http://www.dcreutz.com/news/0
dcreutz
dcreutz.com
July 2018
June 2018

SM122 Calculus II (Fall 2018) 18f_sm122 | Teaching

12:07pm 03 Jun 2018
Instructor for SM122 Calculus II (Fall 2018) at U.S. Naval Academy.
January 2018

SM122 Calculus II (Spring 2018) 18s_sm122 | Teaching

10:27pm 08 Jan 2018
Instructor for SM122 Calculus II (Spring 2018) at U.S. Naval Academy.
October 2017

Talk: Relativized Furstenberg Entropy and an Information Theor... Mathematics

04:09pm 12 Oct 2017
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.
August 2017

SM121 Calculus I (Fall 2017) 17f_sm121 | Teaching

08:32pm 22 Aug 2017
Instructor for SM121 Calculus I (Fall 2017) at U.S. Naval Academy.
March 2017

Talk: Set Theory Past ZFC: Large Cardinals, Forcing, and Indep... Mathematics

05:24pm 20 Mar 2017
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.
January 2017

Talk: The Normal Subgroup Theorem for Lattices in Products (Pr... Mathematics

04:59pm 23 Jan 2017
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.

Talk: The Normal Subgroup Theorem for Lattices in Products (Am... Mathematics

08:20pm 20 Jan 2017
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.

Handout: Homework and Quiz Dates 17s_sm212

November 2016

SM212 Differential Equations (Spring 2017) 17s_sm212 | Teaching

10:27pm 04 Nov 2016
Instructor for SM212 Differential Equations (Spring 2017) at U.S. Naval Academy.

Talk: Random Walks and Harmonic Functions on Groups Mathematics

10:13am 02 Nov 2016
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.
October 2016

Homework Assignment: Homework #10 due on 2 December 2016 16f_sm362

11:25am 27 Oct 2016

Homework Assignment: Homework #9 due on 18 November 2016 16f_sm362

11:24am 27 Oct 2016

Homework Assignment: Homework #8 due on 14 November 2016 16f_sm362

11:24am 27 Oct 2016

Homework Assignment: Homework #7 due on 4 November 2016 16f_sm362

03:23pm 24 Oct 2016

Exam info: Test #2 on 28 October 2016 16f_sm362

03:20pm 24 Oct 2016
The Test #2 will be on 28 October 2016 at 13:30-14:20.

Exam info: Test #3 on 2 November 2016 16f_sm221p

01:45am 17 Oct 2016
The Test #3 will be on 2 November 2016 at 14:30-15:20.

Homework Assignment: Homework #6 due on 24 October 2016 16f_sm362

04:54pm 14 Oct 2016

Exam info: Final Exam on 14 December 2016 16f_sm362

03:54pm 14 Oct 2016
The Final Exam will be on 14 December 2016 at 13:30-16:30.

Exam info: Final Exam on 15 December 2016 16f_sm221p

03:54pm 14 Oct 2016
The Final Exam will be on 15 December 2016 at 13:30-16:30.

Talk: Ergodic Actions of Lattices in Higher-Rank Semisimple Gr... Mathematics

03:32pm 05 Oct 2016
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.
September 2016

Talk: Mixing and Rank-One Transformations Mathematics

12:44pm 29 Sep 2016
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.

Homework Assignment: Homework #5 due on 5 October 2016 16f_sm362

10:38am 26 Sep 2016

Exam info: Test #4 on 6 December 2016 16f_sm221p

06:45pm 13 Sep 2016
The Test #4 will be on 6 December 2016 at 14:30-15:20.

Exam info: Test #2 on 4 October 2016 16f_sm221p

06:45pm 13 Sep 2016
The Test #2 will be on 4 October 2016 at 14:30-15:20.

Exam info: Test #1 on 20 September 2016 16f_sm221p

06:44pm 13 Sep 2016
The Test #1 will be on 20 September 2016 at 14:30-15:20.

Homework Assignment: Homework #4 due on 19 September 2016 16f_sm362

05:19pm 11 Sep 2016

Homework Assignment: Homework #3 due on 14 September 2016 16f_sm362

05:18pm 11 Sep 2016
August 2016

Homework Assignment: Homework #2 due on 7 September 2016 16f_sm362

05:25pm 29 Aug 2016

Talk: Character Rigidity for Lattices in Lie Groups Mathematics

05:31pm 21 Aug 2016
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.

Homework Assignment: Homework #1 due on 2 September 2016 16f_sm362

03:23pm 20 Aug 2016

Exam info: Test #3 on 5 December 2016 16f_sm362

03:17pm 20 Aug 2016
The Test #3 will be on 5 December 2016 at 13:30-14:20.

Exam info: Test #1 on 21 September 2016 16f_sm362

03:17pm 20 Aug 2016
The Test #1 will be on 21 September 2016 at 13:30-14:20.

SM362 Modern Algebra (Fall 2016) 16f_sm362 | Teaching

03:15pm 20 Aug 2016
Instructor for SM362 Modern Algebra (Fall 2016) at U.S. Naval Academy.

SM221P Calculus III with Vector Fields (Fall 2016) 16f_sm221p | Teaching

02:53pm 20 Aug 2016
Instructor for SM221P Calculus III with Vector Fields (Fall 2016) at U.S. Naval Academy.
April 2016

Handout: Midterm Exam #2 - Take-Home Portion 16s_math3641

March 2016

Homework Assignment: Homework #6 due on 11 April 2016 16s_math3641

Homework Assignment: Homework #5 due on 28 March 2016 16s_math3641

Handout: Efficiency of MLEs 16s_math3641

February 2016

Homework Assignment: Homework #4 due on 16 March 2016 16s_math3641

Exam info: Final Exam (Take-Home) on Due 5 May 2016 16s_math3641

02:59pm 19 Feb 2016
The Final Exam (Take-Home) will be on Due 5 May 2016 at 5:00pm in  .

