Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, March 4, 2014.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, March 4, 2014

11:00 am in 243 Altgeld Hall,Tuesday, March 4, 2014

Coarse moduli spaces of stacks over manifolds

Jordan Watts (UIUC math)

Abstract: Consider a Lie group acting properly on a manifold. In the literature, the orbit space of the action has been equipped with various definitions of "smooth structure" for the purpose of extending differential geometry/topology to this space. Examples include differential structures and diffeologies. However, these structures often forget certain invariants induced by the group action. Stacks, on the other hand, encode many of these invariants into the so-called quotient stack. In this talk, I will show how any stack over manifolds has an underlying coarse moduli space equipped with a diffeology which, in the case of a geometric stack, corresponds to the orbit space of a representative Lie groupoid equipped with the quotient diffeology. Moreover, there is a fully faithful functor from diffeological spaces into stacks. This gives us a unifying category in which we can directly compare, in the case of a Lie group action for instance, information encoded by the diffeology versus information encoded by the quotient stack. Time permitting, I will give an example of one such invariant.

1:00 pm in 347 Altgeld Hall,Tuesday, March 4, 2014

Gibbs' measure and almost sure global well-posedness for one dimensional periodic fractional Schr\"odinger equation.

Seckin Demirbas (UIIUC Math)

Abstract: In this talk we will present recent local and global well-posedness results on the one dimensional periodic fractional Schr\"odinger equation. We will also talk about construction of Gibbs' measures on certain Sobolev spaces and how we can prove almost sure global well-posedness using this construction.

2:00 pm in 241 Altgeld Hall,Tuesday, March 4, 2014

2:00 pm in Altgeld Hall 347,Tuesday, March 4, 2014

Harmonic measure for $\Delta+\Delta^{\alpha/2}$ on $C^{1,1}$ open sets and its applications

Hyunchul Park (William and Mary Math)

Abstract: In this talk we investigate the harmonic measure $P_{x}\left(X_{\tau_{D}}\in \cdot\right)$ for (independent) sum of Brownian motions and (rotationally) symmetric $\alpha-$stable processes with the stability index $\alpha\in (0,2)$ for bounded $C^{1,1}$ open sets $D$ in $R^{d}$, $d\geq 3$. Unlike Brownian motions or $\alpha-$stable processes the harmonic measure is supported on $\partial D$ as well as $\bar{D}^{c}$ and we will show that each parts can be represented as an integral against the Martin kernel $M_{D}(x,z)$, $x\in D$, $z\in \partial D$ and the Poisson kernel $P_{D}(x,z)$, $x\in D$, $z\in R^{d}\setminus D$, respectively. This is a joint work with Renming Song (UIUC).

3:00 pm in Altgeld Hall,Tuesday, March 4, 2014

Homomorphism extension problem for Maltsev digraphs is easy

Alexand Kazda   [email] (Vanderbilt Math)

Abstract: A digraph $G$ is Maltsev if the edge relation of $G$ is invariant under a Maltsev operation, which is a mapping $p:V(G)^3\to V(G)$ that satisfies $p(u,u,v)=p(v,u,u)=v$ for all vertices $u$ and $v$. The homomorphism extension problem with the target structure $G$ asks if we can extend a given partial mapping of digraphs $H \to G$ to a homomorphism. If $G$ is a Maltsev digraph, then $G$ has another useful operation called a majority and the homomorphism extension problem with the target $G$ can be solved in logarithmic space. We sketch a proof of this statement, giving a quick introduction to universal algebraic methods of classifying homomorphism extension problem's complexity along the way.

4:00 pm in Altgeld Hall,Tuesday, March 4, 2014

What does a "right" cohomology for rings look like?

Juan S. Villeta-Garcia (UIUC Math)

Abstract: Motivated by the aforementioned question, we introduce Andre-Quillen (co)-homology for commutative algebras using methods of homotopy theory. We connect the theory to the cotangent comples, and prove certain vanishing theorems characterizing classes of maps. We end with some examples in the rational case, and mention a topological characterization.