to the ring whose elements were formal sums of isomorphism classes of $\mathfrak{o}$modules, and where multipication was given by counting submodules of given type and cotype. See, for example, arxiv.org/abs/math/0611617 . Also, as I imagine you know, Derksen and Weyman gave a far reaching generalization of the saturation conjecture in terms of this sort of Hall algebra  see ams.org/journals/jams/20001303/S0894034700003313 
Jul 25 
awarded  Nice Answer 
Jul 24 
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Intuition behind the Kodaira Vanishing Theorem?
Thank you for explaining this! I read through that proof pretty carefully in order to teach it a few years ago, and didn't come up with nearly as good a way to summarize it. (See math.lsa.umich.edu/~speyer/632Old/apr12.pdf if you are curious.) One thing I do find helpful is to point out the stronger result $H^q(X, L \otimes \Omega^p)=0$, for $p+q>n$. That makes the $(p,q)$ symmetry much more clear. 
Jul 24 
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Are there congruence subgroups other than $SL_2(\mathbb Z)$ with exactly 1 cusp?
Indeed, in each row, the first two generators lie in $SL_2$ and, I believe, generate the desired subgroup of $SL_2$. 
Jul 24 
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Are there congruence subgroups other than $SL_2(\mathbb Z)$ with exactly 1 cusp?
You're welcome! Thanks for making me actually think through the details. I just realized that these are very close to Zassenhaus's classification of groups $G \subset GL_2(\mathbb{F}_p)$ which act freely and transitively on $\mathbb{F}_p^2 \setminus \{ (0,0) \}$. You can find explicit generators for those at en.wikipedia.org/wiki/… . I would guess that intersecting with $SL_2$ and quotienting to $PSL_2$ gives generators for the current problem. 
Jul 24 
comment 
A question on representation of graphs
Okay, but that doesn't help you much. The number of odd cycles is $\approx \frac{e + (1)^{n+1} e^{1}}{2} (n1)!$, if I didn't make any errors, so you still need $d \approx \log_2 n!$. 