Exam info: Midterm Exam #2 (Take-Home) on Due 15 April 2016 16s_math3641

02:58pm 19 Feb 2016
The Midterm Exam #2 (Take-Home) will be on Due 15 April 2016 at 10:10am in  .

Exam info: Midterm Exam #1 (Take-Home) on Due 19 February 2016 16s_math3641

02:58pm 19 Feb 2016
The Midterm Exam #1 (Take-Home) will be on Due 19 February 2016 at 10:10am in  .

Exam info: Midterm Exam #2 on 15 April 2016 16s_math3641

02:58pm 19 Feb 2016
The Midterm Exam #2 will be on 15 April 2016 at 10:10am-11:00am in SC 1310.

Handout: Midterm Exam #1 - Take-Home Portion 16s_math3641

Exam info: Midterm Exam #1 on 19 February 2016 16s_math3641

05:28pm 11 Feb 2016
The Midterm Exam #1 will be on 19 February 2016 at 10:10am-11:00am in SC 1310.

Talk: Rigidty Theory of Lattices in Semisimple Groups Mathematics

10:41pm 08 Feb 2016
Rigidty Theory of Lattices in Semisimple Groups
U.S. Naval Academy
3 Feb 2016

Homework Assignment: Homework #3 due on 15 February 2016 16s_math3641

January 2016

Homework Assignment: Homework #2 due on 5 February 2016 16s_math3641

Handout: Course Description – Math 5641 16s_math3641

Talk: The Information Theory of Joinings Mathematics

06:21pm 04 Jan 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.
December 2015

Homework Assignment: Homework #1 due on 27 January 2016 16s_math3641

Handout: Course Description – Math 3641 16s_math3641

Math 3641 Mathematical Theory of Statistics (Spring 2016) 16s_math3641 | Teaching

02:32pm 14 Dec 2015
Instructor for Math 3641 Mathematical Theory of Statistics (Spring 2016) at Vanderbilt University.

Handout: Final Exam Formula Sheet 15f_math1200

Handout: Exam #4 Formula Sheet 15f_math1200

November 2015

Handout: Final Project Descripition 15f_math1010

Handout: Exam #3 Formula Sheet 15f_math1200

September 2015

Exam info: Final Exam on 14 December 2015 15f_math1200

10:25pm 08 Sep 2015
The Final Exam will be on 14 December 2015 at 7:00pm-9:00pm in Wilson Hall 103.

Exam info: Exam #4 on 3 December 2015 15f_math1200

10:25pm 08 Sep 2015
The Exam #4 will be on 3 December 2015 at 7:00pm-8:15pm in Wilson Hall 103.

Exam info: Exam #3 on 5 November 2015 15f_math1200

10:25pm 08 Sep 2015
The Exam #3 will be on 5 November 2015 at 7:00pm-8:15pm in Wilson Hall 103.

Exam info: Exam #2 on 8 October 2015 15f_math1200

10:25pm 08 Sep 2015
The Exam #2 will be on 8 October 2015 at 7:00pm-8:15pm in Wilson Hall 103.

Exam info: Exam #1 on 17 September 2015 15f_math1200

10:25pm 08 Sep 2015
The Exam #1 will be on 17 September 2015 at 7:00pm-8:15pm in Wilson Hall 103.

Publication: Stabilizers of Actions of Lattices in Products of... Mathematics

04:40pm 08 Sep 2015
Stabilizers of Actions of Lattices in Products of Groups
Darren Creutz
Ergodic Theory and Dynamical Systems
August 2015

Homework Assignment: Homework #1 due on 2 September 2015 15f_math1200

Homework Assignment: Homework #2 due on 9 September 2015 15f_math1200

Homework Assignment: Homework #3 due on 16 September 2015 15f_math1200

Homework Assignment: Homework #4 due on 23 September 2015 15f_math1200

Homework Assignment: Homework #5 due on 30 September 2015 15f_math1200

Homework Assignment: Homework #6 due on 7 October 2015 15f_math1200

Homework Assignment: Homework #7 due on 14 October 2015 15f_math1200

Homework Assignment: Homework #8 due on 21 October 2015 15f_math1200

Homework Assignment: Homework #9 due on 28 October 2015 15f_math1200

Homework Assignment: Homework #10 due on 4 November 2015 15f_math1200

Homework Assignment: Homework #11 due on 11 November 2015 15f_math1200

Homework Assignment: Homework #12 due on 18 November 2015 15f_math1200

Homework Assignment: Homework #13 due on 2 December 2015 15f_math1200

Homework Assignment: Homework #14 due on 9 December 2015 15f_math1200

Handout: Request for Exam Time Change Form 15f_math1200

Homework Assignment: Homework #12 due on 9 December 2015 15f_math1010

Homework Assignment: Homework #11 due on 2 December 2015 15f_math1010

Homework Assignment: Homework #10 due on 16 November 2015 15f_math1010

Homework Assignment: Homework #9 due on 2 November 2015 15f_math1010

Homework Assignment: Homework #8 due on 26 October 2015 15f_math1010

Homework Assignment: Homework #7 due on 19 October 2015 15f_math1010

Homework Assignment: Homework #6 due on 12 October 2015 15f_math1010

Homework Assignment: Homework #5 due on 5 October 2015 15f_math1010

Homework Assignment: Homework #4 due on 21 September 2015 15f_math1010

Homework Assignment: Homework #3 due on 14 September 2015 15f_math1010

Exam info: Final Exam on 18 December 2015 15f_math1010

03:00pm 07 Aug 2015
The Final Exam will be on 18 December 2015 at 9:00am-11:00am in SC 1206.

