Abstract: 

It is natural to ask what geometric/dynamical restrictions on discrete subgroups of Lie groups produce restrictions on the isomorphism type of the group. For example, Canary and Tsouvalas showed that certain growth conditions on singular values of group elements give rise to bounds on the cohomological dimension of the group. Farre, Pozzetti  and Viaggi introduced a restriction on subgroups of PSL(d,C) which guaranteed that they must be isomorphic to convex cocompact subgroups of PSL(2,C). 

We introduce a slightly stronger condition which guarantees that the  subgroup of PSL(d,C)  is isomorphic to a uniform lattice in PSL(2,C). If, in addition, the subgroup is strongly irreducible, then we show that it is the image of a uniform lattice in PSL(2,C) by an irreducible representation of PSL(2,C) into PSL(d,C). We may regard this as a global version of a classical local rigidity result of Ragunathan.

This is joint work with Tengren Zhang and Andy Zimmer.

Seminar talk is supported in part by the Mrs. Hepsa Ely Silliman Memorial Fund.

Abstract:

A self-interacting random walk is a random process evolving in an environment which depends on its history. In this talk, we will discuss a few examples of these walks including the Lorentz gas, the mirror walk, the once-reinforced walk and the cyclic walk in the interchange process. I will present methods to analyze these walks in high dimensions and prove that they behave diffusively. The talk is based on joint works with Allan Sly, Felipe Hernandez, Antoine Gloria, Gady Kozma and Lenya Ryzhik.

We have impromptu and (sometimes) scheduled talks, on topics in probability, combinatorics, geometry, and dynamics.

Everyone is welcome! 

Abstract:  The uniqueness of infinity plays a crucial role in understanding solutions to geometric PDEs and the geometric and topological properties of manifolds with Ricci curvature bounds. A major progress is made by Colding—Minicozzi who studied one type of infinity, called the asymptotic cones, of a Ricci flat manifold with Euclidean volume growth and proved the uniqueness of asymptotic cones if one cross section is smooth. Their result generalizes an earlier uniqueness result by Cheeger—Tian which requires integrability of the cross section among other things. 

In this talk I will talk about some recent results on the study of the similar uniqueness problem of another type of infinity called the asymptotic limit spaces for Ricci flat manifold with linear volume growth, following the path of Cheeger—Tian and Colding—Mincozzi for cones. Here one considers translation limits instead of rescaling limits and the limits are no longer cones but cylinders. I will highlight the similarities and differences between the two settings. This is joint work with Zetian Yan.

Complex abelian varieties are compact complex tori that can be embedded as complex subvarieties of the complex projective space. They are among the most fundamental algebraic varieties and their study goes back to Riemann. In the 50’s Hodge introduced a necessary condition for a cohomology class of a smooth algebraic variety to be represented by a linear combination of classes of algebraic subvarieties. He conjectured that the condition is sufficient, i.e., that Hodge classes are algebraic. The Hodge conjecture is known in dimension at most three, but until recently it was not known even for 4-dimensional abelian varieties. Furthermore, Weil identified Hodge classes on 4-dimensional abelien varieties with complex multiplication and proposed them in the late 70’s as test cases for the Hodge conjecture. In the late 90’s it was realized that the algebraicity of the Weil classes would imply the Hodge conjecture for abelian varieties of dimension 4 and 5. We will explain the terms  above and the further developments that were needed for the proof earlier this year that the Weil classes on abelian 4-folds are algebraic.

In 2002, Farb and Mosher introduced the notion of convex cocompactness in the mapping class group to capture coarse geometric information of the associated surface group extensions. Convex cocompact subgroups are necessarily finitely generated and purely pseudo-Anosov, but it is an open question whether the converse is true. Several partial results are known in certain settings, however. For example, work of Dowdall, Kent, Leininger, Russell, and Schleimer give a positive answer for subgroups of fibered 3-manifold groups (aka surface-by-cyclic extensions) naturally embedded in punctured mapping class groups via the Birman exact sequence. We present a generalization of this in the setting of surface-by-abelian extensions.

We discuss recent work that pinpoints the spectral decomposition for the Anderson tight-binding model with an unbounded random potential on the Bethe lattice of sufficiently large degree. We prove that there exist a finite number of mobility edges separating intervals of pure-point spectrum from intervals of absolutely continuous spectrum, confirming a prediction of Abou-Chacra, Thouless, and Anderson. A central component of the proof is a monotonicity result for the leading eigenvalue of a certain transfer operator, which governs the decay rate of fractional moments for the tight-binding model’s off-diagonal resolvent entries. This is joint work with Amol Aggarwal. We will also draw connections to the problem of establishing a mobility edge in the spectrum of heavy-tailed random matrices, which was considered in recent work with Amol Aggarwal and Charles Bordenave.

We have impromptu and (sometimes) scheduled talks, on topics in probability, combinatorics, geometry, and dynamics.

Everyone is welcome! 

Abstract: An important quantity in the study of discrete groups of isometries of Riemannian manifolds, Gromov hyperbolic spaces, and other interesting geometric objects is the critical exponent. For a discrete subgroup of isometries of the quaternionic hyperbolic space or octonionic projective plane, Kevin Corlette established in 1990 that the critical exponent detects whether a discrete subgroup is a lattice or has infinite covolume. Precisely, either the critical exponent equals the volume entropy, in which case the discrete subgroup is a lattice, or the critical exponent is less than the volume entropy by some definite amount, in which case the discrete subgroup has infinite covolume. In 2003, Leuzinger extended this gap theorem for the critical exponent to any discrete subgroup of a Lie group having Kazhdan’s property (T) (for instance, a discrete subgroup of SL(n,R), where n is at least 3).

    In this talk, I will present a result which shows that no such gap phenomenon holds for discrete semigroups of Lie groups. More precisely, for any Zariski dense discrete subgroup of a Lie group, there exist free, finitely generated, Zariski dense subsemigroups whose critical exponents are arbitrarily close to that of the ambient discrete subgroup.

    As an application, we show that the critical exponent is lower semicontinuous in the Chabauty topology whenever the Chabauty limit of a sequence of Zariski dense discrete subgroups is itself a Zariski dense discrete subgroup.

Given a finite-type surface S, it is still unknown exactly which numbers occur as stretch factors of pseudo-Anosov maps on S. It appears there is more flexibility in producing a homeomorphism with a given stretch factor if we allow for the surface to be infinite-type. I’ll discuss current work with Marissa Loving and Chenxi Wu showing every weak Perron number is the stretch factor of some end-periodic homeomorphism on some infinite-type surface.

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We have impromptu and (sometimes) scheduled talks, on topics in probability, combinatorics, geometry, and dynamics.

Everyone is welcome!