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**Local Langlands correspondence**

*Pure Mathematics Seminar*

**by **Masoud Kamgarpour

**Institution: **

**Date: **07/09/2016 1:00pm

**Abstract: **

**Local class Field Theory**

*Pure Mathematics Seminar*

**by **Owen Colman

**Institution: **

**Date: **31/08/2016 1:00pm

**Abstract: **

**Geometric Langlands**

*Pure Mathematics Seminar*

**by **Kari Vilonen

**Institution: **University of Melbourne

**Date: **24/08/2016 1:00pm

**Abstract: **

**Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing nonlinearity**

*Pure Mathematics Seminar*

**by **Florica Cîrstea

**Institution: **School of Mathematics and Statistics, The University of Sydney

**Date: **19/08/2016 3:15pm

**Abstract: **Let \(B_1(0)\) denote the open unit ball in \( \mathbb{R}^n \) with \(n \geq 3\) and \( B^{*}:=B_{1}(0)\setminus\{0\}\).
In this talk, we classify all the singular profiles near zero of the positive solutions \( u \in C^{\infty}(B^{*})\) of nonlinear elliptic equations such as
\begin{equation}
-\Delta u=\frac{u^{2^\star(s)-1}}{|x|^s}-\mu \, u^q\hbox{ in }B^*,
\end{equation}
where \(q> 1\), \(\mu>0\) and \( s \in (0,2)\) are fixed. Here, \( 2^\star(s):=\frac{2(n-s)}{n-2}\) is critical for Sobolev embeddings. When \( \mu=0 \) and \(s=0 \), the profile at the singularity \( 0 \) was fully described by Caffarelli--Gidas--Spruck. We prove that when \( \mu>0 \) and \(s>0\), besides this profile, two new profiles might occur. Special attention is given to solutions satisfying \(\liminf_{|x|\to 0} |x|^{\frac{n-2}{2}} u(x)=0\) and \(\limsup_{|x|\to 0} |x|^{\frac{n-2}{2}} u(x)
\in (0,\infty)\). We treat the special case \(q=\frac{n+2}{n-2}\) separately.
This is joint work with Frédéric Robert (University of Lorraine).

**Special values of zeta functions**

*Pure Mathematics Seminar*

**by **Thomas Geisser

**Institution: **Rikkyo University Tokyo

**Date: **12/08/2016 3:15pm

**Abstract: **

**K-theory of toric surfaces over any field**

*Pure Mathematics Seminar*

**by **Fei Xie

**Institution: **UCLA

**Date: **11/08/2016 3:15pm

**Abstract: **Merkurjev and Panin studied the K-theory of toric varieties over any field in their K-motivic category (or K-correspondences). They showed in this category that a smooth projective toric variety X is a direct summand of some separable algebra and X is isomorphic to one if and only if after base change to separable closure of the base field, K_0(X_sep) is a Galois permutation module. But whether K_0(X_sep) is always a permutation module remains unknown. I confirmed the condition for toric surfaces and constructed the separable algebra they are isomorphic to. In this talk, I will give a background on the motivic category and toric varieties and discuss the case for toric surfaces.

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