C2

 Voevodsky asked what the topology of the universe is in a 
continuous interpretation of type theory, such as Johnstone's 
topological topos. We can actually give a model-independent answer: it 
is indiscrete. I will briefly introduce "intensional Martin-Loef type 
theory" (MLTT) and formulate and prove this in type theory (as opposed 
to as a meta-theorem about type theory). As an application or corollary, 
I will also deduce an analogue of Rice's Theorem for the universe: the 
universe (the large type of all small types) has no non-trivial 
extensional, decidable properties. Topologically this is the fact that 
it doesn't have any clopens other than the trivial ones.

In 1937 Vinogradov showed that every sufficiently large odd number is the sum of three primes, using bounds on the sums of additive characters taken over the primes. He was improving, rather dramatically, on an earlier result of Schnirelmann, which showed that every sufficiently large integer is the sum of at most 37 000 primes. We discuss a natural analogue of this question in the multiplicative group (Z/pZ)* and find that, although the current unconditional character sum technology is too weak to use Vinogradov's approach, an idea from Schnirelmann's work still proves fruitful. We will use a result of Selberg-Delange, an application of a small sieve, and a few easy ideas from additive combinatorics. 

In this talk I will discuss the notion of o-minimality, which can be approached from either a model-theoretic standpoint, or an algebraic one.  I will exhibit some o-minimal structures, focussing on those most relevant to number theorists, and attempt to explain how o-minimality can be used to attain an assortment of results.

In this talk I will describe Arthur's classification of automorphic representations of symplectic and orthogonal groups using automorphic representations of $\mathrm{GL}_N$.