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Upcoming Defenses

Ph.D. Program in Mathematics upcoming Defense of Dissertations:

Note:  If you are interested in “attending” a remote defense, send an email to the student and/or Dissertation Chair (Advisor) for remote access details.  If no response, contact Mathematics@gc.cuny.edu. 

 
Name: Qian Chen
Title: Spectral Sequences For Almost Complex Manifolds
Defense Date: 8/28/2020
Time: 1pm-3pm
Location: Zoom (Contact Student or Committee Chair for link)
 
Committee Chair: Scott Wilson, The Graduate Center & Queens College
Committee Member 2: Luis Fernandez The Graduate Center &  Bronx Community College
Committee Member 3: Bianca Santoro  The Graduate Center & City College
 
Abstract:  In recent work, two new cohomologies were introduced for almost complex manifolds: the so-called J-cohomology and N-cohomology, [CKT17]. For the case of integrable (complex) structures, the former cohomology was already considered in [DGMS75], and the latter agrees with de Rham cohomology. In this dissertation, using ideas from [JC18], we introduce spectral sequences for these two cohomologies, showing the two cohomologies have natural bigradings. We show the spectral sequence for the J-cohomology converges at the second page whenever the almost complex structure is integrable, and explain how both fit in a natural diagram involving Bott-Chern cohomology and the Fr ̀ˆolicher spectral sequence. Using explicit formulas that we derive for the pages, as well as topology in some cases, we deduce several properties of the groups and the natural maps in various degrees. As applications, we study the Kodaira- Thurston and Iwasawa manifolds, as well as a hypothetical complex structure of the six-sphere. 

Name: Christopher Natoli
Title: Some Model Theory of Free Groups
Defense Date: Tuesday, June 30, 2020
Time: 2pm-4pm
Location: Zoom (Contact Student or Committee Chair for link)
 
Committee Chair: Olga Kharlampovich, The Graduate Center & Hunter College
Committee Member 2: Ilya Kapovich, The Graduate Center & Hunter College
Committee Member 3: Vladimir Shpilrain, The Graduate Center & City College
 
Abstract:  There are two main sets of results, both pertaining to the model theory of free groups. In the first set of results, we prove that non-abelian free groups of finite rank at least 3 or of countable rank are not universally homogeneous. We then build on the proof of this result to show that two classes of groups, namely finitely generated free groups and finitely generated elementary free groups, fail to form universal Fraisse classes and that the class of non-abelian limit groups fails to form a strong universal Fraisse class. The second main result is that if a countable group is elementarily equiv-alent to a non-abelian free group and all of its finitely generated abelian subgroups are cyclic, then the group is a union of a chain of regular NTQ groups (i.e., hyperbolic towers).