Exam info: Exam #2 on 9 November 2015 15f_math1010

02:59pm 07 Aug 2015
The Exam #2 will be on 9 November 2015 at 9:10am-10:00am in SC 1206.

Exam info: Exam #1 on 28 September 2015 15f_math1010

02:59pm 07 Aug 2015
The Exam #1 will be on 28 September 2015 at 9:10am-10:00am in SC 1206.

Homework Assignment: Homework #2 due on 7 September 2015 15f_math1010

Homework Assignment: Homework #1 due on 31 August 2015 15f_math1010

Publication: Contractive Spaces and Relatively Contractive Maps Mathematics

06:04pm 06 Aug 2015
Contractive Spaces and Relatively Contractive Maps
Darren Creutz
AMS Contemporary Mathematics

Handout: Final Exam Formula Sheet 15su_math216

July 2015

Homework Assignment: Homework #7 due on 6 August 2015 15su_math216

Homework Assignment: Homework #6 due on 5 August 2015 15su_math216

Homework Assignment: Homework #5 due on 29 July 2015 15su_math216

Homework Assignment: Homework #4 due on 24 July 2015 15su_math216

Math 1200 Single-Variable Calculus I (Fall 2015) 15f_math1200 | Teaching

06:05pm 17 Jul 2015
Instructor for Math 1200 Single-Variable Calculus I (Fall 2015) at Vanderbilt University.

Math 1010 Probability and Statistical Inference (Fall 2015) 15f_math1010 | Teaching

06:05pm 17 Jul 2015
Instructor for Math 1010 Probability and Statistical Inference (Fall 2015) at Vanderbilt University.

Homework Assignment: Homework #3 due on 21 July 2015 15su_math216

Homework Assignment: Homework #2 due on 15 July 2015 15su_math216

Homework Assignment: Homework #1 due on 13 July 2015 15su_math216

June 2015

Exam info: Final Exam on 7 August 2015 15su_math216

01:11pm 29 Jun 2015
The Final Exam will be on 7 August 2015.

Exam info: Exam #2 on 31 July 2015 15su_math216

01:11pm 29 Jun 2015
The Exam #2 will be on 31 July 2015.

Exam info: Exam #1 on 17 July 2015 15su_math216

01:11pm 29 Jun 2015
The Exam #1 will be on 17 July 2015.

Math 216 Probability and Statistics for Engineering (Summer 2015) 15su_math216 | Teaching

01:07pm 29 Jun 2015
Instructor for Math 216 Probability and Statistics for Engineering (Summer 2015) at Vanderbilt University.

Talk: Co-Organizer, Special Session: Classification Problems i... Mathematics

01:04pm 29 Jun 2015
Co-Organizer, Special Session: Classification Problems in Operator Algebras
AMS Joint Mathematics Meetings
11 January 2015

Talk: Harmonic Maps on Groups and Property (T) Mathematics

01:01pm 29 Jun 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).
April 2015

Handout: Final Exam Formula Sheet 15s_math127b

Handout: Final Exam Formula Sheet 15s_math196

March 2015

Handout: Exam #2 Formula Sheet 15s_math196

Homework Assignment: Homework #9 due on 20 April 2015 15s_math127b

Homework Assignment: Homework #8 due on 13 April 2015 15s_math127b

Homework Assignment: Homework #7 due on 6 April 2015 15s_math127b

Homework Assignment: Homework #11 due on 17 April 2015 15s_math196

Homework Assignment: Homework #10 due on 10 April 2015 15s_math196

Homework Assignment: Homework #9 due on 30 March 2015 15s_math196

Homework Assignment: Homework #6 due on 23 March 2015 15s_math127b

Homework Assignment: Homework #8 due on 20 March 2015 15s_math196

Homework Assignment: Homework #5 due on 16 March 2015 15s_math127b

Homework Assignment: Homework #7 due on 16 March 2015 15s_math196

February 2015

Homework Assignment: Homework #4 due on 9 March 2015 15s_math127b

Homework Assignment: Homework #6 due on 27 February 2015 15s_math196

Handout: Exam #1 Formula Sheet 15s_math196

Homework Assignment: Homework #3 due on 23 February 2015 15s_math127b

Homework Assignment: Homework #5 due on 13 February 2015 15s_math196

Homework Assignment: Homework #2 due on 5 February 2015 15s_math127b

Homework Assignment: Homework #4 due on 6 February 2015 15s_math196

January 2015

Homework Assignment: Homework #3 due on 30 January 2015 15s_math196

Homework Assignment: Homework #2 due on 21 January 2015 15s_math196

Homework Assignment: Homework #1 due on 26 January 2015 15s_math127b

Award: Co-Organizer, AMS Special Session on Classification Pro... Mathematics

12:51pm 10 Jan 2015
Co-Organizer, AMS Special Session on Classification Problems in Operator Algebras
Joint Mathematics Meetings
2015

Homework Assignment: Homework #0 due on 12 January 2015 15s_math127b

December 2014

Exam info: Final Exam on 24 April 2015 15s_math196

08:23pm 08 Dec 2014
The Final Exam will be on 24 April 2015 at 3:00pm-5:00pm in SC 1120.

