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December 2011

Math 260 Introduction to Analysis (Spring 2012) 12s_math260 | Teaching

01:18pm 16 Dec 2011
Instructor for Math 260 Introduction to Analysis (Spring 2012) at Vanderbilt University.
November 2011

Talk: SAT Actions and Rigidity of Lattices Mathematics

11:38am 27 Nov 2011
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.

Talk: Normal Subgroups of Commensurators of Lattices Mathematics

11:45am 14 Nov 2011
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.
August 2011

Talk: Normal Subgroups of Commensurators and Rigidity of SAT A... Mathematics

11:38am 23 Aug 2011
Normal Subgroups of Commensurators and Rigidity of SAT Actions
Vanderbilt University
26 Aug & 2 Sep & 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.

Math 155B Accelerated Single-Variable Calculus II (Fall 2011) 11f_math155b | Teaching

11:10am 17 Aug 2011
Instructor for Math 155B Accelerated Single-Variable Calculus II (Fall 2011) at Vanderbilt University.
June 2011

Publication: Commensurated Subgroups and the Dynamics of Group... Mathematics

03:03pm 11 Jun 2011
Commensurated Subgroups and the Dynamics of Group Actions on Quasi-Invariant Measure Spaces
Darren Creutz
Doctoral Dissertation
May 2011

Assistant Professor at Vanderbilt Mathematics | Teaching

03:09pm 26 May 2011
I will be an Assistant Professor of Mathematics at Vanderbilt University from August 2011 onward.

Doctor of Philosophy Mathematics

03:02pm 26 May 2011
Awarded the PhD in Pure Mathematics from UCLA awarded 9 June 2011.
April 2011

Talk: Dynamics of SAT Actions Mathematics

08:40pm 13 Apr 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.

Talk: Normal Subgroups of Commensurators and Rigidity of SAT A... Mathematics

01:25am 01 Apr 2011
Normal Subgroups of Commensurators and Rigidity of SAT Actions
UCLA
6 & 13 April 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.
March 2011

Postdoctoral Position at Vanderbilt Mathematics

12:47pm 24 Mar 2011
Next year I will be a postdoc at Vanderbilt University in Nashville, Tennessee.
February 2011

Publication: Mixing Properties of Random Sequences and "Stocha... Mathematics

09:27pm 16 Feb 2011
Mixing Properties of Random Sequences and “Stochastic Staircase” Transformations
Darren Creutz
(in review)
January 2011

Talk: Quasi-Invariant Group Actions Mathematics

08:16pm 28 Jan 2011
Quasi-Invariant Group Actions
UCLA Logic Seminar
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.

Talk: Normal Subgroups and Rigidity for Commensurators Mathematics

08:11pm 15 Jan 2011
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.

MFE Probability & Programming Review (Winter 2011) Teaching | 11w_mfe

06:45pm 11 Jan 2011
Instructor for MFE Probability & Programming Review (Winter 2011) at UCLA.

Talk: Normal Subgroups of Commensurators and SAT Actions Mathematics

04:19pm 06 Jan 2011
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.
November 2010

Publication: Normal Subgroups of (Dense) Commensurators of Lat... Mathematics

09:35pm 01 Nov 2010
Normal Subgroups of (Dense) Commensurators of Lattices
Darren Creutz and Yehuda Shalom
(in preparation)

Publication: Harmonic Cocycles and Commensurated Subgroups Mathematics

09:34pm 01 Nov 2010
Harmonic Cocycles and Commensurated Subgroups
Darren Creutz and Yehuda Shalom
(in preparation)

Talk: A Normal Subgroup Theorem for Commensurators Mathematics

07:36pm 01 Nov 2010
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.

Talk: Mixing, Random Sequences and Rank-One Transformations Mathematics

07:35pm 01 Nov 2010
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.

Talk: Normal Subgroup and Factor Theorems for Commensurators Mathematics

07:34pm 01 Nov 2010
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.
September 2010

Talk: A Normal Subgroup Theorem for Commensurators of Lattices Mathematics

09:29pm 25 Sep 2010
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).
August 2010

Math 00 Undergraduate “Boot Camp” (Summer 2010) Teaching | 10su_math00

07:51pm 29 Aug 2010
Instructor for Math 00 Undergraduate “Boot Camp” (Summer 2010) at UCLA.

