Department of

Mathematics


Seminar Calendar
for events the day of Thursday, December 1, 2016.

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Thursday, December 1, 2016

11:00 am in 241 Altgeld Hall,Thursday, December 1, 2016

Chebyshev's bias for products of $k$ primes

Xianchang Meng (Illinois Math)

Abstract: For any $k\geq 1$, we study the distribution of the difference between the number of integers $n\leq x$ with $\omega(n)=k$ or $\Omega(n)=k$ in two different arithmetic progressions, where $\omega(n)$ is the number of distinct prime factors of $n$ and $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity . Under some reasonable assumptions, we show that, if $k$ is odd, the integers with $\Omega(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Hudson. However, the integers with $\omega(n)=k$ always have preference for quadratic residue classes. Moreover, as $k$ increases, the biases become smaller and smaller for both of the two cases.

12:00 pm in 243 Altgeld Hall,Thursday, December 1, 2016

Higgs bundles, Lagrangians and mirror symmetry.

Lucas Branco ((Oxford University))

Abstract: The moduli space M(G) of G-Higgs bundles on a compact Riemann surface carries a natural hyperkahler structure, for a complex Lie group G. Also, it comes equipped with an algebraically completely integrable system through the Hitchin map. Motivated by mirror symmetry, we will discuss certain complex Lagrangians (BAA-branes) in M(G) coming from real forms of G and give a proposal for the mirror (BBB-brane) in the moduli space of Higgs bundles for the Langlands dual group of G. For the real groups SU^*(2m), SO^*(4m) and Sp(m,m), which are of particular interest as the corresponding Lagrangian never intersect smooth fibres of the Hitchin map, this involves describing the whole fibre which they intersect. We give two different descriptions of such singular fibres and explain how they are related. If time permits we will discuss complex Lagrangians coming from symplectic representations of G and how they relate to other moduli spaces such as the moduli of pairs.

12:30 pm in 3110 ESB (Note room change),Thursday, December 1, 2016

Entanglement, Holography and Causal Diamonds

Michal Haller (Perimeter Physics)

Abstract: We argue that the degrees of freedom in a d-dimensional CFT can be re-organized in an insightful way by studying observables on the moduli space of causal diamonds (or equivalently, the space of pairs of timelike separated points). This 2d-dimensional space naturally captures some of the fundamental nonlocality and causal structure inherent in the entanglement of CFT states. For any primary CFT operator, we construct an observable on this space, which is defined by smearing the associated one-point function over causal diamonds. Known examples of such quantities are the entanglement entropy of vacuum excitations and its higher spin generalizations. We show that in holographic CFTs, these observables are given by suitably defined integrals of dual bulk fields over the corresponding Ryu-Takayanagi minimal surfaces. Furthermore, we explain connections to the operator product expansion and the first law of entanglement entropy from this unifying point of view. We demonstrate that for small perturbations of the vacuum, our observables obey linear two-derivative equations of motion on the space of causal diamonds. In two dimensions, the latter is given by a product of two copies of a two-dimensional de Sitter space. For a class of universal states, we show that the entanglement entropy and its spin-three generalization obey nonlinear equations of motion with local interactions on this moduli space, which can be identified with Liouville and Toda equations, respectively. This suggests the possibility of extending the definition of our new observables beyond the linear level more generally and in such a way that they give rise to new dynamically interacting theories on the moduli space of causal diamonds. Various challenges one has to face in order to implement this idea are discussed.

2:00 pm in 243 Altgeld Hall,Thursday, December 1, 2016

Martin Caspers (Delft )

Abstract: Absence of Cartan subalgebras for right angled Hecke von Neumann algebras Abstract: Hecke algebras are *-algebras generated by self-adjoint operators T_s with s in some generating set that satisfy the Hecke relation (T_s + q^{1/2})(T_s - q^{-1/2}) = 0 as well as suitable types of commutation relations. They generate a von Neumann algebra called the Hecke von Neumann algebra firstly studied by J. Dymara around 10 years ago. We prove that under natural conditions right-angled Hecke von Neumann algebras are non-amenable, have the completely bounded approximation property and in the hyperbolic case are strongly solid. We also prove (partly as a consequence of the previous results) that they do not possess a Cartan subalgebra.

3:00 pm in 243 Altgeld Hall,Thursday, December 1, 2016

Random Toric Surfaces and a Threshold for Smoothness

Jay Yang (University of Wisconsin)

Abstract: I will present a construction of a random toric surface inspired by the construction of a random graph. With this construction we show a threshold result for smoothness of the surface. The hope is that this inspires further application of randomness to Algebraic Geometry and Commutative Algebra.