Actions of Lattice Subgroups of Higher-Rank Lie Groups
Vanderbilt University
20 Feb 2026

Harmonic Functions on Groups
Vanderbilt University
3 Dec 2025

Implicit in the work of Mok (1995) and Kleiner (2010) is the statement that a finitely generated group is infinite if and only if there exists a nonconstant harmonic (with respect to the counting measure on a generating set) function on the group. I will present (sort of) recent work generalizing this to locally compact groups under a more general class of measures: such a group is noncompact if and only if it admits a nonconstant Lipschitz-bounded harmonic function (for any/every reasonable probability measure). The proof involves an improvement of Mok/Kleiner's energy argument. I will also present how this improvement makes progress towards the Margulis-Zimmer conjecture on commensurates subgroups of lattices in higher-rank semisimple groups.

Instructor for Math 1301 Accelerated Single-Variable Calculus II (Spring 2026) at Vanderbilt University.

Instructor for Math 3310 Mathematical Logic (Spring 2026) at Vanderbilt University.

Instructor for Math 8997 Graduate Independent Study: Ergodic Theory (Fall 2025) at Vanderbilt University.

Instructor for Math 1301 Accelerated Single-Variable Calculus II (Fall 2025) at Vanderbilt University.

Instructor for Math 1300 Accelerated Single-Variable Calculus I (Fall 2025) at Vanderbilt University.

Harmonic Functions on Compactly Generated Groups
Vanderbilt University
12 Sep 2025

Implicit in the work of Mok (1995) and Kleiner (2010) is the statement that a finitely generated group is infinite if and only if there exists a nonconstant harmonic function on the group for any (equivalently every) reasonable probability measure on the group. I will present (sort of) recent work generalizing this to locally compact second countable compactly generated groups: such a group is noncompact if and only if it admits a nonconstant Lipschitz-bounded harmonic function (for any/every reasonable probability measure). The proof involves Margulis' technique of amenability+(T) as well as Mok/Kleiner's energy argument.

Subshifts of Non-Superlinear Word Complexity cannot be Mixing
Vanderbilt University
17 Jan 2025

I will present the companion result to my proof of the existence of mixing subshifts with word complexity arbitrarily close to linear, namely that any subshift which does not have superlinear word complexity must be partially rigid hence cannot be (strongly) mixing. More specifically, if p(q) is the number of words of length q appearing in a subshift and \liminf p(q)/q < infty then there exists a constant 0 < c \leq 1 such that every invariant probability measure \mu for the subshift, there is measure-theoretic partial rigidity of at least c: for all measurable sets B, \limsup \mu(T^{n} B \cap B) \geq c \mu(B). The tools involved range from graph theory and combinatorics to classical ergodic theory.

Instructor for Math 1301 Accelerated Single Variable Calculus II (Spring 2025) at Vanderbilt University.

Symbolic Dynamics: Connecting short-term and long-term complexity in dynamical systems
Vanderbilt University Colloquium
17 Oct 2024

Given a dynamical system and a (reasonable) partition of the space, there is a natural map from the system to a symbolic system: assign each element of the partition a distinct 'letter', look at the orbit of a given point and 'read off the infinite word' by writing the letter of the element of the partition at each time. Symbolic dynamics is the study of subshifts -- closed, shift-invariant subspaces of \mathcal{A}^{\mathbb{Z}} where \mathcal{A} is the 'alphabet' -- exactly what one obtains from a dynamical system and a partition. To study the complexity of a subshift (and by extension the complexity of the underlying system it arose from), there are both quantitative and qualitative notions. Quantitatively, there is the complexity function p(q) = the number of distinct words of length q appearing in any of the infinite words in the subshift. Qualitatively, there are various notions of mixing and asymptotic independence. I will present my work (some joint with R. Pavlov and S. Rodock) on word complexity cutoffs for various qualititative mixing properties and (briefly) explore some of the consequences for 'low complexity' systems.

Word Complexity of Partially Mixing Symbolic Dynamical Systems
Vanderbilt University
25 Oct 2024

A subshift is a closed shift-invariant subset of \mathcal{A}^{\mathbb{Z}} where \mathcal{A} is some finite set, the 'alphabet'. Endowing a subshift with a probability measure, there are then natural questions about how the asymptotic mixing properties of the system relate to quantitative aspects such as the number of distinct words p(q) of a given length q appearing anywhere in the subshift. In joint work with R. Pavlov, we established that weak mixing can manifest in systems with \lim p(q)/q = 1.5 but that any system with \limsup p(q)/q < 1.5 is isomorphic to a rotation on a compact abelian (adelic) group. Relatedly, I established that strong mixing can manifest in systems with p(q) < qf(q) for arbitrary f(q) \to \infty but that \liminf p(q)/q < \infty precludes strong mixing. I will present recent work, joint with Terry Adams, on the word complexity behavior of partially mixing systems. Specifically, we show the existence of partially mixing systems with \liminf p(q)/q = 2 and establish that, for rank-one systems, this is optimal.

Instructor for Math 1300 Accelerated Single Variable Calculus I (Fall 2024) at Vanderbilt University.

Actions of Lattices in Semisimple Lie Groups
University of Denver
1 Mar 2024

A lattice is a subgroup of a locally compact group which is discrete (in the ambient group's topology) and has finite covolume (Haar measure). Lattices in semisimple Lie groups arise naturally in fields ranging from number theory to theoretical physics to dynamical systems. Margulis' celebrated Normal Subgroup Theorem states that every normal subgroup of a lattice in a (center-free) semisimple Lie group is of finite index, e.g. that such lattices are 'almost' simple. I will present work, some joint with J. Peterson (Vanderbilt), generalizing Margulis' theorem to actions: every minimal continuous action of such a lattice on an infinite compact metric space is free (meaning the stabilizer subgroup of every point is trivial).

TBA
University of Canterbury NZ
TBA

Adelic Structure of Low Complexity Subshifts
Vanderbilt University
22 Mar 2024

A subshift is a closed shift-invariant subset of $\{ 0, 1 \}^{\mathbb{Z}}$ or more generally $\mathcal{A}^{\mathbb{Z}}$ for some finite set $\mathcal{A}$, the alphabet. I will present a somewhat surprising result, joint with R. Pavlov, that subshifts of very low complexity are all measurably isomorphic to rotations on one-dimensional adelic compact groups. Consequences of this include that Sarnak's Conjecture holds for all low complexity subshifts and the resolution of an open question of Ferenczi about the minimal complexity for weak mixing.

Adelic Subshifts
Pennsylvania State University
29 Mar 2024

Mixing Properties of Subshifts and Word Complexity Cutoffs
University of Denver Colloquium
8 Mar 2024

In the realm of dynamical systems, there are both qualitative and quantitative notions of `complexity'. Qualitatively we may say a system is, for example, mixing or chaotic, and quantitatively we may speak of, for instance, its entropy. It has long been known that mixing behavior can manifest with zero entropy but one still expects there should be some means of relating these qualitative notions to a quantitative measurement. Subshifts are a natural class of dynamical systems possessing a much finer-grained quantitative measure of complexity than entropy known as word complexity. Answering multiple open questions due to Ferenczi in the 90s, I will present recent work, some joint with Ronnie Pavlov, on the exact word complexity cutoffs at which strong mixing and weak mixing (sometimes referred to as chaotic behavior) can manifest.