Exam info: Final Exam on 28 April 2015 15s_math127b

08:23pm 08 Dec 2014
The Final Exam will be on 28 April 2015 at 3:00pm-5:00pm in SC 1320.

Exam info: Exam #2 on 25 March 2015 15s_math127b

08:23pm 08 Dec 2014
The Exam #2 will be on 25 March 2015 at 9:10am-10:00am in SC 1320.

Exam info: Exam #1 on 11 February 2015 15s_math127b

08:22pm 08 Dec 2014
The Exam #1 will be on 11 February 2015 at 9:10am-10:00am in SC 1320.

Math 127B Probability and Statistical Inference (Spring 2015) 15s_math127b | Teaching

08:20pm 08 Dec 2014
Instructor for Math 127B Probability and Statistical Inference (Spring 2015) at Vanderbilt University.

Homework Assignment: Homework #1 due on 12 January 2015 15s_math196

Exam info: Exam #2 on 3 April 2015 15s_math196

08:15pm 08 Dec 2014
The Exam #2 will be on 3 April 2015 at 11:10am-Noon in SC 1120.

Exam info: Exam #1 on 20 February 2015 15s_math196

08:15pm 08 Dec 2014
The Exam #1 will be on 20 February 2015 at 11:10am-Noon in SC 1120.

Math 196 Differential Equations with Linear Alegbra (Spring 2015) 15s_math196 | Teaching

08:14pm 08 Dec 2014
Instructor for Math 196 Differential Equations with Linear Alegbra (Spring 2015) at Vanderbilt University.

Handout: Final Exam Formula Sheet 14f_math196

November 2014

Homework Assignment: Homework #12 due on 3 December 2014 14f_math127a

Handout: Exam #2 Scores Statistical Analysis 14f_math127a

Handout: Exam #2 Formula Sheet 14f_math196

Homework Assignment: Homework #11 due on 17 November 2014 14f_math127a

Homework Assignment: Homework #10 due on 10 November 2014 14f_math127a

Homework Assignment: Homework #12 due on 3 December 2014 14f_math196

Homework Assignment: Homework #11 due on 10 November 2014 14f_math196

October 2014

Homework Assignment: Homework #10 due on 3 November 2014 14f_math196

Homework Assignment: Homework #9 due on 27 October 2014 14f_math127a

Homework Assignment: Homework #9 due on 27 October 2014 14f_math196

Homework Assignment: Homework #8 due on 20 October 2014 14f_math127a

Homework Assignment: Homework #8 due on 20 October 2014 14f_math196

Homework Assignment: Homework #7 due on 13 October 2014 14f_math127a

Homework Assignment: Homework #7 due on 13 October 2014 14f_math196

Homework Assignment: Homework #6 due on 8 October 2014 14f_math127a

September 2014

Handout: Exam #1 Formula Sheet 14f_math196

Handout: Exam #1 Scores Statistical Analysis 14f_math127a

Homework Assignment: Homework #6 due on 29 September 2014 14f_math196

Homework Assignment: Homework #5 due on 24 September 2014 14f_math127a

Homework Assignment: Homework #5 due on 22 September 2014 14f_math196

Homework Assignment: Homework #4 due on 15 September 2014 14f_math196

Homework Assignment: Homework #4 due on 15 September 2014 14f_math127a

Homework Assignment: Homework #3 due on 8 September 2014 14f_math196

Homework Assignment: Homework #3 due on 8 September 2014 14f_math127a

August 2014

Homework Assignment: Homework #2 due on 1 September 2014 14f_math127a

Homework Assignment: Homework #2 due on 1 September 2014 14f_math196

Homework Assignment: Homework #1 due on 25 August 2014 14f_math127a

Exam info: Final Exam on 11 December 2014 14f_math127a

12:06pm 08 Aug 2014
The Final Exam will be on 11 December 2014 at 3:00pm-5:00pm in SC 1206.

Exam info: Exam #2 on 3 November 2014 14f_math127a

12:06pm 08 Aug 2014
The Exam #2 will be on 3 November 2014 at 9:10am-10:00am in SC 1206.

Exam info: Exam #1 on 22 September 2014 14f_math127a

12:05pm 08 Aug 2014
The Exam #1 will be on 22 September 2014 at 9:10am-10:00am in SC 1206.

Exam info: Final Exam on 12 December 2014 14f_math196

12:05pm 08 Aug 2014
The Final Exam will be on 12 December 2014 at 3:00pm-5:00pm in SC 1320.

Exam info: Exam #2 on 14 November 2014 14f_math196

12:04pm 08 Aug 2014
The Exam #2 will be on 14 November 2014 at 10:10am-11:00am in SC 1320.

Exam info: Exam #1 on 3 October 2014 14f_math196

12:04pm 08 Aug 2014
The Exam #1 will be on 3 October 2014 at 10:10am-11:00am in SC 1320.

Homework Assignment: Homework #1 due on 25 August 2014 14f_math196

Handout: Course Description 14f_math196

Handout: Course Schedule 14f_math127a

Handout: Course Description 14f_math127a

Math 196 Differential Equations with Linear Alegbra (Fall 2014) 14f_math196 | Teaching

12:21pm 05 Aug 2014
Instructor for Math 196 Differential Equations with Linear Alegbra (Fall 2014) at Vanderbilt University.

