Mon, 02 Nov 2015

16:00 - 17:00
C2

The Arithmetic of K3 Surfaces

Christopher Nicholls
(Oxford)
Abstract

The study of rational points on K3 surfaces has recently seen a lot of activity. We discuss how to compute the Picard rank of a K3 surface over a number field, and the implications for the Brauer-Manin obstruction.

Mon, 26 Oct 2015

16:00 - 17:00
C2

Some ideas on rational/integral points on algebraic curves

Junghwan Lim
(Oxford)
Abstract

I will introduce classical results on finiteness theorem with a way of connecting them to idea of covering spaces. I will talk about the proof of FLT under this connection.

Mon, 19 Oct 2015

16:00 - 17:00
C2

Algebraic Automorphic Forms and the Langlands Program

Benjamin Green
(Oxford)
Abstract

In this talk I will define algebraic automorphic forms, first defined by Gross, which are objects that are conjectured to have Galois representations attached to them. I will explain how this fits into the general picture of the Langlands program and, giving some examples, briefly describe one method of proving certain cases of the conjecture. 

Wed, 04 Nov 2015
16:00
C1

Isometries of CAT(0) Spaces

Giles Gardam
(Oxford)
Abstract

This talk will be an easy introduction to some CAT(0) geometry. Among other things, we'll see why centralizers in groups acting geometrically on CAT(0) spaces split (at least virtually). Time permitting, we'll see why having a geometric action on a CAT(0) space is not a quasi-isometry invariant.

 

Wed, 28 Oct 2015
16:00
C1

Word fibers in finite p-groups

Ainhoa Iniguez
(Oxford)
Abstract

 

Let $G$ be a finite group and let $w$ be a word in $k$ variables. We write $P_w(g)$ the probability that a random tuple $(g_1,\ldots,g_k)\in G^{(k)}$ satisfies $w(g_1,\ldots,g_k)=g$. For non-solvable groups, it is shown by Abért that $P_w(1)$ can take arbitrarily small values as $n\rightarrow\infty$. Nikolov and Segal prove that for any finite group, $G$ is solvable if and only if $P_w(1)$ is positively bounded from below as $w$ ranges over all words. And $G$ is nilpotent if and only if $P_w(g)$ is positively bounded from below as $w$ ranges over all words that represent $g$.  Alon Amit conjectured  that in the specific case of finite nilpotent groups and for any word, $P_w(1)\ge 1/|G|$.
 
We can also consider $N_w(g)=|G|^k\cdot P_w(g)$, the number of solutions of $w=g$ in $G^{(k)}$. Note that $N_w$ is a class function. We prove that if $G$ is a finite $p$-group of nilpotency class 2, then $N_w$ is a generalized character. What is more, if $p$ is odd, then $N_w$ is a character and for $2$-groups we can characterize when $N_{x^{2r}}$ is a character. What is more, we prove the conjecture of A. Amit for finite groups of nilpotency class 2. This result was indepently proved by M. Levy. Additionally, we prove that for any word $w$ and any finite $p$-group of class two and exponent $p$, $P_w(g)\ge 1/|G|$ for $g\in G_w$. As far as we know, A. Amit's conjecture is still open for higher nilpotency class groups. For $p$-groups of higher nilpotency class, we find examples of words $w$ for which $N_w$ is no longer a generalized character. What is more, we find examples of non-rational words; i.e there exist finite $p$-groups $G$ and words $w$ for which $g\in G_w$ but $g^{i}\not\in G_w$ for some $(i,p)=1$.

    

  


        

                              

                                                                      

  
          

      
  

                                                            

  
          

      
  

                                                            

  
          

      
  


        

                                                                                                                                                              
  




  






  



      


  
  

  
  
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