Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, February 11, 2014.

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Tuesday, February 11, 2014

1:00 pm in 345 Altgeld Hall,Tuesday, February 11, 2014

A general van der Corput lemma and underlying Ramsey theory

Anush Tserunyan (UIUC Math)

Abstract: Consider a measure-preserving action of a countable group G on a standard probability space and suppose that this action is mixing along a given filter F on G; e.g. mild mixing corresponds to F=IP*, while letting F be the filter of sets of upper density 1 gives weak mixing for amenable G. These notions of mixing imply double recurrence for such systems, and it is a major theme in ergodic Ramsey theory to amplify this to multiple recurrence. This is often done using a so-called van der Corput difference (ratio) lemma, which "drops" the degree of the recurrence, thus enabling proofs by induction. Analogues of this lemma had been proven for various filters, but the existing proofs were different in each case. We prove a van der Corput lemma for a general class of filters, which includes those mentioned above, as well as idempotent ultrafilters. This is based on a new Ramsey theorem for semigroups related to labeling edges between the semigroup elements with their ratios.

1:00 pm in 243 Altgeld Hall,Tuesday, February 11, 2014

Continuity methods, canonical metrics and Gromov-Hausdorff limits

Gabriele La Nave (UIUC Math)

Abstract: We intend to show how Cheeger-Colding theory of Gromov-Hausdorff limits coupled with a continuity method of Monge-Ampere type, can be used to generalize and emulate some parts of the minimal model program on projective varieties to Kaehler manifolds. We will then discuss a version of Hormander technique in this context.

2:00 pm in 241 Altgeld Hall,Tuesday, February 11, 2014

Discussion of relevant open questions

Joseph Rosenblatt (UIUC Math)

Abstract: We will discuss a variety of open questions in ergodic theory and descriptive set theory concerning generic/first category sets of functions and maps.

3:00 pm in 243 Altgeld Hall,Tuesday, February 11, 2014

Mapping stacks and the notion of properness in algebraic geometry

Daniel Halpern-Leistner (Columbia University)

Abstract: One essential feature of a scheme X which is flat and proper over a base scheme S is that for any other finite type S scheme, there is a finite type algebraic space Map(X,Y) parameterizing families of maps from X to Y. There have been several extensions of these results to the setting where X is a proper stack, and Y is a stack satisfying various hypotheses. Unfortunately many of the stacks arising in nature, such as global quotient stacks X/G, have affine stabilizer groups and are about as far as possible from being proper. However, we will show that for many non proper X and a large class of Y, the mapping stack Map(X,Y) is still algebraic and finite type. This leads us to introduce new notions of "projective" and "proper" for morphisms between stacks such that "projective" => "proper", and flat and "proper" => Map(X,Y) is algebraic for reasonable X. We discuss a large list of examples of "projective stacks", including X/G where G is reductive and X is projective-over-affine with H^0(O_X)^G finite dimensional, as well as any quotient stack which admits a projective good moduli space. Based on these, we will come up with an even longer list of "proper" stacks, including stacks which are proper over a scheme in the classical definition. Along the way, we will discuss some surprising "derived h-descent" results in derived algebraic geometry.

3:00 pm in 241 Altgeld Hall,Tuesday, February 11, 2014

Beyond Ohba's Conjecture: On the choice number of k-chromatic n-vertex graphs

Douglas B. West   [email] (UIUC Math)

Abstract: Let $ch(G)$ denote the choice number of an $n$-vertex graph $G$. Noel, Reed, and Wu proved Ohba's conjecture that $ch(G)=\chi(G)$ when $n \leq 2\chi(G)+1$. We extend their result to a general upper bound for all graphs: $ch(G) \leq max\{\chi(G), \lceil (n+\chi(G)-1)/3\rceil \}$. Our bound is sharp for $n \leq 3\chi(G)$ using examples provided by Ohba, and it improves the best-known upper bound for $ch(K_{4,\ldots,4})$. (Joint work with Jonathan Noel, Hehui Wu, and Xuding Zhu.)

4:00 pm in 243 Altgeld Hall,Tuesday, February 11, 2014

Connections between the Geometry of Hyperplane Arrangements and their Combinatorics

Nathan Fieldsteel (UIUC Math)

Abstract: From the data of an arrangement $\mathcal{A}$ of hyperplanes, we can construct two toric varieties. The first is determined by the rational fan $\Sigma(\mathcal{A})$ which has as its maximal cones the sectors of the complement of $\mathcal{A}$. The second is determined by $\Sigma(\mathcal{L}(\mathcal{A}),G)$, a rational fan determined by intersection lattice of the arrangement, together with a choice of building set. This second construction follows the work of Feichtner and Yuzvinsky in which they associate a smooth toric variety to any atomic lattice. We are interested in finding a relationship between these two fans, especially when $\mathcal{A}$ is the arrangement of type $A_n$, $B_n$, or $D_n$.

4:00 pm in 245 Altgeld Hall,Tuesday, February 11, 2014

Dongyang Li, Austin Rochford, and Yat-Sen Wong (UIUC Math)

Abstract: An informal panel discussion with Mathematics graduate students who have done internships both on and off campus. How did they find their internship, what was rewarding about it, and did it change their longer-term career plan? Come and ask what ever is on your mind...