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If you would like to give a talk, please send the title, abstract and related papers (if available) of your proposed talk to one of the organizers by email.
Organizers: Amir Mohammadi, Brandon Seward, Nattalie Tamam
Title: Density at integer points of an inhomogeneous quadratic form and linear form
Abstract: In 1987, Margulis solved an old conjecture of Oppenheim which states that for a nondegenerate, indefinite and irrational quadratic form Q in n≥3 variables, Q(ℤn) is dense in ℝ. Following this, Dani and Margulis proved the simultaneous density at integer points for a pair consisting of quadratic and linear form in 3 variables when certain conditions are satisfied. We prove an analogue of this for the case of an inhomogeneous quadratic form and a linear form. This is based on joint work with Anish Ghosh.
Title: Factor of IID for the free Ising model on the d-regular tree
Abstract: It is known that there are factors of IID for the free Ising model on the d-regular tree when it has a unique Gibbs measure and not when reconstruction holds (when it is not extremal). We construct a factor of IID for the free Ising model on the d-regular tree in (part of) its intermediate regime, where there is non-uniqueness but still extremality. The construction is via the limit of a system of stochastic differential equations. This is a joint work with Danny Nam and Allan Sly.
Title: On the Mozes-Shah phenomenon for horocycle flows on moduli spaces
Abstract: The Mozes-Shah phenomenon on homogeneous spaces of Lie groups asserts that the space of ergodic measures under the action by subgroups generated by unipotents is closed. A key input to their work is Ratner's fundamental rigidity theorems. Beyond its intrinsic interest, this result has many applications to counting problems in number theory. The problem of counting saddle connections on flat surfaces has motivated the search for analogous phenomena for horocycle flows on moduli spaces of flat structures. In this setting, Eskin, Mirzakhani, and Mohammadi showed that this property is enjoyed by the space of ergodic measures under the action of (the full upper triangular subgroup of) SL(2,ℝ). We will discuss joint work with Jon Chaika and John Smillie showing that this phenomenon fails to hold for the horocycle flow on the stratum of genus two flat surfaces with one cone point. As a corollary, we show that a dense set of horocycle flow orbits are not generic for any measure; in contrast with Ratner's genericity theorem.