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bio website math.lsa.umich.edu/~speyer
location Ann Arbor
age 34
visits member for 5 years
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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.


4h
comment scheme of commuting matrices
See arxiv.org/abs/math/0306275
Oct
16
awarded  Enlightened
Oct
16
awarded  Nice Answer
Oct
16
revised A question from the proof of affine algebraic group is a linear
deleted 1 character in body
Oct
16
answered A question from the proof of affine algebraic group is a linear
Oct
15
comment Isotypic components of the action of the symmetric group on polynomials
continued later, student just arrived.
Oct
15
comment Isotypic components of the action of the symmetric group on polynomials
@NicholasProudfoot Because $R:=\mathbb{C}[x_1, \ldots, x_n]^{S_n}$ is a polynomial ring, and $S:=\mathbb{C}[x_1,\ldots, x_n]$ is Cohen-Macualay, $S$ is a free $R$-module. Moreover, each isotypic summand of $S$ is likewise a free $R$-module. (A graded summand of a graded free module over a polynomial ring is graded free.) continued..
Oct
14
comment Gauss's Cirlce Problem #lattice points in circle
The original source is G.H. Hardy, On the Expression of a Number as the Sum of Two Squares, Quart. J. Math. 46, (1915), pp.263–283 . Doesn't seem to be online. This undergrad thesis math.rochester.edu/undergraduate/sums/reu/… repeats the argument.
Oct
10
awarded  Yearling
Oct
9
revised Asking for an English version of a paper
added 2 characters in body
Oct
9
answered Asking for an English version of a paper
Oct
8
comment If the direct image of f preserves coherent sheaves on noetherian schemes, how to show f is proper?
@DavidRydh That specific version doesn't work: Take $Y = \mathbb{A}^2$ and $X$ to be $\mathbb{A}^2$ with a doubled origin. The two obvious sections agree on the affine open $U=\{ x \neq 0 \}$ (and on the larger non-affine $\{ (x,y) \neq (0,0) \}$), but $R^1 f_{\ast} \mathcal{O}_X = \mathcal{O}_Y$. In this case, $R^2 f_{\ast} \mathcal{O}_X$ is non-coherent. But I think an easier approach would probably be to use the valuative criterion to find a curve with doubled point in $X$ and use the structure sheaf of that curve.
Oct
8
comment Does this construction yield the surreal numbers?
Or, if you mean to just choose one $p(x)$ for each $(A,B)$, how does your construction know that the root of $x^2-t$ between $1$ and $t$ cubes to the root of $x^2-t^3$ between $t$ and $t^2$?
Oct
8
comment Does this construction yield the surreal numbers?
(2) seems potentially problematic as phrased. Let $F=\mathbb{Q}(t)$, with $t$ larger than every element of $\mathbb{Q}$. Let $A = \{ x : x < N \ \mbox{for some} \ N \in \mathbb{Q} \}$ and $B = F \setminus A$, this is the prototypical wide gap. Let $p(x) = x^2-t$ and $q(x)=x^2- 2t$. Both of these change signs across $(A,B)$. You need to adjoin two roots (call them $x_p$ and $x_q$), not just one. Moreover, your construction needs to know that $1.41421356 < x_q/x_p < 1.41421357$.
Oct
8
comment Unimodular triangulation and Ehrhart polynomials
De Loera, Haws and Koppe have some fascinating conjectures about coefficients of Erhart polynomials of matroid polytopes in the monomial basis. arxiv.org/abs/0710.4346
Oct
8
comment Minimum word length for an unusual set of generators of the symmetric group
The standard name for this problem is "Pancake Sorting" and the Wikipedia article is pretty good en.wikipedia.org/wiki/Pancake_sorting . One fun fact is that this is the subject of Bill Gates only published paper in math cs.berkeley.edu/~christos/papers/…
Oct
8
comment Lower semicontinuity of naive fiber size
You're right, changed.
Oct
8
revised Lower semicontinuity of naive fiber size
edited tags
Oct
8
asked Lower semicontinuity of naive fiber size
Oct
7
comment A quadratic algebra with four generators and four relations
So, a basis is $y_{i_1} y_{i_2} \cdots y_{i_r} x_{j_1} x_{j_2} \cdots x_{j_s}$, for any binary sequences $(i_1, \ldots, i_r)$, $(j_1, \ldots, j_s)$.