UIUC Dept. of Mathematics Seminar Calendar

    February 2012            March 2012             April 2012     
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Thursday, March 1, 2012

Number Theory Seminar
11:00 am in 243 Altgeld Hall, Thursday, March 1, 2012

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Submitted by ford.

Xiannan Li (Department of Mathematics, University of Illinois at Urbana-Champaign)
The least prime that does not split in a number field
Abstract: I will begin by describing some classical work on unconditionally bounding the least quadratic non-residue dating back to Vinogradov and Burgess. A generalization of this problem is to bound the least prime that does not split completely in a number field, which was studied by K. Murty and then by Vaaler and Voloch. I will describe two different approaches, one based on zeros of L-functions, and the other on the theory of multiplicative functions, which give the best known bounds here when the degree of the number field is larger than 2.

Group Theory Seminar
1:00 pm in 347 Altgeld Hall, Thursday, March 1, 2012

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Submitted by dsrobins.

Derek Robinson (Department of Mathematics, University of Illinois at Urbana-Champaign)
Groups with few isomorphism types of derived subgroup.
Abstract: A derived subgroup in a group G is the derived (or commutator) subgroup of some subgroup of G. Recently there has been interest in trying to understand the significance of the set of derived subgroups within the lattice of all subgroups of G. In particular one can ask about the effect on the group structure of imposing restrictions on the set of derived subgroups. In this talk we will describe recent work on groups in which there are at most two isomorphism types of derived subgroup. While this may sound like a very special class of groups, it contains groups of many diverse types. We will describe some of these types of group and show how their construction involves some interesting number theoretic problems.

Graduate Geometry and Topology Seminar
2:00 pm in 241 Altgeld Hall, Thursday, March 1, 2012

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Submitted by lukyane2.

Juan Villeta-Garcia (Department of Mathematics, University of Illinois at Urbana-Champaign)
The Harder-Narasimhan stratification of quiver representations
Abstract: A quiver is a finite graph with orientations, and their representations are defined by assigning vector spaces to each vertex and linear maps to each arrow. The theory of quiver representations is incredibly broad, with applications to such areas as quantum physics, Lie theory and invariant theory. We will give a brief overview of the category of quiver representations, and use a construction of Reineke that mimics the Harder-Narasimhan filtration for vector bundles, to analyze this category.

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, March 1, 2012

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Submitted by kapovich.

Daniel Král' (Charles University, Czech Republic)
Testing first order logic properties in sparse combinatorial structures
Abstract: Algorithmic metatheorems guarantee that certain types of problems have efficient algorithms. A classical example is the theorem of Courcelle asserting that every monadic second-order logic (MSOL) property can be tested in linear time for graphs with bounded tree-width. As examples of MSOL properties let us mention 3-colorability, hamiltonicity, etc., all well-known NP-hard problems. In this talk, we focus on simpler properties, those that can be expressed in first order logic (FOL). An example of FOL property is an existence of a fixed substructure. While it is not hard to show that every FOL property can be decided in polynomial time, our desire is to design algorithms with faster running time (e.g. linear time). We recall a recent notion of graph classes with bounded expansion, which include classes of graphs with bounded maximum degree and proper-minor closed classes of graphs. We then apply structural results to show that FOL properties can be tested in linear time for classes of graphs with bounded expansion and we will discuss extensions to other structures. At the end of the talk, we will mention several open problems as well as directions for future research. This talk is based on joint work with Zdenek Dvorák and Robin Thomas.