BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:David Mehrle (Cornell University) DTSTART:20211027T194500Z DTEND:20211027T210000Z DTSTAMP:20260404T111134Z UID:rts/1 DESCRIPTION:Title: When free algebras are not free modules and the weird world of equivari ant algebra\nby David Mehrle (Cornell University) as part of Rochester topology seminar\n\nLecture held in Hylan 1106A.\n\nAbstract\nIn homologi cal algebra\, we take for granted that the free $\\mathbb{Z}$-algebra $\\m athbb{Z}$[x] is also free as a $\\mathbb{Z}$-module. This fact is crucial for certain computations in homotopy theory. We want to make some of these same computations in equivariant homotopy theory\, where $\\mathbb{Z}$-al gebras are replaced by incomplete Tambara functors and $\\mathbb{Z}$-modul es are replaced by Mackey functors. However\, a free incomplete Tambara fu nctor is almost never (i.e. with probability zero) free as a Mackey functo r! In this talk\, I will explain this oddity of equivariant homotopy theor y and one way to resolve it. This is joint work with Mike Hill and J.D. Qu igley.\n LOCATION:https://stable.researchseminars.org/talk/rts/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Carissa Slone (University of Kentucky) DTSTART:20211110T204500Z DTEND:20211110T220000Z DTSTAMP:20260404T111134Z UID:rts/2 DESCRIPTION:Title: Characterizing 2-slices over $C_2$ and $K_4$\nby Carissa Slone (Uni versity of Kentucky) as part of Rochester topology seminar\n\nLecture held in Hylan 1106A.\n\nAbstract\nThe slice filtration focuses on producing ce rtain irreducible spectra\, called slices\, from a genuine $G$-spectrum $X $ over a finite group $G$. We have a complete characterization of all 1-\, 0-\, and (-1)-slices for any such $G$. We will characterize 2-slices over $C_2$ and expand this characterization to $K_4 = C_2 \\times C_2$.\n\nZoo m meeting ID: 933 8835 4137\nPasscode: 212814\n LOCATION:https://stable.researchseminars.org/talk/rts/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Jeff Carlson (Imperial College London) DTSTART:20211117T204500Z DTEND:20211117T220000Z DTSTAMP:20260404T111134Z UID:rts/3 DESCRIPTION:Title: Products on Tor\, homogeneous spaces\, and Borel cohomology\nby Jef f Carlson (Imperial College London) as part of Rochester topology seminar\ n\nLecture held in Hylan 1106A.\n\nAbstract\nThe Eilenberg-Moore spectral sequence converges from the classical Tor of a span of cohomology rings to the differential Tor of a span of cochain algebras (which is the cohomolo gy of the homotopy pullback). These are both rings\, the first classically and the second as a corollary of the Eilenberg-Zilber theorem. \n\nOne mi ght well ask when a more general differential Tor of DGAs admits a ring st ructure\, though apparently no one did. We will show that when the DGAs in question admit a certain sort of $E_3$-algebra structure\, Tor is a commu tative graded algebra. \n\nWe have not done this out of an innocent intere st in homotopy-commutative algebras. In 1960s and '70s there was a flurry of activity developing A-infinity-algebraic techniques with an aim toward computing the Eilenberg–Moore spectral sequence (for example\, of a loop space or homogeneous space). Arguably the most powerful result this progr am produced was the 1974 theorem of Munkholm that the sequence collapses w hen the three input spaces have polynomial cohomology over a given princip al ideal domain\, which however only gives the story on cohomology groups. Our result shows that Munkholm's map is in fact an isomorphism of rings. \n\nThe proof hinges on homotopy properties of the (1-)category of augment ed DGAs. This work is all joint with several large commutative diagrams\, who should be considered the true authors.\n\nZoom meeting ID: 988 2359 98 95 passcode: 553391\n LOCATION:https://stable.researchseminars.org/talk/rts/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Cihan Okay (Bilkent University) DTSTART:20211203T193000Z DTEND:20211203T210000Z DTSTAMP:20260404T111134Z UID:rts/4 DESCRIPTION:Title: Homotopy classification of operator solutions of linear systems\nby Cihan Okay (Bilkent University) as part of Rochester topology seminar\n\n Lecture held in Hylan 1106A.\n\nAbstract\nLinear systems of equations over a finite field play an important role in quantum information theory. Inst ead of looking for solutions over the base field one can look for solution s (in a certain sense) over the unitary group\, which are called operator solutions. The data of this system of equations can be expressed using a h ypergraph and the operator solutions can be studied from a topological poi nt of view by considering certain topological realizations of these hyperg raphs. In this talk I will describe how homotopical methods provide a way to classify operator solutions of linear systems. Our basic approach is to interpret operator solutions as maps from a topological realization of th e hypergraph to a certain classifying space first introduced by Adem-Cohen -Torres-Giese.\n\nZoom Meeting ID: 972 4663 8781 Passcode: 031045\n LOCATION:https://stable.researchseminars.org/talk/rts/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Matt Feller (University of Virginia) DTSTART:20211208T204500Z DTEND:20211208T220000Z DTSTAMP:20260404T111134Z UID:rts/5 DESCRIPTION:Title: Generalizing quasi-categories via model structures on simplicial sets\nby Matt Feller (University of Virginia) as part of Rochester topology seminar\n\nLecture held in Hylan 1106A.\n\nAbstract\nQuasi-categories are particular simplicial sets which behave like categories up to homotopy. Th eir theory has been massively developed in the past two decades\, thanks l argely due to Joyal and Lurie\, and they have become vital tools in many a reas of algebraic topology\, algebraic geometry\, and beyond. Due to the s uccess of quasi-categories\, it would be nice to extend the theory to up-t o-homotopy versions of objects more general than categories\, such as the 2-Segal sets of Dyckerhoff-Kapranov and Gàlvez-Kock-Tonks. Such a general ization would ideally come with an associated model structure on the categ ory of simplicial sets\, but finding a model structure with a more general class of fibrant objects than a given model structure is a nontrivial and open-ended task. In this talk\, I will explain how to use Cisinski's mach inery to construct model structures on the category of simplicial sets who se fibrant objects generalize quasi-categories. In particular\, one of the se model structures has fibrant objects precisely the simplicial sets that satisfy a lifting condition which captures the homotopical behavior of qu asi-categories without the algebraic aspects.\n\nZoom Meeting ID: 954 8701 7543\nPasscode: 123708\n LOCATION:https://stable.researchseminars.org/talk/rts/5/ END:VEVENT END:VCALENDAR