UIUC Dept. of Mathematics Seminar Calendar

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Tuesday, October 5, 2010

Topology Seminar
11:00 am in 241 Altgeld Hall, Tuesday, October 5, 2010

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Submitted by mando.

Emily Riehl (University of Chicago)
Algebraic model structures
Abstract: A model structure on a category provides a framework for doing homotopy theory whose main constructive tool consists of four classes of maps - the trivial cofibrations, fibrations, cofibrations, and trivial fibrations - which are defined so as to satisfy certain lifting properties with respect to each other. We'll present an algebraicization of Quillen's model structures that transform these properties into algebraic structure associated to the arrows in each of these classes. More precisely, the trivial cofibrations and fibrations are coalgebras and algebras for a comonad and monad, respectively, arising from the functorial factorization produced by a modified form of Quillen's small object argument. The same is true for the other pair, and there is a natural comparison between these two factorizations. Whenever a model structure is cofibrantly generated, this structural presentation exists and is equivalent to the usual property one, but this new perspective has several interesting features, which we describe. We conclude with a description of an "algebraic Quillen adjunction", which again exists in many familiar situations.

Logic seminar
1:00 pm in Altgeld Hall 345, Tuesday, October 5, 2010

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Submitted by vddries.

Philipp Hieronymi (UIUC Math)
Open definable sets in expansions of topological structures
Abstract: I consider the question of when an expansion of a topological structure has the property that every open set definable in the expansion is definable in the original structure. In joint work with Gareth Boxall, we answered the question in a fairly general setting and provided conditions which in practice are often easy to check. In this talk, I will discuss these results and some applications. On the way, I will introduce several basic notions and results from model theory and o-minimality. This talk will not assume much more than a very basic knowledge of first-order model theory and should hopefully be very understandable for new graduate students (and maybe even senior undergraduates).

Graph Theory and Combinatorics
3:00 pm in 241 Altgeld Hall, Tuesday, October 5, 2010

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Submitted by west.

Seog-Jin Kim (Konkuk University)
Dynamic coloring of sparse graphs
Abstract: A dynamic coloring of a graph G is a proper coloring of V(G) such that each vertex neighborhood of size at least 2 receives at least two distinct colors.� The list dynamic chromatic number�ch_d(G) is the least integer k such that for every assignment of k-element lists to each vertex, there is a dynamic coloring of G with each vertex color chosen from its list. We prove that ch_d(G)≤4 if Mad(G)<8/3, where Mad(G) is�the maximum average degree over subgraphs of G.� Also, ch_d(G)≤4 if G is a planar graph of girth at least 7.� Furthermore, if k≥4, then ch_d(G)≤k if Mad(G)<4k/(k+2).� All these results are sharp and are joint work with Won-Jin Park.