The seminar takes place on Mondays, 4:10-5:30pm, in Stevenson 1320. (Note that we are no longer meeting in Stevenson 1308.)

February 18, 2019

No seminar (time conflict with a grad student’s oral qualifying exam)

February 11, 2019

Title: **Lattices in SL(n,R) and Kazhdan’s Property (T)**

Speaker: Jun Yang (Vanderbilt University)

Abstract: First, we will prove that Property (T) is inherited by lattices. Then we will show SL(n,Z) is a lattice in SL(n,R). (This is Lecture 4 in a semester-long series on SL(3,Z) and Property (T).)

February 4, 2019

Title: **The Howe–Moore Property**

Speaker: Srivatsav Kunnawalkam Elayavalli (Vanderbilt University)

Abstract: We will state the Howe–Moore property and prove it for SL(n,R). (This is Lecture 3 in a semester-long series on SL(3,Z) and Property (T).)

January 28, 2019

Title: **Relative Property (T) for semi-direct products**

Speaker: Brent Nelson (Vanderbilt University)

Abstract: We will define relative property (T) for semi-direct products and show that $(\mathbb{Z}^2, SL_2(\mathbb{Z}) \ltimes \mathbb{Z}^2)$ and $(\mathbb{R}^2, SL_2(\mathbb{Z}) \ltimes \mathbb{R}^2)$ satisfy this definition. (This is Lecture 2 in a semester-long series on $SL_3(\mathbb{Z})$ and Property (T).)

Sources:

Cornulier and Tessera, A characterization of relative Kazhdan property T for semidirect products with abelian groups, Ergodic Theory and Dynamical Systems, 2010.

Shalom, Bounded generation and Kazhdan’s property (T), Publications Mathématiques de l’IHÉS, 1999.

Chifan and Ioana, On relative property (T) and Haagerup’s property, Transactions of the AMS, 2011.

Peterson, Notes on operator algebras, online lecture notes, 2015.

January 21, 2019

No seminar (Martin Luther King Jr. Day)

January 14, 2019

Title: **Equivalent definitions of Property (T)**

Speaker: Lauren C. Ruth (Vanderbilt University)

Abstract: We will give some definitions of Property (T) and show that they are equivalent. (This is the first talk in a semester-long series on SL(3,Z) and Property (T).)

Source:

Bekka, de la Harpe, Valette, Kazhdan’s Property (T), online book, 2007.

January 7, 2019

Title: **Organizational meeting: Semester on SL(3,Z) and Property (T)**

Abstract: This semester, we will be running a special series featuring different proofs that SL(3,Z) has Property (T). At this organizational meeting, we will list the known proofs and provide references, and participants can choose which proofs they would like to present.

(Winter Break)

December 3, 2018

Speaker: Srivatsav Kunnawalkam Elayavalli (Vanderbilt University)

Title: **Group Stability and Property (T)**

Abstract: We present the very recent paper of Becker and Lubotzky with the above title. The various notions of group approximations will be considered along with several examples. The main theorem linking group stability and property (T) will be discussed.

Source:

Becker and Lubotzky, Group stability and Property (T), arXiv, 2018.

November 26, 2018

Speaker: Brent Nelson (Vanderbilt University)

Title: **Crossed products, dual weights, and Takesaki-duality**

Abstract: Given an action of a locally compact abelian group G on a von Neumann algebra M, one can form the crossed product von Neumann algebra. This von Neumann algebra contains both M and L(G), and the group action is encoded via commutation relations. If M is equipped with a faithful normal state, then the crossed product is naturally equipped with a dual weight. Moreover, the crossed product also admits an action of the dual group to G. Takesaki-duality states that if one repeats the crossed product construction with this dual group action, then one obtains a tensor product of the original von Neumann algebra M and B(L^2(G)). I will given an overview of these concepts and results.

Source:

Takesaki, Theory of Operator Algebras II, Chapters 7 and 10.

November 19, 2018

No seminar (Thanksgiving break)

November 12, 2018

Speaker: Jesse Peterson (Vanderbilt University)

Title: **The Furstenberg boundary and C*-simplicity**

Abstract: We will continue the discussion from October 1. In particular, we will discuss Kalantar and Kennedy’s result that a group is C*-simple if and only if the action on its Furstenberg boundary is free, and we will discuss Breuillard, Kalantar, Kennedy, and Ozawa’s result that a group C*-algebra has unique trace if and only if the action on its Furstenberg boundary if faithful.

