In-person meetings are held in AP&M 7218 and Zoom meetings are held here. Please send an email to one of the organizers to get the the password.
If you would like to be included or removed from our email announcements, please email Brandon Seward.
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, Anthony Sanchez, Brandon Seward
Spring 2024 Speakers Past Speakers
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.
video
Title: Which Linear Groups have bounded harmonic functions?
Abstract: The
Poisson boundary of a group is a measure space which serves a dual
purpose. From the perspective of random walks it represents the range
of distinct asymptotic possibilities that a random walk on the group
might take. From the perspective of harmonic analysis it classifies the
space of bounded harmonic functions on that group. Understanding the
Poisson boundary of a group is intimately related to the algebraic and
geometric properties of that group. The most basic question one can ask
about the Poisson boundary is whether it is trivial (equivalently
whether there are any non-constant bounded harmonic functions on that
group). In this talk I will survey some core ideas around the Poisson
boundary and then focus on the case of linear groups. In particular I
will give a complete characterization of when linear groups over
positive characteristic fields admit any non-constant bounded harmonic
functions. This is joint work with Anna Erschler.
video
Title: Anti-classification results for the Kakutani equivalence relation
Abstract: Dating
back to the foundational paper by John von Neumann, a fundamental theme
in ergodic theory is the \emph{isomorphism problem} to classify
invertible measure-preserving transformations (MPT's) up to
isomorphism. In a series of papers, Matthew Foreman, Daniel Rudolph and
Benjamin Weiss have shown in a rigorous way that such a classification
is impossible. Besides isomorphism, Kakutani equivalence is the best
known and most natural equivalence relation on ergodic MPT's for which
the classification problem can be considered. In joint work with
Marlies Gerber we prove that the Kakutani equivalence relation of
ergodic MPT's is not a Borel set. This shows in a precise way that the
problem of classifying such transformations up to Kakutani equivalence
is also intractable.
video
Title: Expanding measures: Random walks and rigidity on homogeneous spaces
Abstract: We
will start by reviewing recent developments in random walks on
homogeneous spaces. In a second part, we will discuss the notion of a
H-expanding probability measure on a connected semisimple Lie group H.
As we shall see, for a H-expanding μ with H < G, on the one hand,
one can obtain a description of μ-stationary probability measures on
the homogeneous space G/Λ (G Lie group, Λ lattice) using the measure
classification results of Eskin-Lindenstrauss, and on the other hand,
the recurrence techniques of Benoist-Quint and
Eskin-Mirzakhani-Mohammadi can be adapted to this setting. With some
further work, these allow us to deduce equidistribution and orbit
closure description results simultaneously for a class of subgroups
which contains Zariski-dense subgroups and further epimorphic subgroups
of H. If time allows, we will see how, utilizing an idea of
Simmons-Weiss, these also allow us to deduce Birkhoff genericity of a
class of fractal measures with respect to certain diagonal flows,
which, in turn, has applications in diophantine approximation problems.
Joint work with Roland Prohaska and Ronggang Shi.
video
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.
video
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(ℤ^{n})
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.
video
Title: Translational tilings in lattices
Abstract: Let F be a finite subset of ℤ^{d}. We say that F is a translational tile of ℤ^{d} if it is possible to cover ℤ^{d} by translates of F without any overlaps. The periodic tiling conjecture, which is perhaps the most well-known conjecture in the area, suggests that any translational tile admits at least one periodic tiling. In the talk, we will motivate and discuss the study of this conjecture. We will also present some new results, joint with Terence Tao, on the structure of translational tilings in lattices and introduce some applications.
Title: A backward ergodic theorem and its forward implications
Abstract: In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation T, one takes averages of a given integrable function over the intervals {x, T(x), T^{2}(x),..., T^{n}(x)} in front of the point x. We prove a "backward" ergodic theorem for a countable-to-one pmp T, where the averages are taken over subtrees of the graph of T that are rooted at x and lie behind x (in the direction of T^{-1}). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank, where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. This strengthens Bufetov's theorem from 2000, which was the most general result in this vein. This is joint work with Anush Tserunyan.