Math 127A Probability and Statistical Inference (Fall 2014) 14f_math127a | Teaching

12:20pm 05 Aug 2014
Instructor for Math 127A Probability and Statistical Inference (Fall 2014) at Vanderbilt University.
June 2014

Handout: Course Schedule 14su_math150a

Handout: Course Description 14su_math150a

Math 150A Single Variable Calculus (Summer 2014) 14su_math150a | Teaching

04:47pm 02 Jun 2014
Instructor for Math 150A Single Variable Calculus (Summer 2014) at Vanderbilt University.
January 2014
December 2013

Handout: Course Description 14s_math394

Publication: A Normal Subgroup Theorem for Commensurators of L... Mathematics

09:35pm 22 Dec 2013
A Normal Subgroup Theorem for Commensurators of Lattices
Darren Creutz and Yehuda Shalom
Groups, Geometry and Dynamics

Math 394 Ergodic Theory of Group Actions (Spring 2014) 14s_math394 | Teaching

02:15am 21 Dec 2013
Instructor for Math 394 Ergodic Theory of Group Actions (Spring 2014) at Vanderbilt University.

Handout: Final Exam Formula Sheet 13f_math175

November 2013

Talk: Operator Algebraic Superrigidity for Lattices and Commen... Mathematics

05:40pm 11 Nov 2013
Operator Algebraic Superrigidity for Lattices and Commensurators
Northwestern University
3 Dec 2013

Handout: Exam 3 Formula Sheet 13f_math175

October 2013

Talk: Rigidity for Characters on Lattices and Commensurators Mathematics

06:39pm 30 Oct 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.

Publication: Character Rigidity for Lattices and Commensurators Mathematics

05:26pm 30 Oct 2013
Character Rigidity for Lattices and Commensurators
Darren Creutz and Jesse Peterson
(in review)

Homework Assignment: No Homework Due 20 Nov due on 20 Nov 2013 13f_math175

Homework Assignment: Homework #11 due on 4 Dec 2013 13f_math175

Homework Assignment: Homework #10 due on 13 Nov 2013 13f_math175

Homework Assignment: Homework #9 due on 6 Nov 2013 13f_math175

Homework Assignment: Homework #8 due on 30 Oct 2013 13f_math175

Handout: Exam 2 Formula Sheet 13f_math175

Homework Assignment: No Homework Due 23 Oct due on 23 Oct 2013 13f_math175

Homework Assignment: Homework #7 due on 16 Oct 2013 13f_math175

Homework Assignment: Homework #6 due on 9 Oct 2013 13f_math175

September 2013

Talk: Operator-Algebraic Superrigidity for Lattices Mathematics

04:38pm 24 Sep 2013
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.

Talk: Character Rigidity for Lattices and Commensurators Mathematics

04:37pm 24 Sep 2013
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.

Homework Assignment: Homework #5 due on 2 Oct 2013 13f_math175

Handout: Exam 1 Formula Sheet 13f_math175

Homework Assignment: No Homework Due 25 Sep due on 25 Sep 2013 13f_math175

Publication: Mixing on Stochastic Staircase Transformations Mathematics

09:27pm 09 Sep 2013
Mixing on Stochastic Staircase Transformations
Darren Creutz
(in review)

Homework Assignment: Homework #4 due on 18 Sep 2013 13f_math175

Homework Assignment: Homework #3 due on 11 Sep 2013 13f_math175

August 2013

Exam info: Final Exam on 12 Dec 2013 13f_math175

04:07pm 22 Aug 2013
The Final Exam will be on 12 Dec 2013 at 9:00am-11:00am.

Exam info: Exam #3 on 15 Nov 2013 13f_math175

04:07pm 22 Aug 2013
The Exam #3 will be on 15 Nov 2013 at 10:10am-11:00am in SC 1210.

Exam info: Exam #2 on 18 Oct 2013 13f_math175

04:07pm 22 Aug 2013
The Exam #2 will be on 18 Oct 2013 at 10:10am-11:00am in SC 1210.

Exam info: Exam #1 on 20 Sep 2013 13f_math175

04:07pm 22 Aug 2013
The Exam #1 will be on 20 Sep 2013 at 10:10am-11:00am in SC 1210.

Homework Assignment: Homework #2 due on 4 Sep 2013 13f_math175

Homework Assignment: Homework #1 due on 28 Aug 2013 13f_math175

Handout: Course Description 13f_math175

July 2013

Math 175 Multivariable Calculus (Fall 2013) 13f_math175 | Teaching

05:30pm 26 Jul 2013
Instructor for Math 175 Multivariable Calculus (Fall 2013) at Vanderbilt University.

Handout: Syllabus 13su_math155a

Handout: Course Description 13su_math155a

Math 155A Single-Variable Calculus (Summer 2013) 13su_math155a | Teaching

05:23pm 26 Jul 2013
Instructor for Math 155A Single-Variable Calculus (Summer 2013) at Vanderbilt University.
May 2013

Talk: Stabilizers of Actions of Groups and Invariant Random Su... Mathematics

08:23am 02 May 2013
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.
April 2013

Talk: Stabilizers of Ergodic Actions of Product Groups and Lat... Mathematics

05:33pm 11 Apr 2013
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.

Homework Assignment: HW 9 due on 15 Apr 2013 13s_math260

Talk: Stabilizers of Actions of Product Groups and Lattices in... Mathematics

06:49pm 04 Apr 2013
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.
March 2013

Homework Assignment: HW 8 due on 22 Mar 2013 13s_math260

Homework Assignment: HW 7 due on 15 Mar 2013 13s_math260

February 2013

Homework Assignment: HW 6 due on 1 Mar 2013 13s_math260

Homework Assignment: HW 5 due on 22 Feb 2013 13s_math260

January 2013

Homework Assignment: HW 4 due on 1 Feb 2013 13s_math260

Homework Assignment: HW 3 due on 25 Jan 2013 13s_math260

Talk: Mixing on Rank-One Transformations Mathematics

12:05pm 15 Jan 2013
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.

