UIUC Dept. of Mathematics Seminar Calendar

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                        30 31

Thursday, October 27, 2011

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, October 27, 2011

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Submitted by kapovich.

Ben McReynolds (Purdue University)
Controlling symmetries
Abstract: It is not uncommon in inverse and rigidity problems to encounter unexpected symmetries. In this talk, I will discuss a particular problem that highlights this issue and introduce a method for dealing with these symmetries. Specifically, I will discuss the inverse problem for isometry groups of closed hyperbolic 2-manifolds; given a finite group Q, can we realize Q as the isometry group of a closed hyperbolic 2-manifold? The answer is yes and due to Greenberg. I will discuss a method for dealing with this and similar questions using group actions on spaces of representations.

Number Theory
1:00 pm in 241 Altgeld Hall, Thursday, October 27, 2011

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Xiannan Li (Department of Mathematics, University of Illinois)
On the size of L-functions at the edge of the critical strip
Abstract: I will talk about finding upper bounds on L(1) where L(s) is an L-function. The value of an L-function at 1 has been an object of great historical interest. For instance, the value of the classical Dirichlet L-functions at 1 is linked to the class number of quadratic fields. With the conception of the Langland's program and the conjectures therein, there is now a much larger class of L-functions which may be studied. Finding upper bounds for these L-functions at 1 presents new obstacles and yields many interesting applications. I will first describe some examples and applications to motivate the discussion and present a new upper bound. This work improves and generalizes previous results of Iwaniec, Molteni, and Brumley. Time permitting, I will briefly describe some recent work of Blomer which uses this result.

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, October 27, 2011

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Eugeniu Spinu (University of Alberta, Edmonton)
Domination problems on some Banach spaces with a normal generating real cone
Abstract: Let $X$ and $Y$ be Banach spaces with normal generating real cones $K$ and $L$, respectively. We say that $T:X \to Y$ is positive if $T(K) \subseteq L$ and $T \le S$ if $(S-T)$ is positive. We will consider the following question. Let $0\le T \le S :X \to Y$ and $S$ is either (weakly) compact, or strictly singular operator. Is $T$ (weakly) compact or strictly singular, respectively? We will pay special attention to the case when $X$ and $Y$ are Banach lattices and also discuss the case when one of the spaces is a $C^{*}$-algebra or a Schatten class.

CR Geometry Seminar
3:00 pm in 345 Altgeld Hall, Thursday, October 27, 2011

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Submitted by tumanov.

Ilya Kossovskiy (University of Western Ontario)
Mappings of 2-nondegenerate hypersurfaces in dimension three
Abstract: Let (M,p) and (M',p') be two real hypersurfaces with distinguished points in complex affine n-space and let H(M,p;M',p') be the space of local biholomorphic mappings of the ambient space preserving the hypersurfaces and the distinguished points. How "rich" can the space H(M,p;M',p') be? Poincare (for n=2) and later Chern and Moser (for arbitrary n) in their famous papers gave the answer to this question for Levi non-degenerate hypersurfaces. Their results generated a big stream of further papers on CR-geometry and led to remarkable theorems in complex analysis. Using a new approach, we avoid the difficulties which occur in the Levi-degenerate case and reproduce the Poincare-Chern-Moser theory for the case of 2-nondegenerate hypersurfaces in complex 3-space. This is a joint work with Valery Beloshapka.

Mathematics Colloquium
4:00 pm in Altgeld Hall 245, Thursday, October 27, 2011

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Wilhelm Schlag (University of Chicago)
Invariant manifolds and Hamiltonian evolution equations
Abstract: The field of nonlinear dispersive evolutions equations has undergone rapid changes in recent years. The equations in question are Hamiltonian and encompass a wide class ranging from nonlinear Klein-Gordon, wave, and Schroedinger equations on the one hand, to more geometric equations such as wave maps and other so-called 'field equations' of physics on the other hand. These equations have traditionally been studied from the point of view of the fundamental well-posedness problem locally in time which often requires large amounts of analytical machinery. The question of global in time properties of the evolution is the subject of much ongoing research in nonlinear evolution equations. Within the past five to six years, several open problems have been settled in the field by introducing new ideas such as concentration-compactness for evolution equations, and the use of invariant manifolds from hyperbolic dynamics. We shall give an overview over some of these developments.