November 5, 2018

Speaker: Krishnendu Khan (Vanderbilt University)

Title: **Bi-exact groups and Lacunary hyperbolic groups**

Abstract: In this talk I’ll talk about examples of bi-exact groups arising from geometric group theory shown by Ozawa and a subclass of lacunary hyperbolic groups constructed by Olshanskii, Osin, Sapir.

October 29, 2018

No seminar (time conflict with special Subfactor Seminar)

October 22, 2018

Speaker: Lauren C. Ruth (Vanderbilt University)

Title: **Equivalent definitions of invariant random subgroups**

Abstract: An invariant random subgroup (IRS) of a locally compact second-countable group G is a conjugation-invariant probability measure on the space of subgroups of G. We will show how every IRS arises from a measure-preserving action on a probability space via the stabilizer map, comparing the proofs in Creutz–Peterson (2016) and Abért–Bergeron–Biringer–Gelander–Nikolov–Raimbault–Samet (2017), after mentioning the proof for discrete groups in Abért–Glasner–Virág (2014).

Sources:

Abért, Glasner, and Virág, Kesten’s theorem for invariant random subgroups, Duke Mathematical Journal, 2014.

Creutz and Peterson, Stabilizers of ergodic actions of lattices and commensurators, Transactions of the AMS, 2016.

Abért, Bergeron, Biringer, Gelander, Nikolov, Raimbault, and Samet, On the growth of L²-invariants for sequences of lattices in Lie groups, Annals of Mathematics, 2017.

October 15, 2018

Speaker: Srivatsav Kunnawalkam Elayavalli (Vanderbilt University)

Title: **On Amenability and Tubularity**

Abstract: We shall discuss the notion of tubularity due to Jung, and discuss about how this provides an interesting geometric characterization of injectivity for von Neumann algebras.

Source:

Jung, Amenability, tubularity, and embeddings into $\mathcal{R}^\omega$, arXiv 2006.

October 8, 2018

No seminar.

October 1, 2018

Speaker: Jesse Peterson (Vanderbilt University)

Title: **The Furstenberg boundary and C*-simplicity**

Abstract: We will continue the discussion the last two weeks by Lauren Ruth. In particular, we will discuss Kalantar and Kennedy’s result that a group is C*-simple if and only if the action on its Furstenberg boundary is free, and we will discuss Breuillard, Kalantar, Kennedy, and Ozawa’s result that a group C*-algebra has unique trace if and only if the action on its Furstenberg boundary if faithful.

September 24, 2018

Speaker: Lauren C. Ruth (Vanderbilt University)

Title: **Results on boundary actions, cont’d**

Abstract: Last time, we defined boundary actions and saw how the Gleason–Yamabe theorem (crucial in Montgomery–Zippin’s solution to Hilbert’s 5th Problem) gives rise to the group splitting in Furman’s result. This time, we will take a closer look at the maximal G-boundary, also called the Furstenberg boundary. After showing existence and uniqueness, we will prove that G acts faithfully on its Furstenberg boundary if and only if its amenable radical is trivial, following notes of Ozawa. Time permitting, we will go through Haagerup’s proof of Breuillard–Kalantar–Kennedy–Ozawa’s result that the amenable radical of G is trivial if and only if G has the unique trace property.

Sources:

Haagerup, A new look at C*-simplicity and the unique trace property of a group, arXiv 2016.

Ozawa, Lecture on the Furstenburg boundary and C*-simplicity, online lecture notes.

September 17, 2018

Speaker: Lauren C. Ruth (Vanderbilt University)

Title: **A result of Furman on boundary actions**

Abstract: After giving the definition and examples of boundary actions, we will sketch Furman’s proof that if a boundary action of a locally compact group H satisfies certain hypotheses (related to the “no small subgroups” property), then we may conclude that either H is discrete infinite countable, or H is a connected semi-simple real Lie group with trivial center.

Sources:

Furman, On minimal, strongly proximal actions of locally compact groups, Israel Journal of Mathematics 2003.

Tao, Hilbert’s Fifth Problem and Related Topics, online book.

September 10, 2018

Speaker: Srivatsav Kunnawalkam Elayavalli (Vanderbilt University)

Title: **1-Bounded Entropy, Part I: Free entropy for pedestrians**

Abstract: In this talk I aim to discuss the notions of free entropy and free entropy dimension invented by Dan Voiculescu. Time permitting, we will also talk about free group factors and other wild animals. We will not go in depth into the proofs, but refer to the attached notes for a careful exposition.

Sri’s lecture notes:

1-bounded entropy and applications, Part I