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bio website math.lsa.umich.edu/~speyer
location Ann Arbor
age 35
visits member for 5 years, 10 months
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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
comment 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
comment 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 $k=n-k$, then my choice of $(k-1,1)+(1,k-1)$ giving $k^2$ is much better than your choice of $(1,1)+(1,1) + \cdots + (1,1)$ giving $2^k$.
Jul
27
comment Number of bases of a matroid
@TimothyChow Thanks! And that means that the case of no loops and no co-loops reduces to a hopefully easy optimization problem: Minimize $\prod (a_i b_i+1)$, subject to $\sum a_i = k$, $\sum b_i = n-k$, and $a_i$, $b_i \geq 1$. My claim is that the optimum is $((a_1, b_1), (a_2, b_2)) = ((k-1,1), (1,n-k-1))$.
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 co-loops. (I.e. all singletons are independent in the dual matroid.)
Jul
27
comment Determining whether $k(x + x^{-1})$ is post-critically 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 post-critically finite for $0 < |k| < 1$
Jul
27
comment Determining whether $k(x + x^{-1})$ is post-critically finite for $0 < |k| < 1$
I also missed it. Of course, there are no non-archimedean norms for which $|1/2|<1$.
Jul
27
revised A question on representation of graphs
added 3542 characters in body
Jul
26
comment Determining whether $k(x + x^{-1})$ is post-critically finite for $0 < |k| < 1$
If $k=1/2$, then $\pm 1$ are fixed points. I suspect a typo here?
Jul
26
comment 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 co-type. 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/2000-13-03/S0894-0347-00-00331-3
Jul
25
awarded  Nice Answer
Jul
24
comment 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/apr-12.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
comment 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
comment 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 $\approx \frac{e + (-1)^{n+1} e^{-1}}{2} (n-1)!$, if I didn't make any errors, so you still need $d \approx \log_2 n!$.