Word Complexity Cutoffs for Mixing Properties of Subshifts
University of California: San Diego
8 Feb 2024

Actions of Lattices in Semisimple Lie Groups
Pennsylvania State University
8 Dec 2023

Co-Organizer, Workshop on Low Complexity Dynamical Systems
Brin Mathematics Research Institute at the University of Maryland
2023

Seminar Lead, Operator Algebras and Dynamics Seminar
U.S. Naval Academy
2018–2023

Co-Organizer, Shanks Workshop: von Neumann Algebras and Ergodic Theory
Vanderbilt University
2013

Word Complexity Cutoffs for Mixing Properties of Subshifts
Pennsylvania State University
12 Dec 2023

Subshift Word Complexity and Mixing Properties
University of Denver
14 Nov 2023

Workshop on Low Complexity Dynamical Systems (co-organizer)
University of Maryland
2-6 Oct 2023

Multiple notions of low complexity exist for dynamical systems: low structural complexity, e.g. symbol substitution systems and interval exchange maps; low dynamical complexity, e.g. (partial) rigidity, countable (or finite) ergodic measures, loosely Bernoulli propertt, lack of weak mixing and discrete spectrum; and low word complexity subshifts.

Recent work, by various authors, established that (sub)linear word complexity implies low complexity in a variety of other senses, e.g. being substitutive of finite alphabet rank, having at most countably many ergodic measures, necessarily being partially rigid, and so on.

Two major conjectures relate forms of low complexity: the S-adic conjecture, which asserts that there is an explicit relationship between (sub)linear word complexity and a substitutive structure, and the Pisot conjecture, which asserts that, in the context of substitution systems, discrete spectrum is equivalent, roughly, to a specific form of algebraic substitutive structure (and presumably these are also implied, in some sense, by a word complexity property).

The goal of the workshop is to bring together experts in various different areas of low complexity systems research to survey the recent progress, aiming to enhance cross-subfield collaboration and connect the various research programs, ideally leading to potential progress on the aforementioned and other longstanding conjectures.

Low Complexity Systems
U.S. Naval Academy
18 Apr 2023

Instructor for SM221 Calculus III (Spring 2023) at U.S. Naval Academy.

Word Complexity Cutoffs for Mixing Properties of Subshifts
University of Maryland
13 Oct 2022

Word Complexity Cutoffs for Mixing Properties of Subshifts
U.S. Naval Academy
3 Oct 2022

Word Complexity of Rank-One Subshifts
University of Denver
26 May 2022

Instructor for SM221 Calculus III (Fall 2022) at U.S. Naval Academy.

Mixing Subshifts with Low Word Complexity
Univeristy of Virginia
11 May 2022

(joint work with Ronnie Pavlov and Shaun Rodock) It is well-known that high dynamical complexity--(measure-theoretic) strong mixing--can coexist with low information complexity--zero entropy. Within the class of zero entropy transformations, subshifts (closed shift invariant subsets of { 0, 1 \}^Z) have the more fine-grained notion of word complexity. We show that strong mixing can coexist with very low word complexity, specifically that for any F : N --> N such that Sum 1/f(q) < infinity, there exists a strongly mixing rank-one transformation whose associated subshift has word complexity p such that p(q) = o(f(q)).

Low Complexity Subshifts admitting Mixing Measures
AMS Special Session on Zero-Dimensional Dynamics, University of Denver (Virtual)
14 May 2022

We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations, and prove they are (strongly) mixing. For any f : N to N with f(n)/n increasing and Sum 1/f(n) < infinity, we prove there exists an extremely elevated staircase transformation with word complexity p satisfying p(q) = o(f(q)). This improves the previous best known lower bound for mixing subshifts, resolving a conjecture of Ferenczi. This is joint work with Ronnie Pavlov and Shaun Rodock.

Low Complexity Subshifts
U.S. Naval Academy
11 Apr 2022

Instructor for SM411 Complex Variables (Spring 2022) at U.S. Naval Academy.

Instructor for SM221 Calculus III (Spring 2022) at U.S. Naval Academy.

Ergodic Theory of Group Actions
Darren Creutz
Lecture Notes

Continuous Actions of Lattices (in Higher-Rank Semisimple Groups) are Free
U.S. Naval Academy
14 Oct 2021

Let \Gamma be an irreducible lattice in a higher-rank semisimple group G. Generalizing Margulis' Normal Subgroup theorem in the connected case, Stuck and Zimmer (1994) showed that every nonatomic probability-preserving action of \Gamma in a connected higher-rank Lie group G is essentially free. The general case was proven by the speaker and J. Peterson (2017). However, being (very) nonamenable, not every action of such a lattice on a compact metric space admits an invariant measure, leaving open the question of whether every minimal action of such a lattice on an infinite compact metric space is (topologically) free. I will present the complete result: every action of such a lattice is indeed (topologically) free. The main idea is to replace invariant measure by stationary measure (which always exist) and show that every stationary action is essentially free. This answers the general form of a question of Glasner and Weiss on URS's for such lattices.

Continuous Actions of Lattices (in Higher-Rank Semisimple Groups) are Free
University of Maryland
30 Sep 2021

Let \Gamma be an irreducible lattice in a higher-rank semisimple group G. Generalizing Margulis' Normal Subgroup theorem in the connected case, Stuck and Zimmer (1994) showed that every nonatomic probability-preserving action of \Gamma in a connected higher-rank Lie group G is essentially free. The general case was proven by the speaker and J. Peterson (2017). However, being (very) nonamenable, not every action of such a lattice on a compact metric space admits an invariant measure, leaving open the question of whether every minimal action of such a lattice on an infinite compact metric space is (topologically) free. I will present the complete result: every action of such a lattice is indeed (topologically) free. The main idea is to replace invariant measure by stationary measure (which always exist) and show that every stationary action is essentially free. This answers the general form of a question of Glasner and Weiss on URS's for such lattices.

Ergodic Theory: The Mathematics of Experiment, Probability and Time
Colloquium, U.S. Naval Academy
13 Nov 2019

An overview of the field of ergodic theory, originally developed to systematize the ideas underlying the scientific method of repeated experiment and observation, focusing on the interplay of probability and time. The presentation will assume very little in the way of background and everyone is encouraged to attend. Rudimentary knowledge of probability and some familiarity with matrices is all that is necessary. Towards the end, I will present (in a high-level form) some of my own results on the behavior of matrices in the probabilistic setting, specifically on the nature of discretizing matrix actions (as is necessary and commonly done when utilizing computers).

Complexity from Simplicity: Dimensions, Fractals and Chaos
U.S. Naval Academy (online)
23 Feb 2021

Picture a (equilateral) triangle. Choose a point at random inside the triangle. Now pick a corner of the triangle at random and move halfway from your point toward that corner. Repeat: pick a corner at random and move halfway to it. Repeat this indefinitely. Where can we end up? Can it be anywhere inside the triangle? (The answer may surprise you; it certainly got my attention). By a series of examples, I will present the sorts of simple systems that give rise to complicated behavior, leading us to conceptual questions about dimensionality, chaos, and the oft-mentioned but not always well-explained fractals. The talk will be *very* accessible, assuming minimal mathematical knowledge (e.g. if you know what complex numbers that would be good but not necessary).

