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Probability, Geometry, and algorithms  ,[object Object]
Igor Rivin ,[object Object]
How do we make a simple object ,[object Object],[object Object],[object Object]
Why is linear good? ,[object Object],hard
Why is continuous good? ,[object Object],[object Object]
Abelian is good ,[object Object]
Now, apply the above to groups: ,[object Object],[object Object],[object Object]
How easy is  ,[object Object],is known as harmonic analysis
Obviously too easy. ,[object Object],[object Object]
The Prime Number Theorem ,[object Object],[object Object]
(Some confusion) ,[object Object]
Another continuous, linear, abelian group ,[object Object],[object Object]
Analogue of prime number theorem? ,[object Object],[object Object],[object Object]
The study of  Z n ,[object Object]
Now, let’s remove commutativity ,[object Object]
What is a lattice? ,[object Object],[object Object]
Geometry of lattices: examples ,[object Object]
Here is an example
Here is another
Everyone’s favorite lattice ,[object Object]
More generally ,[object Object]
More favorites ,[object Object],[object Object]
Why “principal”? ,[object Object]
(Hard) exercises ,[object Object],[object Object],[object Object],[object Object]
Congruence subgroup property (CSP) ,[object Object]
Is a random matrix in SL(n,  Z )   irreducible? ,[object Object],[object Object]
Second question first... ,[object Object]
First question is harder: ,[object Object],[object Object]
[object Object],[object Object]
[object Object],[object Object],Luckily, in both cases the answer is the same
[object Object],[object Object]
Experimental results
Generating sets ,[object Object],[object Object]
How are results like this proved? ,[object Object],[object Object]
To get convergence rates ,[object Object],[object Object],[object Object]
Property T, continued ,[object Object]
Going even further from continuous ,[object Object],[object Object]
Is a group Zariski dense? ,[object Object],[object Object]
But ,[object Object]
Unlike for lattices ,[object Object],[object Object],[object Object]
Why undecidable? ,[object Object],[object Object]

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