The seminar takes place on Mondays, 4:10-5:30pm, in Stevenson 1320. (Note that we are no longer meeting in Stevenson 1308.)<\/p>\n
April 15, 2019 <\/p>\n
Title: Property (T) and $\\epsilon$-orthogonal subgroups<\/strong> <\/p>\n Speaker: Krishnendu Khan (Vanderbilt University)<\/p>\n Abstract: In this talk we’ll be discussing the notion of angle between subgroups of group G introduced by M. Kassabov and how the small angle (small orthogonality) helps to lift relative property (T) of G with respect to generating subgroups to the entire group G (result of M. Ershov and A. Jaikin-Japirain). (This is Lecture 10 in a semester-long series on SL(3,Z) and Property (T).) <\/p>\n April 8, 2019 <\/p>\n Seminar rescheduled for next week<\/p>\n April 1, 2019 <\/p>\n Title: Property (T) without bounded generation<\/strong> <\/p>\n Speaker: Srivatsav Kunnawalkam Elayavalli (Vanderbilt University)<\/p>\n Abstract: We will present the work of Mimura proving (T) for the elementary groups EL_n(R) which uses ideas of Shalom’s argument but does it without the bounded generation aspects. (This is Lecture 9 in a semester-long series on SL(3,Z) and Property (T).) <\/p>\n March 25, 2019 <\/p>\n Title: Bounded generation by subgroups and Property (T)<\/strong> <\/p>\n Speaker: Srivatsav Kunnawalkam Elayavalli (Vanderbilt University)<\/p>\n Abstract: I will continue our ongoing discussion and discuss Shalom’s proof of (T) for SL(3,Z). We will prove a more general result that any group G that contains three subgroups H, K_1 and K_2 satisfying some conditions (and K_1 and K_2 possessing relative property T with respect to G), has Property (T). (This is Lecture 8 in a semester-long series on SL(3,Z) and Property (T).) <\/p>\n March 18, 2019 <\/p>\n Title: Property (T), affine actions and (reduced) cohomology, Part III<\/strong> <\/p>\n Speaker: Jesse Peterson (Vanderbilt University)<\/p>\n Abstract: We will present several equivalent conditions for property (T) in terms of affine actions and (reduced) cohomology. This follows the work of Delorme, Guichardet, Shalom, and Ozawa. (This is Lecture 7 in a semester-long series on SL(3,Z) and Property (T).) <\/p>\n March 11, 2019 <\/p>\n Title: Property (T), affine actions and (reduced) cohomology, Part II<\/strong> <\/p>\n Speaker: Jesse Peterson (Vanderbilt University)<\/p>\n Abstract: We will present several equivalent conditions for property (T) in terms of affine actions and (reduced) cohomology. This follows the work of Delorme, Guichardet, Shalom, and Ozawa. (This is Lecture 6 in a semester-long series on SL(3,Z) and Property (T).) <\/p>\n March 4, 2019 <\/p>\n No seminar (spring break)<\/p>\n February 25, 2019 — talk will begin at 4:30pm instead of the usual 4:10pm<\/em><\/p>\n Title: Property (T), affine actions and (reduced) cohomology, Part I<\/strong> <\/p>\n Speaker: Jesse Peterson (Vanderbilt University)<\/p>\n Abstract: We will present several equivalent conditions for property (T) in terms of affine actions and (reduced) cohomology. This follows the work of Delorme, Guichardet, Shalom, and Ozawa. (This is Lecture 5 in a semester-long series on SL(3,Z) and Property (T).) <\/p>\n February 18, 2019<\/p>\n No seminar (time conflict with a grad student’s oral qualifying exam)<\/p>\n February 11, 2019<\/p>\n Title: Lattices in SL(n,R) and Kazhdan\u2019s Property (T)<\/strong> <\/p>\n Speaker: Jun Yang (Vanderbilt University)<\/p>\n 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).) <\/p>\n February 4, 2019<\/p>\n Title: The Howe–Moore Property<\/strong> <\/p>\n Speaker: Srivatsav Kunnawalkam Elayavalli (Vanderbilt University)<\/p>\n 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).) <\/p>\n January 28, 2019<\/p>\n Title: Relative Property (T) for semi-direct products<\/strong> <\/p>\n Speaker: Brent Nelson (Vanderbilt University)<\/p>\n 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).) <\/p>\n Sources:<\/p>\n Cornulier and Tessera, A characterization of relative Kazhdan property T for semidirect products with abelian groups<\/a>, Ergodic Theory and Dynamical Systems, 2010.<\/p>\n Shalom, Bounded generation and Kazhdan’s property (T)<\/a>, Publications Math\u00e9matiques de l’IH\u00c9S, 1999.<\/p>\n Chifan and Ioana, On relative property (T) and Haagerup’s property<\/a>, Transactions of the AMS, 2011. <\/p>\n Peterson, Notes on operator algebras<\/a>, online lecture notes, 2015.<\/p>\n January 21, 2019<\/p>\n No seminar (Martin Luther King Jr. Day)<\/p>\n January 14, 2019<\/p>\n Title: Equivalent definitions of Property (T)<\/strong> <\/p>\n Speaker: Lauren C. Ruth (Vanderbilt University)<\/p>\n Abstract: We will give some definitions of Property (T) and show that they are equivalent. (This is Lecture 1 in a semester-long series on SL(3,Z) and Property (T).) <\/p>\n Source:<\/p>\n Bekka, de la Harpe, Valette, Kazhdan’s Property (T)<\/a>, online book, 2007.