Mixing of Stochastic Staircase Transformations
U.S. Naval Academy
21 Oct 2019

Mixing of Stochastic Staircase Transformations
U.S. Naval Academy
30 Sep 2019

Fractals and Dimension
Annapolis Civitan Club
15 Aug 2019

Harmonicity and energy of cocycles: (T) without ultraproducts nor conditionally negative type
University of Virginia
5 Apr 2019

I will present a new characterization of Kaxhdan's Property (T) in terms of the absence harmonic cocyles, providing the natural analogue for (T) to Furstenberg-Kaimanavich-Vershik characterizing amenability as the absence of bounded harmonic functions. This approach gives new, vastly simplified, proofs of many of Shalom's results, including that the reduced cohomology of a product of groups splits as the product of the reduced cohomologies. An immediate corollary of our work, one apparently overlooked in the literature, is: let G be a locally compact second countable group; then TFAE: (1) there exists an admissible probability measure on G such that every at most Lipschitz harmonic function is constant; (2) the same holds for every admissible measure; and (3) G is compact. Time permitting, I will also explain how this approach makes progress on a well-known long-standing conjecture of Margulis and Zimmer about commensurated subgroups of lattices in Lie groups.

Mathematical Existence: A Philosophical Discussion
U.S. Naval Academy
16 Apr 2019

“Everyone knows” that ZFC is the foundation of modern mathematics, yet virtually no one actually knows what those axioms are (except choice of course which is the one most often questioned, albeit erroneously). The reality of the situation is more nuanced: we all know what a proof is in that we recognize it when we see it. But do we really all agree? The issue centers on the meaning of "there exists" and the various interpretations that can be given. I will present an overview of the various schools of mathematical philosophy, and their accompanying notions of proof, ranging from the classical logic ZFC approach to constructivism/intuitionism (it only exists if you can construct it) to the bizarre ultrafinitism (there is a largest number) over to game formalism (proofs are all a game of symbol pushing). Holding some not-so-standard views myself, my intent is for the “talk” to be much more a guided group discussion rather than a traditional talk. All are welcome, no background (other than having done some mathematics) will be assumed.

Rigidity for actions and characters of lattices (in products of Lie groups)
Colloquium, University of Virginia
4 Apr 2019

Let Gamma be a lattice in G, a (product of) simple center-free Lie groups, e.g. SL_3(Z) in SL_3(R) or SL_3(Z[1/p]) in SL_3(R) cross SL_3(Q_p). Margulis' Normal Subgroup Theorem states that if N is a normal subgroup of Gamma then N has finite index in Gamma, i.e. that Gamma is "just infinite". This "rigidity" that Gamma inherits some of the simplicity of G. An invariant random subgroup (IRS) of Gamma is a conjugation-invariant probability measure on subgroups of Gamma, a natural generalization of normal subgroup. I will present joint work with J. Peterson showing that Gamma admits only finite index IRS's, equivalently that every measure-preserving action of Gamma on a nonatomic probability space is essentially free, another rigidity phenomena of Gamma inheriting simplicity of actions from G. A character on a group is a conjugation-invariant positive definite function tau on Gamma with tau(e) = 1. Generalizing our result on actions, I will discuss our work that Gamma admits only the regular character tau = delta_e and characters coming from finite-dimensional representations.

How discontinuous can an integrable function be?
U.S. Naval Academy
22 Jan 2019

An intro to measure theory as the natural extension of calculus. One of the few problems Riemann could not solve was a question about his own integrals: if f(x) is a bounded function on [a,b], when does the integral from a to b of f(x) dx exist (when are the lower and upper sums equal)? If f is continuous then the integral exists, and indeed f can have jump discontinuities. But if f is discontinuous at every point on a subinterval then the integral won't exist, so where is the line? Can an integrable function have a infinitely many discontinuities? Could it be discontinuous on something as large as a Cantor set? I will answer these questions by presenting Lebesgue's notion of measurable sets, developed, initially, precisely to answer Riemann's question. No background beyond calculus will be assumed. If you haven't seen measure theory before, or it's just been a very long time, and you'd be interested in knowing what us analysts are doing, please join.

Relative Entropy and an Entropy Gap for Some Groups Without (T)
Purdue University
15 Feb 2019

Furstenberg introduced entropy for stationary (not necessarily measure-preserving) actions of groups which Kaimanavich and Vershik used in a crucial way to show that a group is amenable iff it admits a trivial Poisson Boundary. Nevo showed that for groups with (T) there is a strong opposite condition: every stationary of a (T) group is either measure-preserving or has entropy at least some positive constant (independent of the action). I will present a generalization of Furstenberg entropy to maps between stationary spaces and from this, show that certain product groups not having (T) also have an entropy gap for their actions.

Completeness, Incompleteness, Consistency and Independence: Gödel's revolutionizing of logic
U.S. Naval Academy
11 Oct 2018

Most mathematicians have heard of Gödel's incompleteness theorem but few can say what it says beyond the imprecise and somewhat misleading "there are true but unprovable statements". What does that actually mean? What is "true" here as it is evidently different than "provable"? The answer to this lies in the far more important completeness theorem, also due to Gödel, which is at the heart of the modern study of logic. We will discuss the completeness theorem at a very high level, assuming no background, and its connection to the proper formulation of the incompleteness theorem as well as how these ideas lead to proofs about consistency and independence, specifically the famous theorems of Gödel and Cohen that (1) both the axiom of choice and its negation are consistent with ZF and (2) that the continuum hypothesis is independent of ZFC. The overarching theme will be the contributions of Gödel to the field of logic and how his work laid out the path of virtually all of modern set theory.

Amenability of Stationary Actions of Lattices
University of Virginia
30 Mar 2018

Amenability of Stationary Actions of Lattices
Vanderbilt University
6 Apr 2018

A (potential) topological/C*-algebra approach to rigidity for lattices (joint w/ M. Kalantar)
U.S. Naval Academy
5 Mar 2018

Stationary Actions of Lattices
Brazos Analysis Seminar, University of Houston
4 Nov 2017

Instructor for SM221 Calculus III (Fall 2021) at U.S. Naval Academy.

Instructor for SM221 Calculus III (split classroom hybrid) (Spring 2021) at U.S. Naval Academy.

Harmonic Functions on Compactly Generated Groups
Darren Creutz
(in review)

Stabilizers of Stationary Actions of Lattices in Semisimple Groups
Darren Creutz
(in review)

Superstability and Finite Time Extinction for C₀-Semigroups
Darren Creutz and Manuel Mazo, Jr.

Instructor for SM122 Calculus II (online) (Fall 2020) at U.S. Naval Academy.

Instructor for SM221 Calculus III (online) (Summer 2020) at U.S. Naval Academy.

Instructor for SM221 Calculus III (Spring 2020) at U.S. Naval Academy.

Instructor for SM331H Real Analysis (Honors) (Fall 2019) at U.S. Naval Academy.

Instructor for SM122 Calculus II (Fall 2019) at U.S. Naval Academy.

Instructor for SM221 Calculus III (Spring 2019) at U.S. Naval Academy.

Instructor for SM122 Calculus II (Fall 2018) at U.S. Naval Academy.

Instructor for SM122 Calculus II (Spring 2018) at U.S. Naval Academy.