Homework Assignment: HW 2 due on 18 Jan 2013 13s_math260

December 2012

Exam info: Final Exam on 10 Dec 2012 12f_math208

11:30am 03 Dec 2012
The Final Exam will be on 10 Dec 2012 at 3:00-5:00pm in SC 1432.

Exam info: Exam 2 on 14 Nov 2012 12f_math208

11:30am 03 Dec 2012
The Exam 2 will be on 14 Nov 2012 at 2:10-3:00pm in SC 1432.

Exam info: Exam 1 on 1 Oct 2012 12f_math208

11:30am 03 Dec 2012
The Exam 1 will be on 1 Oct 2012 at 2:10-3:00pm in SC 1432.
November 2012

Handout: Syllabus 13s_math260

Homework Assignment: HW 1 due on 11 Jan 2013 13s_math260

Exam info: Midterm Exam #2 on 29 Mar 2013 13s_math260

10:15am 19 Nov 2012
The Midterm Exam #2 will be on 29 Mar 2013 at 1:10pm-2pm in SC 1120.

Exam info: Take Home Midterm #2 on 5 Apr 2013 13s_math260

10:04am 19 Nov 2012
The Take Home Midterm #2 will be on 5 Apr 2013.

Exam info: Take Home Midterm #1 on 15 Feb 2013 13s_math260

10:03am 19 Nov 2012
The Take Home Midterm #1 will be on 15 Feb 2013.

Exam info: Midterm Exam #1 on 8 Feb 2013 13s_math260

10:03am 19 Nov 2012
The Midterm Exam #1 will be on 8 Feb 2013 at 1:10pm-2pm in SC 1120.

Exam info: Final Exam on 30 Apr 2013 13s_math260

12:57pm 18 Nov 2012
The Final Exam will be on 30 Apr 2013 at 3pm-5pm in SC 1120.

Math 260 Introduction to Analysis (Spring 2013) 13s_math260 | Teaching

12:50pm 18 Nov 2012
Instructor for Math 260 Introduction to Analysis (Spring 2013) at Vanderbilt University.

Homework Assignment: HW 11 due on 3 Dec 2012 12f_math208

Talk: Stabilizers of Ergodic Actions of Lattices and Commensur... Mathematics

03:06pm 08 Nov 2012
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 PSLn(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.

Talk: Poisson Boundaries, Harmonic Functions and Random Walks ... Mathematics

01:02pm 03 Nov 2012
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.
October 2012

Homework Assignment: HW 10 due on 7 Nov 2012 12f_math208

Homework Assignment: HW 9 due on 2 Nov 2012 12f_math208

Homework Assignment: HW 8 due on 24 Oct 2012 12f_math208

Homework Assignment: HW 7 due on 17 Oct 2012 12f_math208

September 2012

Homework Assignment: HW 6 due on 10 Oct 2012 12f_math208

Homework Assignment: HW 5 due on 26 Sep 2012 12f_math208

Homework Assignment: HW 4 due on 19 Sep 2012 12f_math208

Talk: Stabilizers of Ergodic Actions of Lattices and Commensur... Mathematics

03:18pm 05 Sep 2012
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.

Homework Assignment: HW 3 due on 12 Sep 2012 12f_math208

August 2012

Homework Assignment: HW 2 due on 5 Sep 2012 12f_math208

Homework Assignment: HW 1 due on 29 Aug 2012 12f_math208

July 2012

Talk: Stabilizers of Ergodic Actions of Lattices and Commensur... Mathematics

11:09am 10 Jul 2012
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 PSLn(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.

Math 208 Ordinary Differential Equations (Fall 2012) 12f_math208 | Teaching

03:25pm 03 Jul 2012
Instructor for Math 208 Ordinary Differential Equations (Fall 2012) at Vanderbilt University.
June 2012

Publication: Stabilizers of Ergodic Actions of Lattices and Co... Mathematics

04:09pm 28 Jun 2012
Stabilizers of Ergodic Actions of Lattices and Commensurators
Darren Creutz and Jesse Peterson
Transactions of the AMS
May 2012

Talk: Stabilizers of Ergodic Actions of Lattices and Commensur... Mathematics

11:04am 31 May 2012
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.
March 2012

Talk: Stabilizers for Ergodic Actions of Commensurators Mathematics

07:20pm 26 Mar 2012
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.

Homework Assignment: HW 10 due on 20 Apr 2012 12s_math260

Homework Assignment: HW 9 due on 13 Apr 2012 12s_math260

Exam info: Final Exam on 30 Apr 2012 12s_math260

03:11pm 20 Mar 2012
The Final Exam will be on 30 Apr 2012 at 9am-11am in SC 1313.

Exam info: Take-Home Midterm #2 on 4 Apr 2012 12s_math260

03:10pm 20 Mar 2012
The Take-Home Midterm #2 will be on 4 Apr 2012 at 1:10pm in SC 1313.

Exam info: Midterm #2 on 30 Mar 2012 12s_math260

03:10pm 20 Mar 2012
The Midterm #2 will be on 30 Mar 2012 at 1:10pm-2pm in SC 1313.

