With the support of this grant, the PI solved a conjecture of Arone, Dwyer, Kuhn, Lesh, and Mahowald for the prime 2 - that the Goodwillie spectral sequence for the circle collapses at the E₂-page.  Working with Charles Rezk, the PI established a generalized version of Quillen and Sullivan's rational homotopy theory which applies to unstable v_n-periodic homotopy.  With Kyle Ormsby, the PI performed new computations in the theory of topological modular forms (tmf) with level structures.  The ring of cooperations for tmf was studied in collaboration with Ormsby, Nathaniel Stapleton, and Vesna Stojanoska.  This paved the way for the study of the Adams spectral sequence based on tmf.  With Agnes Beaudry, Prasit Bhattacharya, Dominic Culver, and Zhouli Xu, the PI developed a method to study Adams spectral sequences based on non-flat cohomology theories.  This technique was employed in the case of connective K-theory to completely compute the corresponding Adams spectral sequence for the sphere spectrum through the 40-stem.  This technique was also adapted to the study of the tmf-based Adams spectral sequence of a certain type 2 complex.  In this context, the PI and his collaborators proved an analog of Mahowald's "Bounded Torsion Theorem", isolating the possibilitites for counterexamples to the telescope conjecture at chromatic height 2.  The PI and his collaborators also performed low dimensional computations of this tmf-based Adams spectral sequence.  These low dimensional computations prove a spectral sequence studied by Bhattacharya-Egger for the K(2)-local homotopy groups of the aforementioned type 2 complex collapses.  This results in the first example of a computation of the K(2)-local homotopy groups of a type 2 complex at the prime 2.  The tmf-based Adams spectral sequence was also employed by the PI, with Michael Hill, Michael Hopkins, and Mark Mahowald, to show that the only even dimensions d less than 140 for which the d-dimensional sphere  admits a unique differentiable structure are 2, 6, 12, 56, and perhaps 4 (the odd dimensional cases were solved by a combination of the work of Kervaire, Milnor, Hill, Hopkins, Ravenel, Xu, and Wang).  With the support of this grant, the PI created a mentoring program at MIT to increase access to graduate study in mathematics to underrepresented groups.  The PI also started a bridge program for incoming graduate students to Notre Dame to help equalize the program for students coming from a diverse range of undergraduate institutions.  The PI advised seven PhD students with the support of this grant, and coorganized five conferences.

Last Modified: 12/30/2017
Modified by: Mark J Behrens