Relativized Furstenberg Entropy and an Information Theory of Joinings
U.S. Naval Academy
16 Oct 2017

The Furstenberg entropy of a nonsingular action of a group on a probability space (a G-space) is a numerical measure of how far the action is from being measure-preserving. This entropy has found many applications due to its close connection to the Poisson boundary; in particular, Kaimanavich and Vershik proved a Shannon-McMillan-Breiman theorem for actions centered on this concept. In this talk, I will present a generalization of the Furstenberg to entropy to maps between G-spaces. This generalization enjoys all of the properties one would expect from an entropy and forms the basis for an information-theoretic approach to joinings of G-spaces. In particular, I introduce concepts of mutual and conditional information which satisfy inequalities corresponding to their classical counterparts. Some applications to groups with property (T) will be discussed (and some conjectures), time permitting.

Instructor for SM121 Calculus I (Fall 2017) at U.S. Naval Academy.

Set Theory Past ZFC: Large Cardinals, Forcing, and Independence
U.S. Naval Academy
3 April 2017

Most everyone "knows" that math is built on the foundation of Zermelo-Frankl (ZF) set theory, but most of us don't know about the axioms that go beyond ZF(C). Godel's proof that the Axiom of Choice is consistent with ZF, followed by Cohen's proof that its negation is also consistent with ZF, was the first indication that there is more going on than was first thought. The same result (by the same people) showing that the Continuum Hypothesis is independent of ZFC led to a foundational crisis in mathematical logic. I will give a high-level overview of these results and then explain the various additional axioms that have been proposed (many of which are now taken as "standard" by set theorists) to resolve these issues. Time permitting (or in a follow-up talk) I will discuss how this relates to Godel's Incompleteness Theorem and model theory.

The Normal Subgroup Theorem for Lattices in Products (Property (T))
U.S. Naval Academy
6 Feb 2017

Margulis' Normal Subgroup Theorem states that if Gamma is an irreducible lattice in a higher-rank semisimple Lie group with trivial center then every nontrivial normal subgroup of Gamma has finite index. Moving away from Lie groups, Bader and Shalom proved that the same result holds for lattices in products of arbitrary simple nondiscrete locally compact groups. I will present joint work with Y. Shalom which gives a new proof of this result and explain how it leads into my work on a conjecture of Margulis and Zimmer about the nature of commensurated subgroups of lattices. The proof is in two distinct halves (as was Margulis'): we prove Gamma / N is finite by showing it is both amenable and has Kazhdan's property (T). The first discussed the amenability proof; this talk will present the (T) half. In particular, this talk will be a standalone talk and will not assume knowledge of what was talked about in part one.

The Normal Subgroup Theorem for Lattices in Products (Amenability)
U.S. Naval Academy
23 Jan 2017

Margulis' Normal Subgroup Theorem states that if Gamma is an irreducible lattice in a higher-rank semisimple Lie group with trivial center then every nontrivial normal subgroup of Gamma has finite index. Moving away from Lie groups, Bader and Shalom proved that the same result holds for lattices in products of arbitrary simple nondiscrete locally compact groups. I will present joint work with Y. Shalom which gives a new proof of this result and explain how it leads to my work with J. Peterson that every ergodic action of an irreducible lattice in a product of higher-rank semisimple groups on a nonatomic probability space is essentially free. The proof is in two distinct halves (as was Margulis'): we prove Gamma / N is finite by showing it is both amenable and has Kazhdan's property (T). Part one will discuss the amenability proof; part two (to be scheduled) will discuss (T). Note: part two will not rely on part one; each talk will be stand-alone.

Instructor for SM212 Differential Equations (Spring 2017) at U.S. Naval Academy.

Random Walks and Harmonic Functions on Groups
U.S. Naval Academy
7 November 2016

A natural question in geometric group theory is to study the random walk of a finitely generated group. Specifically, for a probability distribution mu on a finite generating set S, one considers the behavior of the random walk on the Cayley graph built from S with law mu (meaning at each step in the walk, we choose which edge in S to follow according to mu). In particular, one considers the exit boundary of the walk--the space of all distinct paths to infinity. Another natural question is to study the space of bounded mu-harmonic functions on G: functions f : G --> Reals such that for each g in G, Sum_{s in S} f(gs) mu(s) = f(g). The classical Dirichlet problem establishes a correspondence between bounded harmonic functions on SL_2 (the fractional linear transformations) and bounded measurable functions on the unit circle. I will present Furstenberg's Poisson Boundary construction which establishes that random walks on groups and harmonic functions are both determined by the bounded measurable functions on the ``boundary" of the random walk. In particular, the bounded harmonic functions are in one-one correspondence with the bounded measurable functions on the boundary. A concrete example of this is the free nonabelian group on two generators F_2: the Cayley graph (for the usual generating set) is the regular 4-tree and the natural weighing is to give all 4 directions equal weight; the boundary here is the ``big circle", the boundary of the 4-tree, and the harmonic functions on F_2 are in one-one correspondence with L^infinity of the big circle.

The Test #2 will be on 28 October 2016 at 13:30-14:20.

The Test #3 will be on 2 November 2016 at 14:30-15:20.

The Final Exam will be on 14 December 2016 at 13:30-16:30.

The Final Exam will be on 15 December 2016 at 13:30-16:30.

Ergodic Actions of Lattices in Higher-Rank Semisimple Groups
University of Maryland
6 October 2016

Lattices in higher-rank semisimple groups arise naturally in many areas of mathematics, and include groups such as SL_n[Z] for n >= 3. These groups exhibit a variety of rigidity properties, most notably the results of Margulis--the Normal Subgroup Theorem that every nontrivial normal subgroup of an irreducible lattice in a center-free higher-rank semisimple group has finite index and the Superrigidity Theorem that every isomorphism of such a lattice into any algebraic group either has precompact image or extends to the ambient semisimple group. I will present work of myself and J. Peterson generalizing both of these theorems. The main focus of the talk will be on our theorem that every ergodic action of such a lattice on a nonatomic probability space is essentially free (taking the action to be the Bernoulli shift on the lattice modulo a normal subgroup recovers the NST); the proof of which involves a careful understanding of the dynamics of the Poisson boundary and of Howe-Moore groups. I will also present (largely without proof) our operator-algebraic superrigidity theorem that any representation of such a lattice as unitary operators on a finite von Neumann algebra is either finite-dimensional (hence coming from a quotient by a finite index normal subgroup) or extends to the entire group von Neumann algebra of the lattice.

Mixing and Rank-One Transformations
U.S. Naval Academy
3 October 2016

Ergodic theory (the classical theory) is the study of transformations on probability spaces. This talk will introduce the basic notions of the theory: ergodicity and various forms of mixing; then introduce a class of transformations constructed by an intuitive process of "cutting and stacking". These transformations (rank-one transformations) have been studied since the 1940s as a means to understand the mixing notions. The talk will present some of the main results beginning with Chacon's proof of weak mixing not implying strong mixing and Ornstein's proof of the existence of zero entropy transformations with no square root which are strong mixing and conclude with the presenter's work (partly joint with C. Silva) on constructing explicit examples of such transformations.

The Test #4 will be on 6 December 2016 at 14:30-15:20.

The Test #2 will be on 4 October 2016 at 14:30-15:20.

The Test #1 will be on 20 September 2016 at 14:30-15:20.