Exam info: Take-Home Midterm #1 on 15 Feb 2012 12s_math260

03:10pm 20 Mar 2012
The Take-Home Midterm #1 will be on 15 Feb 2012 at 1:10pm in SC 1313.

Exam info: Midterm #1 on 10 Feb 2012 12s_math260

03:09pm 20 Mar 2012
The Midterm #1 will be on 10 Feb 2012 at 1:10pm-2pm in SC 1313.

Homework Assignment: HW 8 due on 23 Mar 2012 12s_math260

February 2012

Homework Assignment: HW 7 due on 16 Mar 2012 12s_math260

Homework Assignment: HW 6 due on 2 Mar 2012 12s_math260

Talk: The Property (T) "Half" of the Margulis-Zimmer Conjectur... Mathematics

10:27pm 13 Feb 2012
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.

Talk: Property (T) for Certain Totally Disconnected Groups Rel... Mathematics

09:37pm 09 Feb 2012
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.

Homework Assignment: HW 5 due on 24 Feb 2012 12s_math260

January 2012

Homework Assignment: HW 4 due on 3 Feb 2012 12s_math260

Homework Assignment: HW 3 due on 27 Jan 2012 12s_math260

Homework Assignment: HW 2 due on 20 Jan 2012 12s_math260

Homework Assignment: HW 1 due on 13 Jan 2012 12s_math260

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

Handout: Formula Sheet for the Final Exam 11f_math155b

Handout: Formula Sheet for Exam 4 11f_math155b

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.

Exam info: Final Exam on 15 December 2011 11f_math155b

06:33pm 16 Nov 2011
The Final Exam will be on 15 December 2011 at 7pm-9pm in SC 1307.

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.
October 2011

Handout: Formula Sheet for Exam 2 11f_math155b

September 2011

Exam info: Exam #4 on 01 December 2011 11f_math155b

06:29pm 12 Sep 2011
The Exam #4 will be on 01 December 2011 at 7pm-8:15pm in SC 1307 & 1308.

Exam info: Exam #3 on 03 November 2011 11f_math155b

06:28pm 12 Sep 2011
The Exam #3 will be on 03 November 2011 at 7pm-8:15pm in SC 1307 & 1308.

Exam info: Exam #2 on 13 October 2011 11f_math155b

06:28pm 12 Sep 2011
The Exam #2 will be on 13 October 2011 at 7pm-8:15pm in SC 1307 & 1308.

Exam info: Exam #1 on 15 September 2011 11f_math155b

06:28pm 12 Sep 2011
The Exam #1 will be on 15 September 2011 at 7pm-8:15pm in SC 1307 & 1308.

Handout: Formula Sheet for Exam 1 11f_math155b

Homework Assignment: HW 14 due on 7 Dec 2011 11f_math155b

Homework Assignment: HW 13 due on 30 Nov 2011 11f_math155b

Homework Assignment: HW 12 due on 16 Nov 2011 11f_math155b

Homework Assignment: HW 11 due on 9 Nov 2011 11f_math155b

Homework Assignment: HW 10 due on 2 Nov 2011 11f_math155b

Homework Assignment: HW 9 due on 26 Oct 2011 11f_math155b

Homework Assignment: HW 8 due on 19 Oct 2011 11f_math155b

Homework Assignment: HW 7 due on 12 Oct 2011 11f_math155b

Homework Assignment: HW 6 due on 5 Oct 2011 11f_math155b

Homework Assignment: HW 5 due on 28 Sep 2011 11f_math155b

Homework Assignment: HW 4 due on 21 Sep 2011 11f_math155b

Handout: Study Guide for Exam 1 11f_math155b

Handout: Limit Rules 11f_math155b

Homework Assignment: HW 3 due on 14 Sep 2011 11f_math155b

August 2011

Homework Assignment: HW 2 due on 7 Sep 2011 11f_math155b

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 & 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.

Homework Assignment: HW 1 due on 31 Aug 2011 11f_math155b

Handout: Course Policies 11f_math155b

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
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.
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

Lecture Notes: Probability Lecture 5: Limit Laws and Recursion 11w_mfe

Lecture Notes: Probability Lecture 4: Continuous Variables 11w_mfe

January 2011

Talk: Quasi-Invariant Group Actions Mathematics

08:16pm 28 Jan 2011
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.

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.

Lecture Notes: Probability Lecture 1: Basics of Probability 11w_mfe

Lecture Notes: Probability Lecture 2: Counting and Expectation... 11w_mfe

Lecture Notes: Probability Lecture 3: Conditional Probability 11w_mfe

Lecture Notes: Programming Lecture 1: Overview for Interviews 11w_mfe

Lecture Notes: Programming Lecture 2: Writing C++ Code 11w_mfe

Lecture Notes: Programming Lecture 3: Pointers 11w_mfe

Lecture Notes: Programming Lecture 4: Sorting & Optimality 11w_mfe

Handout: Conditional Expectation via Indicators 11w_mfe

Handout: Infinite Series 11w_mfe

Handout: Sample Probability Problems Solutions 11w_mfe

Handout: Sample Probability Problems 11w_mfe

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

06:45pm 11 Jan 2011
Instructor for MFE Probability & Programming Review (Winter 2011) at University of California: Los Angeles.

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

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).