Title: Equidistribution of expanding translates of lines in SL_{3}(ℝ)/SL_{3}(ℤ)
Abstract: Let X=SL_{3}(ℝ)/SL_{3}(ℤ) and a(t)=diag(t^{2},t^{-1},t^{-1}). The expanding horospherical group U^{+} is isomorphic to ℝ^{2}. A result of Shah tells us that the a(t)-translates of a non-degenerate real-analytic curve in a (U^{+})-orbit get equidistributed in X. It remains to study degenerate curves, i.e. planar lines y=ax+b. In this talk, we give a Diophantine condition on the parameter (a,b) which serves as a necessary and sufficient condition for equidistribution. Joint work with Kleinbock, Saxcé and Shah. If time permits, I will also talk about generalisations to SL_{n}(ℝ)/SL_{n}(ℤ). Joint work with Shah.
Title: Point processes on groups, their cost, and fixed price for G x Z
Abstract: Invariant point processes on groups are a rich class of probability measure preserving (pmp) actions. In fact, every essentially free pmp action of a nondiscrete locally compact second countable group is isomorphic to a point process. The cost of a point process is a numerical invariant that, informally speaking, measures how hard it is to "connect up" the point process. This notion has been very profitably studied for discrete groups, but little is known for nondiscrete groups. This talk will not assume any sophisticated knowledge of probability theory. I will define point processes, their cost, and discuss why every point process on groups of the form G x Z has cost one. Joint work with Miklós Abért.
Title: Topological dynamics beyond Polish groups
Abstract: When G is a Polish group, one way of knowing that it has nice dynamics is to show that M(G), the universal minimal flow of G, is metrizable. For non-Polish groups, this is not the relevant dividing line: the universal minimal flow of the symmetric group of a set of cardinality κ is the space of linear orders on κ-not a metrizable space, but still nice, for example. In this talk, we present a set of equivalent properties of topological groups which characterize having nice dynamics. We show that the class of groups satisfying such properties is closed under some topological operations and use this to compute the universal minimal flows of some concrete groups, like Homeo(ω_{1}). This is joint work with Andy Zucker.
Title: About Borel and almost Borel embeddings for ZD actions
Abstract: Krieger's generator theorem says that all free ergodic measure preserving actions (under natural entropy constraints) can be modelled by a full shift. Recently, in a sequence of two papers Mike Hochman noticed that this theorem can be strengthened: He showed that all free homeomorphisms of a Polish space (under entropy constraints) can be Borel embedded into the full shift. In this talk we will discuss some results along this line from a recent paper with Tom Meyerovitch and ongoing work with Spencer Unger.
Title: Flexibility of the Pressure Function
Abstract: Our settings are one-dimensional compact symbolic systems. We discuss the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics since they correspond to qualitative changes of the characteristics of a dynamical system referred to as phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We show that these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter. This is based on joint work with Anthony Quas.
Title: Rigidity and flexibility phenomenons in isometric orbit equivalence
Abstract: TBAIn an ongoing work, we introduce the notion of isometric orbit equivalence for probability measure preserving actions of marked groups. This notion asks the Schreier graphings defined by the actions of the marked groups to be isomorphic. In the first part of the talk, we will prove that pmp actions of a marked group whose Cayley graph has a discrete automorphisms group are rigid up to isometric orbit equivalence. In a second time, we will explain how to construct pmp actions of the free group that are isometric orbit equivalent but not conjugate.
Title: Marked groups with isomorphic Cayley graphs but different Descriptive combinatorics.
Abstract: We discuss the relationship between the Borel/Baire measurable/measurable combinatorics of the action of a finitely generated group on its Bernoulli shift and the discrete combinatorics of the multiplication action of that group on itself. Our focus is on various chromatic numbers of graphs generated by these actions. We show that marked groups with isomorphic Cayley graphs can have Borel/Baire measurable/measurable chromatic numbers which differ by arbitrarily much. In the Borel two-ended, Baire measurable, and measurable hyperfinite settings, we show our constructions are nearly best possible (up to only a single additional color), and we discuss prospects for improving our constructions in the general Borel setting. Along the way, we will get tightness of some bounds of Conley and Miller on Baire measurable and measurable chromatic numbers of locally finite Borel graphs.
