We give a method to bound the entropy of measures on SL(d,R)/SL(d,Z) which are invariant under one parameter diagonal subgroups, in terms of entropy contributions from regions of the cusp corresponding to different parabolic groups. These bounds depend on an auxiliary linear functional on the Lie algebra of the Cartan group, which can then be optimized depending on the bound one is trying to obtain. We discuss how this method is utilized to obtain new sharp bounds for cusp entropies.

 

Mapping class groups of surfaces are of fundamental importance in dynamics, geometric group theory, and
low-dimensional topology.  The word problem for groups in general, the definition of the mapping class group, its finite generation by twists, and the solution to its word problem were all set out by Dehn [1911, 1922, 1938].  Some of this material was rediscovered by Lickorish [1960’s] and then by Thurston [1970-80’s] – they gave
important applications of the mapping class group to the topology and geometry of three-manifolds.  In the past fifty years, various mathematicians (including Penner, Mosher, Hamidi-Tehrani, Dylan Thurston, Dynnikov) have given solutions to the word problem in the mapping class group, using a variety of techniques.  All of these algorithms are quadratic-time.

We give an algorithm requiring only $O(n \log^3(n))$ time.  We do this by combining Dynnikov’s approach to curves on surfaces, M"oller’s version of the half-GCD algorithm, and a delicate error analysis in interval arithmetic.

This is joint work with Mark Bell.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!

The notion of extended geometrically finite representations introduced by Weisman generalizes Anosov representations by studying the convergence dynamics on group boundary extensions. We prove that divergent, extended geometrically finite representations can be interpreted as admitting dominated splitting over certain flow spaces. In particular, we provide an example of such representation constructed by Tholozan–Wang in the study of simple Anosov representations.

We will give a short overview of the Nielsen-Thurston Classification problem (classifying the homeomorphism type of surfaces) on finite-type surfaces and then move to infinite-type surface mentioning the current state and challenges to extending some known arguments. We will then discuss how current work in progress on potential definitions of pseudo-anosov mapping classes in the infinite-type setting (joint with F.Valdez) and how to address some of the previous challenges (joint with Y.Chandran)

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!

A natural problem in the study of local systems on complex varieties is to characterize those that arise in a family of varieties. We refer to such local systems as motivic. While a classification of motivic local systems is evidently out of reach, Simpson conjectured that for a reductive group G, rigid G-local systems with suitable finiteness conditions at infinity are motivic. This was proven for curves when G = GL_n by Katz, who classified such rigid local systems. In this talk, we discuss our generalization of Katz’s theorem to a general reductive group. Our proof goes through the (tamely ramified) categorical geometric Langlands program in characteristic zero.

In 3-dimension topology, the study of foliations, flows and \pi_1-actions on 1-manifolds are closely related. Given a 3-manifold M, one can construct a \pi_1(M)-action on a circle from either a taut foliation or a pseudo-Anosov flow in M by works of Thurston, Calegari-Dunfield and Fenley. When the foliation is depth-one and the pseudo-Anosov flow is transverse to the foliation, we show (with some additional assumptions) that the circle actions from both settings are topologically conjugate. Moreover, the two circles admit extra structures that are compatible in the most natural sense.

Effective exploration is a vital component in the success of robotic applications in ocean and space exploration, environmental monitoring, and search and rescue tasks. This talk presents a novel formulation of exploration that permits optimality criteria and performance guarantees for robotic exploration tasks. We define the problem of exploration as a coverage problem on continuous (infinite-dimensional) spaces based on ergodic theory and derive control methods that satisfy various notions of optimality and robustness such as asymptotic coverage, set-invariance, time-optimality, and reachability in exploration tasks. Last, we demonstrate successful execution of the approach on a range robotic systems and present an outlook on novel directions.

The Langlands program predicts some surprising connections between number theory and representation theory, most of which are still conjectural. In recent years, powerful new tools in non-archimedean geometry have led to dramatic progress on these questions. One fascinating aspect of these new tools is that in a certain precise sense, they allow us to treat a fixed prime number p as a variable, which opens the door to importing ideas from the geometric Langlands program into more arithmetic settings. I will give a historically motivated introduction to this circle of ideas, and give a big-picture view of some of the most exciting recent and ongoing developments.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!