Character Rigidity for Lattices in Lie Groups
U.S. Naval Academy
26 August 2016

Characters on groups (positive definite conjugation-invariant functions) arise naturally both from probability-preserving actions (the measure of the set of fixed points) and unitary representations on finite factors (the trace); the classical theory of characters is the first step in the classification of finite simple groups and culminates in the Peter-Weyl theorem for compact groups. I will present the results of J. Peterson and myself that the only characters on lattices in semisimple groups are the left-regular character and the classical characters. This is in actuality operator-algebraic superrigidity for lattices, answering a question of Connes. The main idea is to bring dynamics into the operator-algebraic picture; the second half of the talk will focus on the ergodic-theoretic ideas of contractiveness and the Poisson boundary and how these ideas lead to operator-algebraic results.

The Test #3 will be on 5 December 2016 at 13:30-14:20.

The Test #1 will be on 21 September 2016 at 13:30-14:20.

Instructor for SM362 Modern Algebra (Fall 2016) at U.S. Naval Academy.

Instructor for SM221P Calculus III with Vector Fields (Fall 2016) at U.S. Naval Academy.

The Final Exam (Take-Home) will be on Due 5 May 2016 at 5:00pm in  .

The Midterm Exam #2 (Take-Home) will be on Due 15 April 2016 at 10:10am in  .

The Midterm Exam #1 (Take-Home) will be on Due 19 February 2016 at 10:10am in  .

The Midterm Exam #2 will be on 15 April 2016 at 10:10am-11:00am in SC 1310.

The Midterm Exam #1 will be on 19 February 2016 at 10:10am-11:00am in SC 1310.

Rigidty Theory of Lattices in Semisimple Groups
U.S. Naval Academy
3 Feb 2016

The Information Theory of Joinings
Vanderbilt University
22 January 2016

I will present ongoing research into an area I am developing based on the idea of treating joinings of quasi-invariant actions of groups on probability spaces along similar lines are treating random variables as representing information, in particular I consider the ``mutual information" of two spaces in terms of their joinings. Furstenberg entropy is a numerical measure of how far a quasi-invariant action of a group on a probability space is from measure-preserving. The main new tool I introduce is a relative version of this entropy measuring how far a homomorphism between such spaces is from being relatively measure-preserving. I show that it enjoys the properties one would expect such as additivity over compositions and apply this notion to develop an “information theory” of joinings proving analogues of many of the key theorems in the information theory of random variables.

Instructor for Math 3641 Mathematical Theory of Statistics (Spring 2016) at Vanderbilt University.

The Final Exam will be on 14 December 2015 at 7:00pm-9:00pm in Wilson Hall 103.

The Exam #4 will be on 3 December 2015 at 7:00pm-8:15pm in Wilson Hall 103.

The Exam #3 will be on 5 November 2015 at 7:00pm-8:15pm in Wilson Hall 103.

The Exam #2 will be on 8 October 2015 at 7:00pm-8:15pm in Wilson Hall 103.

The Exam #1 will be on 17 September 2015 at 7:00pm-8:15pm in Wilson Hall 103.

Stabilizers of Actions of Lattices in Products of Groups
Darren Creutz
Ergodic Theory and Dynamical Systems

The Final Exam will be on 18 December 2015 at 9:00am-11:00am in SC 1206.

The Exam #2 will be on 9 November 2015 at 9:10am-10:00am in SC 1206.

The Exam #1 will be on 28 September 2015 at 9:10am-10:00am in SC 1206.

Contractive Spaces and Relatively Contractive Maps
Darren Creutz
AMS Contemporary Mathematics

Instructor for Math 1200 Single-Variable Calculus I (Fall 2015) at Vanderbilt University.

Instructor for Math 1010 Probability and Statistical Inference (Fall 2015) at Vanderbilt University.

The Final Exam will be on 7 August 2015.

The Exam #2 will be on 31 July 2015.

The Exam #1 will be on 17 July 2015.

Instructor for Math 216 Probability and Statistics for Engineering (Summer 2015) at Vanderbilt University.

Co-Organizer, Special Session: Classification Problems in Operator Algebras
AMS Joint Mathematics Meetings
11 January 2015

Harmonic Maps on Groups and Property (T)
Noncommutative Geometry and Operator Algebras Spring Institute
6 May 2015

Furstenberg's boundary theory allows us to characterize amenability in terms of the absence of bounded harmonic functions on the group. Building on joint work with Y. Shalom, I will present a similar method for characterizing Kazhdan's Property (T) in terms of the absence of certain harmonic maps on the group. Together, these results give some insight into a potential unified proof of Margulis' Normal Subgroup Theorem (and other Normal Subgroup Theorems).

Co-Organizer, AMS Special Session on Classification Problems in Operator Algebras
Joint Mathematics Meetings
2015

The Final Exam will be on 24 April 2015 at 3:00pm-5:00pm in SC 1120.

The Final Exam will be on 28 April 2015 at 3:00pm-5:00pm in SC 1320.

The Exam #2 will be on 25 March 2015 at 9:10am-10:00am in SC 1320.

The Exam #1 will be on 11 February 2015 at 9:10am-10:00am in SC 1320.

Instructor for Math 127B Probability and Statistical Inference (Spring 2015) at Vanderbilt University.

The Exam #2 will be on 3 April 2015 at 11:10am-Noon in SC 1120.

The Exam #1 will be on 20 February 2015 at 11:10am-Noon in SC 1120.

Instructor for Math 196 Differential Equations with Linear Alegbra (Spring 2015) at Vanderbilt University.

The Final Exam will be on 11 December 2014 at 3:00pm-5:00pm in SC 1206.

The Exam #2 will be on 3 November 2014 at 9:10am-10:00am in SC 1206.

The Exam #1 will be on 22 September 2014 at 9:10am-10:00am in SC 1206.

The Final Exam will be on 12 December 2014 at 3:00pm-5:00pm in SC 1320.

The Exam #2 will be on 14 November 2014 at 10:10am-11:00am in SC 1320.

The Exam #1 will be on 3 October 2014 at 10:10am-11:00am in SC 1320.

Instructor for Math 196 Differential Equations with Linear Alegbra (Fall 2014) at Vanderbilt University.

Instructor for Math 127A Probability and Statistical Inference (Fall 2014) at Vanderbilt University.

Instructor for Math 150A Single Variable Calculus (Summer 2014) at Vanderbilt University.

A Normal Subgroup Theorem for Commensurators of Lattices
Darren Creutz and Yehuda Shalom
Groups, Geometry and Dynamics

Instructor for Math 394 Ergodic Theory of Group Actions (Spring 2014) at Vanderbilt University.

Operator Algebraic Superrigidity for Lattices and Commensurators
Northwestern University
3 Dec 2013

Rigidity for Characters on Lattices and Commensurators
Vanderbilt University
30 Oct 2013

Characters on groups (positive definite conjugation-invariant functions) arise naturally both from probability-preserving actions (the measure of the set of fixed points) and unitary representations on finite factors (the trace). I will present joint work with J. Peterson showing the nonexistence of nontrivial characters for irreducible lattices in semisimple groups and for their commensurators. Consequently, any finite factor representation of such a group generates either the left regular representation or a finite-dimensional representation, answering a question of Connes and generalizing our result that every nonatomic probability-preserving action of such a group is essentially free. The key new idea is to use the contractive nature of the Poisson boundary to bring it into the operator algebraic setting and along with it the rigidity behavior of lattices in their ambient groups.

