Study the parabolic Hitchin fibrations for Langlands dual groups. Sketch the proof of a duality theorem of the natural symmetries on their cohomology.
Abstract. In this talk, I will present joint work with Uri Onn, Mark Berman, and Pirita Paajanen.
Let G be a linear algebraic group defined over the integers. Let O be a compact discrete valuation ring with a finite residue field of cardinality q and characteristic p. The group
G(O) has a filtration by congruence subgroups
G_m(O) (which is by definition the kernel of reduction map modulo P^m where P is the maximal ideal of O).
Let c_m=c_m(G(O)) denote the number of conjugacy classes in the finite quotient group G(O)/G_m(O) (which is called the mth congruence quotient of G(O)). The conjugacy class zeta function of
G(O) is defined to be the Dirichlet series Z_{G(O)}(s)=\sum_{m=0,1,...} c_m q^_{-ms}, where s is a complex number with Re(s)>0. This zeta function was defined by du Sautoy when G is a p-adic analytic group and O=Z_p, the ring of p-adic integers, and he proved that in this case it is a rational function in p^{-s}. We consider the question of dependence of this zeta function on p and more generally on the ring O.
We prove that for certain algebraic groups, for all compact discrete valuation rings with finite residue field of cardinality q and sufficiently large residue characteristic p, the conjugacy class zeta function is a rational function in q^{-s} which depends only on q and not on the structure of the ring. Note that this applies also to positive characteristic local fields.
A key in the proof is a transfer principle. Let \psi(x) and f(x) be resp.
definable sets and functions in Denef-Pas language.
For a local field K, consider the local integral Z(K,s)=\int_\psi(K)
|f(x)|^s dx, where | | is norm on K and dx normalized absolute value
giving the integers O of K volume 1. Then there is some constant
c=c(f,\psi) such that for all local fields K of residue characteristic larger than c and residue field of cardinality q, the integral Z(K,s) gives the same rational function in q^{-s} and takes the same value as a complex function of s.
This transfer principle is more general than the specialization to local fields of the special case when there is no additive characters of the motivic transfer principle of Cluckers and Loeser since their result is the case when the integral is zero.
The conjugacy class zeta function is related to the representation zeta function which counts number of irreducible complex representations in each degree (provided there are finitely many or finitely many natural classes) as was shown in the work of Lubotzky and Larsen, and gives information on analytic properties of latter zeta function.
In this talk, I will present joint work with Uri Onn, Mark Berman, and Pirita Paajanen.
Let G be a linear algebraic group defined over the integers. Let O be a compact discrete valuation ring with a finite residue field of cardinality q and characteristic p. The group
G(O) has a filtration by congruence subgroups
G_m(O) (which is by definition the kernel of reduction map modulo P^m where P is the maximal ideal of O).
Let c_m=c_m(G(O)) denote the number of conjugacy classes in the finite quotient group G(O)/G_m(O) (which is called the mth congruence quotient of G(O)). The conjugacy class zeta function of
G(O) is defined to be the Dirichlet series Z_{G(O)}(s)=\sum_{m=0,1,...} c_m q^_{-ms}, where s is a complex number with Re(s)>0. This zeta function was defined by du Sautoy when G is a p-adic analytic group and O=Z_p, the ring of p-adic integers, and he proved that in this case it is a rational function in p^{-s}. We consider the question of dependence of this zeta function on p and more generally on the ring O.
We prove that for certain algebraic groups, for all compact discrete valuation rings with finite residue field of cardinality q and sufficiently large residue characteristic p, the conjugacy class zeta function is a rational function in q^{-s} which depends only on q and not on the structure of the ring. Note that this applies also to positive characteristic local fields.
A key in the proof is a transfer principle. Let \psi(x) and f(x) be resp.
definable sets and functions in Denef-Pas language.
For a local field K, consider the local integral Z(K,s)=\int_\psi(K)
|f(x)|^s dx, where | | is norm on K and dx normalized absolute value
giving the integers O of K volume 1. Then there is some constant
c=c(f,\psi) such that for all local fields K of residue characteristic larger than c and residue field of cardinality q, the integral Z(K,s) gives the same rational function in q^{-s} and takes the same value as a complex function of s.
This transfer principle is more general than the specialization to local fields of the special case when there is no additive characters of the motivic transfer principle of Cluckers and Loeser since their result is the case when the integral is zero.
The conjugacy class zeta function is related to the representation zeta function which counts number of irreducible complex representations in each degree (provided there are finitely many or finitely many natural classes) as was shown in the work of Lubotzky and Larsen, and gives information on analytic properties of latter zeta function.
The Wulff droplet arises by conditioning a spin system in a dominant
phase to have an excess of signs of opposite type. These gather
together to form a droplet, with a macroscopic Wulff profile, a
solution to an isoperimetric problem.
I will discuss recent work proving that the phase boundary that
delimits the signs of opposite type has a characteristic scale, both
at the level of exponents and their logarithmic corrections.
This behaviour is expected to be shared by a broad class of stochastic
interface models in the Kardar-Parisi-Zhang class. Universal
distributions such as Tracy-Widom arise in this class, for example, as
the maximum behaviour of repulsive particle systems. time permitting,
I will explain how probabilistic resampling ideas employed in spin
systems may help to develop a qualitative understanding of the random
mechanisms at work in the KPZ class.
We define the edge reconnecting model, a random multigraph evolving in time. At each time step we change one endpoint of a uniformly chosen edge: the new endpoint is chosen by linear preferential attachment. We consider a sequence of edge reconnecting models where the sequence of initial multigraphs is convergent in a sense which is a natural generalization of the Lovász-Szegedy notion of convergence of dense graph sequences. We investigate how the limit objects evolve under the edge reconnecting dynamics if we rescale time properly: we give the complete characterization of the time evolution of the limiting object from its initial state up to the stationary state using the theory of exchangeable arrays, the Pólya urn model, queuing and diffusion processes. The number of parallel edges and the degrees evolve on different timescales and because of this the model exhibits “aging”.