Publication: Mixing on Rank-One Transformations Mathematics

09:41pm 15 Aug 2010
Mixing on Rank-One Transformations
Darren Creutz and Cesar Silva
Studia Mathematica
July 2010

Publication: Superstability and Finite Time Extinction for C0-... Mathematics

03:43am 31 Jul 2010
Superstability and Finite Time Extinction for C0-Semigroups
D. Creutz, M. Mazo Jr. and C. Preda
(in review)
June 2010

Award: Robert Sorgenfrey Distinguished Teaching Award Mathematics

10:10pm 04 Jun 2010
Robert Sorgenfrey Distinguished Teaching Award
UCLA
2010
April 2010

Talk: Superstability and Finite-Time Extinction for Semigroups Mathematics

09:29pm 03 Apr 2010
Superstability and Finite-Time Extinction for Semigroups
UCLA
27 Apr 2010
March 2010

Math 31B Integration and Infinite Series (Spring 2010) Teaching | 10s_math31b

01:07am 11 Mar 2010
Instructor for Math 31B Integration and Infinite Series (Spring 2010) at UCLA.
February 2010

Award: VIGRE Instructorship Mathematics

10:09pm 11 Feb 2010
VIGRE Instructorship
UCLA
2010
December 2009

Math 115A Linear Algebra (Winter 2010) Teaching | 10w_math115a

05:12pm 02 Dec 2009
Teaching assistant for Math 115A Linear Algebra (Winter 2010) at UCLA with Professor Yehuda Shalom.
August 2009

Award: VIGRE Fellowship Mathematics

10:09pm 29 Aug 2009
VIGRE Fellowship
UCLA
2005-2009
March 2009

Math 111 Theory of Numbers (Spring 2009) Teaching | 09s_math111

05:45pm 12 Mar 2009
Teaching assistant for Math 111 Theory of Numbers (Spring 2009) at UCLA with Professor Yehuda Shalom.
January 2009

PIC 10A Intro. Programming (Winter 2009) Teaching | 09w_pic10a

06:45pm 05 Jan 2009
Teaching assistant for PIC 10A Intro. Programming (Winter 2009) at UCLA with Professor Wittman.

Talk: Poisson Boundaries and Their Applications Mathematics

09:29pm 04 Jan 2009
Poisson Boundaries and Their Applications
UCLA
Jan 2009
September 2008

Math 134 Nonlinear Differential Equations (Fall 2008) Teaching | 08f_math134

05:41pm 17 Sep 2008
Teaching assistant for Math 134 Nonlinear Differential Equations (Fall 2008) at UCLA with Professor Grossman.
August 2008

Math 170B Intro. Probability Theory (Spring 2008) Teaching | 08s_math170b

12:22am 25 Aug 2008
Teaching assistant for Math 170B Intro. Probability Theory (Spring 2008) at UCLA with Professor Brose.

Math 167 Game Theory (Winter 2008) Teaching | 08w_math167

12:21am 25 Aug 2008
Teaching assistant for Math 167 Game Theory (Winter 2008) at UCLA with Professor Radko.

Math 113 Combinatorics (Fall 2007) Teaching | 07f_math113

12:19am 25 Aug 2008
Teaching assistant for Math 113 Combinatorics (Fall 2007) at UCLA with Professor Duke.

Math 171 Stochastic Processes (Spring 2007) Teaching | 07s_math171

12:17am 25 Aug 2008
Teaching assistant for Math 171 Stochastic Processes (Spring 2007) at UCLA with Professor Schmitz.

Math 170B Intro. Probability Theory (Winter 2007) Teaching | 07w_math170b

12:03am 25 Aug 2008
Teaching assistant for Math 170B Intro. Probability Theory (Winter 2007) at UCLA with Professor Schmitz.

Math 170A Intro. Probability Theory (Fall 2006) Teaching | 06f_math170a

10:57pm 24 Aug 2008
Teaching assistant for Math 170A Intro. Probability Theory (Fall 2006) at UCLA with Professor Schmitz.
March 2007

Talk: Rank-One Actions, Mixing and Singular Spectra Mathematics

09:28pm 01 Mar 2007
Rank-One Actions, Mixing and Singular Spectra
UCLA
Mar 2007
September 2004

Award: SMALL Research Internship Mathematics

05:40pm 26 Sep 2004
SMALL Research Internship
Williams College
2001-2004
March 2004

Publication: Mixing on a Class of Rank-One Transformations Mathematics

05:39pm 15 Mar 2004
Mixing on a Class of Rank-One Transformations
Darren Creutz and Cesar Silva
J. Ergodic Th. & Dyn. Sys.
May 2003

Publication: Rank-One Mixing and Dynamical Averaging Mathematics

04:41am 25 May 2003
Rank-One Mixing and Dynamical Averaging
Darren Creutz
Honors Thesis
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