Character Rigidity for Lattices and Commensurators
Darren Creutz and Jesse Peterson
(in review)

Operator-Algebraic Superrigidity for Lattices
AMS Special Session on Classification Problems in Operator Algebras, Baltimore Maryland
15 Jan 2014

I will present an overview of my recent work, both joint with J. Peterson and solo, classifying the possible actions of lattices in semisimple groups, and more generally, products of groups with the Howe-Moore property. The main result is that, provided at least one simple factor in the ambient group has property (T) (is of higher-rank), every ergodic probability-preserving action of such a lattice on a nonatomic space is essentially free. I will also explain more recent work, joint with J. Peterson, on the rigidity for characters on such lattices, the noncommutative analogue of the statement on actions.

Character Rigidity for Lattices and Commensurators
Vanderbilt University
27 Sep 2013

Characters on groups (positive definite conjugation-invariant functions) arise naturally both from probability-preserving actions (the measure of the set of fixed points) and unitary representations on finite factors (the trace). I will present joint work with J. Peterson showing the nonexistence of nontrivial characters for irreducible lattices in semisimple groups and for their commensurators. Consequently, any finite factor representation of such a group generates either the left regular representation or a finite-dimensional representation, generalizing our earlier result that every nonatomic probability-preserving action of such groups is essentially free. The key new idea is to use the contractive nature of the Poisson boundary to bring it in operator algebraic setting and along with it the rigidity behavior of lattices in their ambient groups.

Mixing on Stochastic Staircase Transformations
Darren Creutz
Studia Mathematica

The Final Exam will be on 12 Dec 2013 at 9:00am-11:00am.

The Exam #3 will be on 15 Nov 2013 at 10:10am-11:00am in SC 1210.

The Exam #2 will be on 18 Oct 2013 at 10:10am-11:00am in SC 1210.

The Exam #1 will be on 20 Sep 2013 at 10:10am-11:00am in SC 1210.

Instructor for Math 175 Multivariable Calculus (Fall 2013) at Vanderbilt University.

Instructor for Math 155A Single-Variable Calculus (Summer 2013) at Vanderbilt University.

Stabilizers of Actions of Groups and Invariant Random Subgroups
Vanderbilt University
26 Apr 2013

As an introduction to the upcoming Shanks workshop on von Neumann Algebras and Ergodic Theory, I will introduce the basic notions involved with invariant random subgroups. Actions of groups give rise to invariant random subgroups via the stabilizer map; I will show how to construct an action that gives a prescribed invariant random subgroup as its stabilizers. Then I will discuss notions such as subgroups of random subgroups (due to myself and J. Peterson) and quotienting out by random subgroups.

Stabilizers of Ergodic Actions of Product Groups and Lattices in Products
Shanks Workshop on von Neumann Algebras and Ergodic Theory, Vanderbilt University
28 Apr 2013

The Margulis Normal Subgroup Theorem states that any normal subgroup of an irreducible lattice in a center-free higher-rank semisimple Lie group is of finite index. Stuck and Zimmer, expanding on Margulis' approach, showed that any properly ergodic probability-preserving ergodic action of a semisimple real Lie group with every simple factor of higher-rank is essentially free and likewise for lattices in such groups. Bader and Shalom, following a different approach, showed that any properly ergodic action of a product of two simple groups with property (T) is essentially free, but their methods do not yield information about lattices.

I will present recent work expanding on the approach of Bader and Shalom generalizing the results of Stuck and Zimmer and of Bader and Shalom to the case when only one factor has (T) and obtaining a classification statement for actions of lattices in products of simple Howe-Moore groups.

Stabilizers of Actions of Product Groups and Lattices in Product Groups
Vanderbilt University
5 April 2013

I will present my recent work on the stabilizers of actions of products of groups and irreducible lattices in products. The main results are a classification of all possible stabilizer groups for actions of products of Howe-Moore groups, at least one of which has (T), and a classification statement for actions of lattices in such products. In contrast to previous work (joint with J. Peterson) on stabilizers, the approach taken here does not involve writing lattices as commensurators and therefore applies even in the case when neither of the ambient groups are totally disconnected and in this sense complement the previous work.

Mixing on Rank-One Transformations
Vanderbilt University
25 Jan 2013

In this talk on a more classical part of ergodic theory, that of Z-actions, I will explain the construction of rank-one transformations via cutting and stacking that goes back to von Neumann and Kakutani and has been used to create examples and counterexamples of various mixing-like properties. Following the explanation of the subject, I will present some of my work on when such transformations are mixing. Some of the results presented are joint work with Cesar Silva.

The Final Exam will be on 10 Dec 2012 at 3:00-5:00pm in SC 1432.

The Exam 2 will be on 14 Nov 2012 at 2:10-3:00pm in SC 1432.

The Exam 1 will be on 1 Oct 2012 at 2:10-3:00pm in SC 1432.

The Midterm Exam #2 will be on 29 Mar 2013 at 1:10pm-2pm in SC 1120.

The Take Home Midterm #2 will be on 5 Apr 2013.

The Take Home Midterm #1 will be on 15 Feb 2013.

The Midterm Exam #1 will be on 8 Feb 2013 at 1:10pm-2pm in SC 1120.

The Final Exam will be on 30 Apr 2013 at 3pm-5pm in SC 1120.

Instructor for Math 260 Introduction to Analysis (Spring 2013) at Vanderbilt University.

Stabilizers of Ergodic Actions of Lattices and Commensurators
University of California: San Diego
16 Nov 2012

The Margulis Normal Subgroup Theorem states that any normal subgroup of an irreducible lattice in a center-free higher-rank semisimple Lie group is of finite index. Stuck and Zimmer, expanding on Margulis' approach, showed that any properly ergodic probability-preserving ergodic action of such a lattice is essentially free.

I will present similar results: my work with Y. Shalom on normal subgroups of lattices in products of simple locally compact groups and normal subgroups of commensurators of lattices, and my work with J. Peterson generalizing this result to stabilizers of ergodic probability-preserving actions of such groups. As a consequence, S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) as do lattices in certain product groups. In particular, any nontrivial ergodic probability-preserving action of PSL_n(Q), for n ≥ 3, is essentially free.

The key idea in the study of normal subgroups is considering nonsingular actions which are the extreme opposite of measure-preserving. Somewhat surprisingly, the key idea in understanding stabilizers of probability-preserving actions also involves studying such actions and the bulk of our work is directed towards properties of these contractive actions.

Poisson Boundaries, Harmonic Functions and Random Walks on Groups
Vanderbilt University
9 Nov & 5 Dec 2012

I will present the construction of the Poisson Boundary of a group, originally defined by Furstenberg, and explain its various properties and applications. The Poisson Boundary can be thought of as the exit boundary of a random walk on the group and can be identified with the space of harmonic functions on the group. The first talk will focus on the construction of the Poisson Boundary and various results due primarily to Furstenberg and Zimmer about boundaries. The second talk will focus on the dynamical behavior of the boundary and its applications to ergodic theory.

Stabilizers of Ergodic Actions of Lattices and Commensurators
Vanderbilt University
19 Sep 2012

A strong generalization of the Margulis Normal Subgroup Theorem, due to Stuck and Zimmer, states that any properly ergodic finite measure-preserving action of an irreducible lattice in a center-free semisimple Lie group with all simple factors of higher-rank is essentially free. We present a similar result generalizing the Normal Subgroup Theorem for Commensurators of Lattices, due to the first author and Shalom, to actions of commensurators. As a consequence, we show that S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) and that lattices in certain product groups do as well. In the second talk, I will explain how the results developed in the first talk lead to the conclusions about S-arithmetic lattices and to lattices in products. The main ideas involve using the Howe-Moore property and property (T) to ensure that actions of the ambient groups satisfy the necessary conditions. Another key idea in studying lattices in products is that most lattices in product groups are isomorphic to the commensurator of a lattice in one of the component groups.