<\/p>\n January 7, 2019<\/p>\n Title: Organizational meeting: Semester on SL(3,Z) and Property (T)<\/strong> <\/p>\n 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. <\/p>\n (Winter Break)<\/p>\n December 3, 2018<\/p>\n Speaker: Srivatsav Kunnawalkam Elayavalli (Vanderbilt University)<\/p>\n Title: Group Stability and Property (T)<\/strong><\/p>\n 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.<\/p>\n Source: <\/p>\n Becker and Lubotzky, Group stability and Property (T)<\/a>, arXiv, 2018.<\/p>\n November 26, 2018<\/p>\n Speaker: Brent Nelson (Vanderbilt University)<\/p>\n Title: Crossed products, dual weights, and Takesaki-duality<\/strong><\/p>\n 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.<\/p>\n Source: <\/p>\n Takesaki, Theory of Operator Algebras II<\/a>, Chapters 7 and 10.<\/p>\n November 19, 2018<\/p>\n No seminar (Thanksgiving break)<\/p>\n November 12, 2018<\/p>\n Speaker: Jesse Peterson (Vanderbilt University)<\/p>\n Title: The Furstenberg boundary and C*-simplicity<\/strong><\/p>\n 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.<\/p>\n November 5, 2018<\/p>\n Speaker: Krishnendu Khan (Vanderbilt University)<\/p>\n Title: Bi-exact groups and Lacunary hyperbolic groups<\/strong><\/p>\n 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. <\/p>\n October 29, 2018<\/p>\n No seminar (time conflict with special Subfactor Seminar<\/a>)<\/p>\n October 22, 2018<\/p>\n Speaker:\u00a0 Lauren C. Ruth (Vanderbilt University)<\/p>\n Title:\u00a0\u00a0Equivalent definitions of invariant random subgroups<\/strong><\/p>\n Abstract:\u00a0\u00a0An 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 Ab\u00e9rt–Bergeron–Biringer–Gelander–Nikolov–Raimbault–Samet (2017), after mentioning the proof for discrete groups in Ab\u00e9rt–Glasner–Vir\u00e1g (2014).<\/p>\n Sources:<\/p>\n Ab\u00e9rt, Glasner, and Vir\u00e1g, Kesten’s theorem for invariant random subgroups<\/a>, Duke Mathematical Journal, 2014.<\/p>\n Creutz and Peterson, Stabilizers of ergodic actions of lattices and commensurators<\/a>, Transactions of the AMS, 2016.<\/p>\n Ab\u00e9rt, Bergeron, Biringer, Gelander, Nikolov, Raimbault, and Samet, On the growth of L\u00b2-invariants for sequences of lattices in Lie groups<\/a>, Annals of Mathematics, 2017.<\/p>\n October 15, 2018<\/p>\n Speaker:\u00a0 Srivatsav\u00a0Kunnawalkam Elayavalli (Vanderbilt University)<\/p>\n Title:\u00a0\u00a0On Amenability and Tubularity<\/strong><\/p>\n Abstract:\u00a0\u00a0We 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.<\/p>\n Source:<\/p>\n Jung, Amenability, tubularity, and embeddings into $\\mathcal{R}^\\omega$<\/a>, arXiv 2006.<\/p>\n October 8, 2018<\/p>\n No seminar.<\/p>\n October 1, 2018<\/p>\n Speaker: Jesse Peterson (Vanderbilt University)<\/p>\n Title: The Furstenberg boundary and C*-simplicity<\/strong><\/p>\n 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.<\/p>\n September 24, 2018<\/p>\n Speaker:\u00a0 Lauren C. Ruth\u00a0(Vanderbilt University)<\/p>\n Title:\u00a0\u00a0Results on boundary actions, cont’d<\/strong><\/p>\n Abstract:\u00a0 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.\u00a0 This time, we will take a closer look at the maximal G-boundary, also called the Furstenberg boundary.\u00a0 After showing existence and uniqueness,\u00a0we will prove\u00a0that G acts faithfully on its Furstenberg boundary if and only if\u00a0its amenable radical is trivial, following notes of Ozawa.\u00a0 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.<\/p>\n Sources:<\/p>\n Haagerup,\u00a0A new look at C*-simplicity and the unique trace property of a group<\/a>, arXiv 2016.<\/p>\n Ozawa,\u00a0Lecture on the Furstenburg boundary and C*-simplicity<\/a>, online lecture notes.<\/p>\n September 17, 2018<\/p>\n Speaker:\u00a0 Lauren C. Ruth\u00a0(Vanderbilt University)<\/p>\n Title:\u00a0 A\u00a0result of Furman on boundary actions<\/strong><\/p>\n Abstract:\u00a0 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.<\/p>\n Sources:<\/p>\n Furman, On minimal, strongly proximal actions of locally compact groups<\/a>, Israel Journal of Mathematics 2003.<\/p>\n Tao, Hilbert’s Fifth Problem and Related Topics<\/a>, online book.<\/p>\n September 10, 2018<\/p>\n Speaker:\u00a0 Srivatsav\u00a0Kunnawalkam Elayavalli (Vanderbilt University)<\/p>\n Title:\u00a0\u00a01-Bounded Entropy, Part I: Free entropy for pedestrians<\/strong><\/p>\n Abstract:\u00a0\u00a0In 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.<\/p>\n Sri’s lecture notes:<\/p>\n 1-bounded entropy and applications, Part I<\/a><\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" The seminar takes place on Mondays, 4:10-5:30pm, in Stevenson 1320. (Note that we are no longer meeting in Stevenson 1308.) April 15, 2019 Title: Property (T) and $\\epsilon$-orthogonal subgroups Speaker: Krishnendu Khan (Vanderbilt University) Abstract: In this talk we’ll be … Continue reading
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