Title: Invariant measures for horospherical actions and Anosov groups.
Abstract: Let Γ be an Anosov subgroup of a connected semisimple real linear Lie group G. For a maximal horospherical subgroup N of G, we show that the space of all non-trivial NM-invariant ergodic and A-quasi-invariant Radon measures on Γ \ G, up to proportionality, is homeomorphic to ℝ^{rank G-1}, where A is a maximal real split torus and M is a maximal compact subgroup which normalizes N. This is joint work with Hee Oh.
Title: A decomposition for measure-preserving near-actions of ergodic full groups
Abstract: Given
a measure-preserving action of a countable group on a standard
probability space, one associates to it a full group which by Dye's
reconstruction theorem completely remembers the associated equivalence
relation whose classes are the action's orbits. A natural question is
then to understand how exactly this full group encodes the properties
of the associated (measure-preserving) equivalence relation. In this
talk, we will see that all non-free ergodic near-actions of the full
group actually come from measure-preserving actions of the equivalence
relation (or of its symmetric powers), paralleling a recent result of
Matte-Bon concerning actions by homeomorphisms of topological full
groups. If time permits, we will explain how this can be used to show
that a measure-preserving ergodic equivalence relation has property (T)
if and only if all the non-free ergodic near-actions of its full group
are strongly ergodic. This talk is based on an ongoing joint work with
Alessandro Carderi and Alice Giraud.
Title: Equidistribution of affine random walks on some nilmanifolds
Abstract: We consider the action of the group of affine transformations on nilmanifolds. Given a probability measure on this group and a starting point x, a random walk on the nilmanifold is defined. Consider the distribution of the point after m random steps. We show that under certain assumptions, that hold for Heisenberg nilmanifolds for example, the distribution of this point converges to the Haar measure on the nilmanifold as m goes to infinity, unless there is the obvious obstruction that the orbit closure of x by the semigroup generated by the support of the random walk measure is a finite homogeneous union of affine sub-nilmanifolds. Furthermore, this result is quantitative and gives a rate for the convergence to Haar measure (equidistribution) depending on how close the starting point and random walk measure are to such an obstruction. This talk is based on joint works with Weikun He and Elon Lindenstrauss.
Title: Multiscale substitution tilings
Abstract: Multiscale substitution tilings are a new family of tilings of Euclidean space that are generated by multiscale substitution rules. Unlike the standard setup of substitution tilings, which is a basic object of study within the aperiodic order community and includes examples such as the Penrose and the pinwheel tilings, multiple distinct scaling constants are allowed, and the defining process of inflation and subdivision is a continuous one. Under a certain irrationality assumption on the scaling constants, this construction gives rise to a new class of tilings, tiling spaces and tiling dynamical systems, which are intrinsically different from those that arise in the standard setup. In the talk I will describe these new objects and discuss various structural, geometrical, statistical and dynamical results. Based on joint work with Yaar Solomon.
Title: Stationary actions of higher rank lattices on non-commutative spaces
Abstract: I will present new results about stationary actions of higher rank semi-simple lattices on compact spaces, in the spirit of Nevo and Zimmer's work. Then I will explain how these results generalize to stationary actions on C*-algebras (i.e. "non-commutative" spaces) and give consequences about unitary representations of these lattices and their characters. All these results can be seen as generalizations of Margulis normal subgroup theorem at different levels. This is based on joint works with Cyril Houdayer, Uri Bader and Jesse Peterson.
Title: Sofic entropy and the (relative) f-invariant
Abstract: In this talk I will explain an interpretation (due to Lewis Bowen) of the f-invariant as a variant of sofic entropy: it is the exponential growth rate of the expected number of “good models” for an action over a random sofic approximation. I will then introduce the relative f-invariant and provide a similar interpretation of this quantity. This provides a formula for the growth rate of the expected number of good models over a type of stochastic block model.