Stabilizers of Ergodic Actions of Lattices and Commensurators
Williams College Ergodic Theory Conference
28 July 2012

The Margulis Normal Subgroup Theorem states that any normal subgroup of an irreducible lattice in a center-free higher-rank semisimple Lie group is of finite index. Stuck and Zimmer, expanding on Margulis' approach, showed that any properly ergodic probability-preserving ergodic action of such a lattice is essentially free. I will present similar results: my work with Y. Shalom on normal subgroups of lattices in products of simple locally compact groups and normal subgroups of commensurators of lattices, and my work with J. Peterson generalizing this result to stabilizers of ergodic probability-preserving actions of such groups. As a consequence, S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) as do lattices in certain product groups. In particular, any nontrivial ergodic probability-preserving action of PSL_n(Q), for n at least 3, is essentially free. The key idea in the study of normal subgroups is considering nonsingular actions which are the extreme opposite of measure-preserving. Somewhat surprisingly, the key idea in understanding stabilizers of probability-preserving actions also involves studying such actions and the bulk of our work is directed towards properties of these contractive, or SAT, actions.

Instructor for Math 208 Ordinary Differential Equations (Fall 2012) at Vanderbilt University.

Stabilizers of Ergodic Actions of Lattices and Commensurators
Darren Creutz and Jesse Peterson
Transactions of the AMS

Stabilizers of Ergodic Actions of Lattices and Commensurators
UCLA Workshop on von Neumann Algebras and Ergodic Theory
26 May 2012

A strong generalization of the Margulis Normal Subgroup Theorem, due to Stuck and Zimmer, states that any properly ergodic finite measure-preserving action of an irreducible lattice in a center-free semisimple Lie group with all simple factors of higher-rank is essentially free. We present a similar result generalizing the Creutz-Shalom Normal Subgroup Theorem for Commensurators of Lattices to actions of commensurators. As a consequence, we show that S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) and that lattices in certain product groups do as well. In particular, any nontrivial ergodic measure-preserving action of PSLn(Q), for n at least three, is essentially free. This is joint work with Jesse Peterson.

Stabilizers for Ergodic Actions of Commensurators
Vanderbilt University
6 April 2012

A strong generalization of the Margulis Normal Subgroup Theorem, due to Stuck and Zimmer, states that any properly ergodic finite measure-preserving action of an irreducible lattice in a center-free semisimple Lie group with all simple factors of higher-rank is essentially free. We present a similar result generalizing the Creutz-Shalom Normal Subgroup Theorem for Commensurators of Lattices to actions of commensurators. As a consequence, we show that S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) and that lattices in certain product groups do as well. In particular, any nontrivial ergodic measure-preserving action of $\mathrm{PSL}_{n}(\mathbb{Q})$, for $n \geq 3$, is essentially free.

The Final Exam will be on 30 Apr 2012 at 9am-11am in SC 1313.

The Take-Home Midterm #2 will be on 4 Apr 2012 at 1:10pm in SC 1313.

The Midterm #2 will be on 30 Mar 2012 at 1:10pm-2pm in SC 1313.

The Take-Home Midterm #1 will be on 15 Feb 2012 at 1:10pm in SC 1313.

The Midterm #1 will be on 10 Feb 2012 at 1:10pm-2pm in SC 1313.

The Property (T) “Half” of the Margulis-Zimmer Conjecture
Vanderbilt University
29 Feb 2012

Generalizing the Margulis Normal Subgroup Theorem, Margulis and Zimmer conjectured that any subgroup of a lattice in a higher-rank Lie group which is commensurated by the lattice is (up to finite index) of a standard form. I will present some of my work on property (T) for totally disconnected groups and countable dense subgroups and explain how it provides "half" of the solution to the conjecture. This is joint work with Yehuda Shalom.

Property (T) for Certain Totally Disconnected Groups Related to a Conjecture of Margulis and Zimmer
Vanderbilt University
17 Feb 2012

I will present some of my work on reduced cohomology and property (T) for totally disconnected groups and dense countable subgroups. The primary application of this work is to show property (T) for a class of totally disconnected groups arising from a conjecture of Margulis and Zimmer regarding the classification of all commensurated subgroups of lattices in higher-rank Lie groups. The key idea in our work is to expand on Kleiner's work on the energy of a cocycle (the idea of which goes back to Mok) and derive a very general result about energy and reduced cohomology. This is joint work with Yehuda Shalom.

Instructor for Math 260 Introduction to Analysis (Spring 2012) at Vanderbilt University.

SAT Actions and Rigidity of Lattices
Vanderbilt University
30 Nov 2011

I will present an overview of SAT actions, a type of quasi-invariant group action on a probability space that is the opposite of measure-preserving, and recent work of Y. Shalom and myself on the rigidity of such actions for lattices in the form of our SAT Factor Theorem. I will then explain how this result plays the key role in the previously presented work on Normal Subgroups of Commensurators of Lattices.

The Final Exam will be on 15 December 2011 at 7pm-9pm in SC 1307.

Normal Subgroups of Commensurators of Lattices
Vanderbilt University
9 Nov 2011

I will present some results of myself and Y. Shalom. I will focus on our Normal Subgroup Theorem for Commensurators of lattices: any normal subgroup of a (dense) commensurator of a lattice in a locally compact group necessarily contains the lattice. Consequences of this theorem will also be discussed: classification of normal subgroups of commensurators; an improved form of Bader-Shalom's normal subgroup theorem for lattices in products; and a partial answer to a question of Lubotzky, Mozes and Zimmer on tree automorphisms.

The Exam #4 will be on 01 December 2011 at 7pm-8:15pm in SC 1307 & 1308.

The Exam #3 will be on 03 November 2011 at 7pm-8:15pm in SC 1307 & 1308.

The Exam #2 will be on 13 October 2011 at 7pm-8:15pm in SC 1307 & 1308.

The Exam #1 will be on 15 September 2011 at 7pm-8:15pm in SC 1307 & 1308.

Normal Subgroups of Commensurators and Rigidity of SAT Actions
Vanderbilt University
26 Aug & 2 & 9 Sep 2011

I will present some results of myself and Y. Shalom in a pair of talks.

During the first talk, I will focus on our Normal Subgroup Theorem for Commensurators of lattices: any normal subgroup of a (dense) commensurator of a lattice in a locally compact group necessarily contains the lattice. Consequences of this theorem will also be discussed: classification of normal subgroups of commensurators; an improved form of Bader-Shalom's normal subgroup theorem for lattices in products; and a partial answer to a question of Lubotzky, Mozes and Zimmer on tree automorphisms.

The second talk will focus on our results on group dynamics for quasi-invariant actions that are the main new ingredient required to prove the normal subgroup theorem. I will discuss strongly approximately transitive actions and their various structural and rigidity properties. The talk will conclude with a discussion of a potential structure theory for quasi-invariant actions.

The second talk should be understandable even without the background presented in the first though it will be helpful.

Instructor for Math 155B Accelerated Single-Variable Calculus II (Fall 2011) at Vanderbilt University.

Commensurated Subgroups and the Dynamics of Group Actions on Quasi-Invariant Measure Spaces
Darren Creutz
Doctoral Dissertation

I will be an Assistant Professor of Mathematics at Vanderbilt University from August 2011 onward.