Solutions: Logic Solutions 10su_math00

Solutions: Limits Solutions 10su_math00

Handout: Lecture 9: Logic 10su_math00

Handout: Lecture 8: Limits 10su_math00

Homework Assignment: Logic Problems due on 22 September 2010 10su_math00

Homework Assignment: Limits Problems due on 21 September 2010 10su_math00

Homework Assignment: Diophantine Equations Problems due on 20 ... 10su_math00

Handout: Lecture 5: GCD & Euclid's Algorithm 10su_math00

Handout: Lecture 6: Unique Factorization 10su_math00

Homework Assignment: Unique Factorization Problems due on 17 S... 10su_math00

Homework Assignment: GCD & Euclid's Algorithm Problems due... 10su_math00

Solutions: Unique Factorization Solutions 10su_math00

Handout: Lecture 1: Set Theory 10su_math00

Handout: Course Outline 10su_math00

Handout: Lecture 2: Functions and Relations 10su_math00

Handout: Lecture 3: Probability 10su_math00

Handout: Lecture 4: Axioms of Integers 10su_math00

Solutions: Extra Problems Solutions 10su_math00

Solutions: Probability Solutions 10su_math00

Solutions: Set Theory Solutions 10su_math00

Homework Assignment: Extra Problems due on 15 September 2010 10su_math00

Homework Assignment: Axioms of Integers Problems due on 15 Sep... 10su_math00

Homework Assignment: Probability Problems due on 13 September ... 10su_math00

Homework Assignment: Functions Problems due on 10 September 2010 10su_math00

Homework Assignment: Set Theory Problems due on 9 September 2010 10su_math00

August 2010

Math 00 Advanced Topics for Undergraduates (Summer 2010) Teaching | 10su_math00

07:51pm 29 Aug 2010
Instructor for Math 00 Advanced Topics for Undergraduates (Summer 2010) at University of California: Los Angeles.

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

Handout: Series Convergence Tests 10s_math31b

Handout: Final Exam Study Guide 10s_math31b

June 2010

Solutions: Final Exam Solutions 10s_math31b

Exam info: Final Exam on 7 June 2010 10s_math31b

12:25am 07 Jun 2010
The Final Exam will be on 7 June 2010 at 3pm-6pm in MS 6627.

Award: Robert Sorgenfrey Distinguished Teaching Award Mathematics

10:10pm 04 Jun 2010
Robert Sorgenfrey Distinguished Teaching Award
University of California: Los Angeles
2010

Homework Assignment: Homework #10 due on Not Due 10s_math31b

Solutions: Homework #10 Solutions 10s_math31b

May 2010

Solutions: Homework #9 Solutions 10s_math31b

Handout: Final Exam Formula Sheet 10s_math31b

Solutions: Midterm #2 Solutions 10s_math31b

Solutions: Homework #8 Solutions 10s_math31b

Solutions: Homework #7 Solutions 10s_math31b

Homework Assignment: Homework #9 due on 28 May 2010 10s_math31b

Handout: Midterm #2 Formula Sheet 10s_math31b

Handout: Practice Midterm #2 Soltuions 10s_math31b

Homework Assignment: Homework #8 due on 21 May 2010 10s_math31b

Solutions: Homework #6 Solutions 10s_math31b

Homework Assignment: Homework #7 due on 14 May 2010 10s_math31b

April 2010

Solutions: Homework #5 Solutions 10s_math31b

Homework Assignment: Homework #6 due on 7 May 2010 10s_math31b

Solutions: Midterm #1 Solutions 10s_math31b

Solutions: Homework #4 Solutions 10s_math31b

Homework Assignment: Homework #5 due on 30 April 2010 10s_math31b

Solutions: Homework #3 Solutions 10s_math31b

Handout: Practice Midterm #1 Solutions 10s_math31b

Handout: Midterm #1 Formula Sheet 10s_math31b

Handout: Rules for Computing Limits 10s_math31b

Solutions: Homework #2 Solutions 10s_math31b

Homework Assignment: Homework #4 due on 23 April 2010 10s_math31b

Handout: Compound Interest, Present Value & Annuities 10s_math31b

Homework Assignment: Homework #3 due on 16 April 2010 10s_math31b

Talk: Superstability and Finite-Time Extinction for Semigroups Mathematics

09:29pm 03 Apr 2010
Superstability and Finite-Time Extinction for Semigroups
University of California: Los Angeles
27 Apr 2010

Solutions: Homework #1 Solutions 10s_math31b

Homework Assignment: Homework #2 due on 9 April 2010 10s_math31b

March 2010

Homework Assignment: Homework #1 due on 2 April 2010 10s_math31b

Handout: Course Information 10s_math31b

Exam info: Midterm #2 on 21 May 2010 10s_math31b

05:41pm 16 Mar 2010
The Midterm #2 will be on 21 May 2010 at During Class in MS 6627.

Exam info: Midterm #1 on 23 April 2010 10s_math31b

05:41pm 16 Mar 2010
The Midterm #1 will be on 23 April 2010 at During Class in MS 6627.

Handout: Course Outline 10s_math31b

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 University of California: Los Angeles.
February 2010

Award: VIGRE Instructorship Mathematics

10:09pm 11 Feb 2010
VIGRE Instructorship
University of California: Los Angeles
2010
August 2009

Award: VIGRE Fellowship Mathematics

10:09pm 29 Aug 2009
VIGRE Fellowship
University of California: Los Angeles
2005-2009
January 2009

Talk: Poisson Boundaries and Their Applications Mathematics

09:29pm 04 Jan 2009
Poisson Boundaries and Their Applications
University of California: Los Angeles
Jan 2009
March 2007

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

09:28pm 01 Mar 2007
Rank-One Actions, Mixing and Singular Spectra
University of California: Los Angeles
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
Ergodic Theory and Dynamical Systems
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
» dcreutz.com » mathematics » All News
© 2018 Darren Creutz