Title: An introduction to the f-invariant
Abstract: The f-invariant was introduced by Lewis Bowen in 2008 and is a real-valued isomorphism invariant that is defined for a large class of probability measure-preserving actions of finite-rank free groups. Most notably, the f-invariant provided the first classification up to isomorphism of Bernoulli shifts over finite-rank free groups. It is also quite useful for the study of finite state Markov chains with values indexed by a finite-rank free group. The f-invariant is conceptually similar to entropy, and it has a formal connection to sofic entropy. In this expository talk, I will introduce the f-invariant and discuss some of its basic properties.
Title: A signature for some subgroups of the permutation group of [0,1[.
Abstract: For every infinite set X we define S(X) as the group of all permutations of X. On its subgroup consisting of all finitely supported permutations there exists a natural group homomorphism signature. However, thanks to an observation of Vitali in 1915, we know that this group homomorphism does not extend to S(X). In the talk we extend the signature on the subgroup of S(X) consisting of all piecewise isometric elements (strongly related to the Interval Exchange Transformation group). This allows us to list all of its normal subgroups and gives also informations about an element of the second cohomology group of some slides
Title: Effective equidistribution of horospherical flows in infinite volume
Abstract: By Ratner's famous equidistribution theorem, we know that unipotent orbits in finite volume quotients of Lie groups equidistribute in their closures. Often, in applications, one needs to know more: specifically, at what rate does the orbit equidistribute? We call a statement that includes a quantitative error term effective. In this talk, I will present an effective equidistribution theorem, with polynomial rate, for horospherical orbits in the frame bundle of certain infinite volume hyperbolic manifolds. This is joint work with Nattalie Tamam.
Title: Gaps of saddle connection directions for some branched covers of tori
Abstract: Holonomy vectors of translation surfaces provide a geometric generalization for higher genus surfaces of (primitive) integer lattice points. The counting and distribution properties of holonomy vectors on translation surfaces have been studied extensively. A natural question to ask is: How random are the holonomy vectors of a translation surface? We motivate the gap distribution of slopes of holonomy vectors as a measure of randomness and compute the gap distribution for the class of translation surfaces given by gluing two identical tori along a slit. No prior background on translation surfaces or gap distributions will be assumed.
Title: Finitary isomorphisms of Poisson point processes
Abstract: As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary. This is joint work with Terry Soo.
Title: Spaces of cut and project quasicrystals: classification and statistics
Abstract: Cut and project sets are well-studied models of almost-periodic discrete subsets of ℝ^{d}. In 2014 Marklof and Strombergsson introduced a natural class of random processes which generate cut and project sets in a way which is invariant under the group ASL(d,ℝ). Using Ratner’s theorem and the theory of algebraic groups we classify all these measures. Using the classification we obtain results analogous to those of Siegel, Rogers, and Schmidt in geometry of numbers: summation formulas and counting points in large sets for typical cut and project sets. Joint work with Rene Ruehr and Yotam Smilansky.
Title: A variational principle in the parametric geometry of numbers
Abstract: We describe an ongoing program to resolve certain problems at the interface of Diophantine approximation and homogenous dynamics. Highlights include computing the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, thereby resolving a conjecture of Kadyrov-Kleinbock-Lindenstrauss-Margulis (2014) as well as answering a question of Bugeaud-Cheung-Chevallier (2016). As a corollary of the Dani correspondence principle, this implies that the set of divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions. Other applications include dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions. This is joint work with David Simmons, Lior Fishman, and Mariusz Urbanski. The reduction of various problems to questions about certain combinatorial objects that we call templates along with a variant of Schmidt's game allows us to answer some of these problems, while leaving plenty that remain open. The talk will be accessible to students and faculty interested in some convex combination of homogeneous dynamics, Diophantine approximation and geometric measure theory.