Awarded the PhD in Pure Mathematics from UCLA awarded 9 June 2011.

Dynamics of SAT Actions
CalTech
2 May 2011

I will present an overview of SAT actions, a class of quasi-invariant actions that are the “opposite” of measure-preserving in a natural way. After presenting key results on SAT and some of my own work (joint with Y. Shalom), I will discuss my new notion of relatively SAT factor maps–the counterpart to relative measure-preserving–and discuss progress toward a structure theory for quasi-invariant actions.

Normal Subgroups of Commensurators and Rigidity of SAT Actions
University of California: Los Angeles
6 Apr & 13 Apr 2011

I will present an overview of my dissertation research in a pair of talks. During the first talk, I will focus on our Normal Subgroup Theorem for Commensurators of lattices: any normal subgroup of a (dense) commensurator of a lattice in a locally compact group necessarily contains the lattice. Consequences of this theorem will also be discussed: classification of normal subgroups of commensurators; an improved form of Bader-Shalom's normal subgroup theorem for lattices in products; and a partial answer to a question of Lubotzky, Mozes and Zimmer on tree automorphisms. The second talk will focus on our results on group dynamics for quasi-invariant actions that are the main new ingredient required to prove the normal subgroup theorem. I will discuss strongly approximately transitive actions and their various structural and rigidity properties. The talk will conclude with a discussion of our progress on two open questions: the Margulis-Zimmer Conjecture on commensurated subgroups of lattices and a potential structure theory for quasi-invariant actions. The second talk should be understandable even without the background presented in the first. This is joint work with Yehuda Shalom.

Next year I will be a postdoc at Vanderbilt University in Nashville, Tennessee.

Quasi-Invariant Group Actions
University of California: Los Angeles
18 Feb 2011

I will present an overview of the ergodic theory of groups acting quasi-invariantly on probability spaces (meaning the measure is not preserved by the action but the null sets are). Such actions arise naturally in the context of Lie groups acting on symmetric space and automorphisms of trees acting on graphs. The bulk of the talk will be background and introductory material; I will conclude with a description of my own research and results in this area.

Normal Subgroups and Rigidity for Commensurators
Vanderbilt University
28 Feb 2011

We present a Normal Subgroup Theorem for (dense) commensurators of lattices in arbitrary locally compact groups (not necessarily Lie). In particular, any normal subgroup of a (dense) commensurator of an (integrable) lattice in a simple topological group necessarily contains (up to finite index) the lattice.
The approach involves new rigidity theorems for commensurators both in the context of representations and in dynamics, in particular a new factor theorem for SAT actions (the natural opposite of measure-preserving) more general than those for boundaries.
This is joint work with Yehuda Shalom.

Instructor for MFE Probability & Programming Review (Winter 2011) at University of California: Los Angeles.

Normal Subgroups of Commensurators and SAT Actions
CalTech
20 Jan 2011

I will present a Normal Subgroup Theorem for (dense) commensurators of lattices in arbitrary locally compact groups. In particular, any normal subgroup of a (dense) commensurator of an (integrable) lattice in a simple topological group necessarily virtually contains the lattice.
SAT actions, the natural opposite of measure-preserving, play a key role and we establish several results about them culminating in a Factor Theorem for SAT actions of lattices.
Some consequences of our work, including a new proof of the Normal Subgroup Theorem for lattices in products, will complete my presentation.
Knowledge of commensurators and Normal Subgroup Theorems will not be assumed.
This is joint work with Yehuda Shalom.

A Normal Subgroup Theorem for Commensurators
Yale University
15 Nov 2010

We present a Normal Subgroup Theorem for (dense) commensurators of lattices in arbitrary locally compact groups (not necessarily Lie). In particular, any normal subgroup of a (dense) commensurator of an (integrable) lattice in a simple topological group necessarily contains (up to finite index) the lattice.
The approach, as in Margulis’ Normal Subgroup Theorem for lattices, involves, on the one hand, using cohomology and rigidity theory to prove a certain group has property (T), and on the other hand, Furstenberg’s Boundary Theory to prove this group is also amenable.
This is joint work with Yehuda Shalom.

Mixing, Random Sequences and Rank-One Transformations
Northwestern University
9 Nov 2010

We present new results on "random" sequences (sufficiently general enough to include deterministic sequences such as polynomials) having various mixing- and ergodic-type properties with respect to transformations having certain mixing-type properties. The main application is a proof of mixing on "stochastic staircase" rank-one transformations, a class which includes all previously known examples of mixing rank-one. The talk will consist of a discussion of the mixing- and ergodic-type properties involved, some indications as to the proofs for random sequences, and an introduction to rank-one transformations with an indication of how one proves mixing.

Normal Subgroup and Factor Theorems for Commensurators
University of Illinois: Chicago
8 Nov 2010

We present a Normal Subgroup Theorem for (dense) commensurators of lattices in arbitrary locally compact groups (not necessarily Lie). In particular, any normal subgroup of a (dense) commensurator of an (integrable) lattice in a simple topological group necessarily contains (up to finite index) the lattice. The approach, as in Margulis’ Normal Subgroup Theorem, involves, on the one hand, using cohomology and rigidity theory to prove a certain group has property (T), and on the other hand, Furstenberg’s Boundary Theory to prove this group is also amenable. We will focus more on the amenability half of the proof, in particular our new ”Factor Theorem” which facilitates the proof (and which is of independent interest). This is join work with Yehuda Shalom.

A Normal Subgroup Theorem for Commensurators of Lattices
AMS Western Meeting
9 Oct 2010

We prove a statement akin to Margulis’ Normal Subgroup Theorem for lattices in Lie groups, but our Theorem applies not to lattices but to commensurators of lattices. We show that any infinite normal subgroup of a (dense) commensurator of a lattice in a Lie group necessarily intersects the lattice in a finite index subgroup. We then develop this into a correspondence between normal subgroups of the commensurator and open normal subgroups of the relative profinite completion.
The approach, as in Margulis’ Theorem, involves, on the one hand, using cohomology and rigidity theory to prove a certain group has property (T), and on the other hand, Furstenberg’s Boundary Theory to prove this group is also amenable. We will focus more on the amenability half of the proof, in particular our new ”Factor Theorem” which facilitates the proof (and which is of independent interest).

Instructor for Math 00 Advanced Topics for Undergraduates (Summer 2010) at University of California: Los Angeles.

Mixing on Rank-One Transformations
Darren Creutz and Cesar Silva
Studia Mathematica

The Final Exam will be on 7 June 2010 at 3pm-6pm in MS 6627.

Superstability and Finite-Time Extinction for Semigroups
University of California: Los Angeles
27 Apr 2010

The Midterm #2 will be on 21 May 2010 at During Class in MS 6627.

The Midterm #1 will be on 23 April 2010 at During Class in MS 6627.

Instructor for Math 31B Integration and Infinite Series (Spring 2010) at University of California: Los Angeles.

Poisson Boundaries and Their Applications
University of California: Los Angeles
Jan 2009

Rank-One Actions, Mixing and Singular Spectra
University of California: Los Angeles
Mar 2007

Mixing on a Class of Rank-One Transformations
Darren Creutz and Cesar Silva
Ergodic Theory and Dynamical Systems

Rank-One Mixing and Dynamical Averaging
Darren Creutz
Honors Thesis