Mathematics

Residue field domination in real closed valued fields

Clifton Ealy   [email] (Western Illinois Univ Math)

Abstract: It has been shown by Haskell, Hrushovski, and Macpherson that in an algebraically closed valued field, the residue field and the value group control the rest of the structure in the following sense: Suppose that L and M are valued fields containing a maximal field C, and suppose that the residue field of L (denoted k(L)) is algebraically independent from k(M) over C and likewise the value group of L (denoted $\Gamma(L)$) is independent from the $\Gamma(M)$ over C in the sense of vector spaces. Then tp(L/ M,k(L),$\Gamma(L)$ ) is implied by tp(L/ k(L),$\Gamma(L)$).
      This behaviour is striking, because it is what typically occurs in a stable structure (where types over algebraically closed sets have unique nonforking extensions) but valued fields are far from stable, due to the order on the value group. Real closed valued fields are even farther from stable since the main sort is ordered, and one might expect the analogous theorem about real closed valued fields to be that tp(L/ M,k(L),$\Gamma(L)$) is implied by tp(L/ k(L)$\Gamma(L)$) together with the order type of $L/M$. In fact we show that the order type is unnecessary, that just as in the algebraically closed case one has that tp(L/ M,k(L),$\Gamma(L)$) is implied by tp(L/ k(L)$\Gamma(L)$). This is joint work with Haskell and Marikova.

No meeting this week

Fractal and Local Properties of Parabolic Stochastic Partial Differential Equations

Yimin Xiao   [email] (Michigan State University)

Abstract: Let $(t,x) \mapsto u_t(x)$ denote the solution to the stochastic PDE \[ \partial_t u = - \frac{1}{2} (-\Delta)^{\alpha/2} u_t(x) + \sigma(u_t(x)) \dot{W}_t(x),\quad x\in\mathbb{R}, t>0, \] subject to $u_0(x):= U_0$ for all $x\in \mathbb{R}$, for some positive and finite constant $U_0$. In the above $-(-\Delta)^{\alpha/2}$ denotes the fractional Laplacian of index $\alpha/2, \sigma:\mathbb{R}\to \mathbb{R}$ is non-random and Lipschitz continuous, and $\dot{W}$ denotes space-time white noise on $(0,\infty)\times \mathbb{R}$. We prove a quantitative version of the statement that ``locally, $t \mapsto u_t(x)$ behaves as a conditionally-Gaussian process.'' We then apply that statement in order to derive a number of detailed results about the local behavior of $t \mapsto u_t(x)$, where $x \in \mathbb{R}$ is fixed. Those results include facts such as iterated-logarithm-type behavior, a fractal analysis of the exceptional times to those LILs, and analysis of sample-function variations. This is based on a joint paper with Davar Khoshnevisan, Jason Swason and Liang Zhang.

Syzygies of rings of invariants (cancelled)

Marc Chardin (Jussieu)

$I,F$-partitions of sparse graphs

Sarah Loeb   [email] (UIUC Math)

Abstract: A star $k$-coloring is a proper $k$-coloring such that the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest, where a 2-independent set is a set of vertices having pairwise distance more than 2. Such a partition is called an $I,F$-partition. We use the discharging method and other techniques to prove that every graph with maximum average degree less than $\frac{5}{2}$ has an $I,F$-partition, which is sharp and answers a question of Cranston and West. This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon previous results of Bu, Cranston, Montassier, Raspaud, and Wang.