BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Duncan Clark (Ohio State University) DTSTART:20200606T181000Z DTEND:20200606T191000Z DTSTAMP:20260423T022734Z UID:GOATS/4 DESCRIPTION:Title: On the Goodwillie Derivatives of the Identity in Structured Ring Spectra \nby Duncan Clark (Ohio State University) as part of GOATS\n\n\nAbstract\n Functor calculus was introduced by Goodwillie as a means for analyzing hom otopy functors between suitable model categories. \n One noteworthy fac et is that "nice" functors $F\\colon \\mathsf{C}\\to \\mathsf{D}$ are dete rmined by a certain symmetric sequence called the derivatives of $F$. \n\n This sequence of derivatives is known to posses much structure: for in stance\, the derivatives of the identity functor on the category of based topological spaces is an operad\, as first shown by Ching. \n It is fur ther expected that a result of this type should hold in any suitable model category\, and in particular conjectured that the derivatives of the iden tity on the category of algebras over an operad $\\mathcal{O}$ in spectra should be equivalent to $\\mathcal{O}$ as operads. \n\n In this talk we produce an intrinsic "homotopy-coherent" operad structure for the derivat ives of the identity which is equivalent to that on $\\mathcal{O}$\, thus resolving the above conjecture. \n Along the way we will discuss the ne cessary background of functor calculus and algebras over operads of spectr a. \n Our method is to induce a homotopy coherent operadic pairing on t he derivatives by a suitable pairing on the cosimplicial resolution offere d by the stabilization adjunction for $\\mathcal{O}$-algebras. \n \n Time permitting\, we will provide some other applications of our techniqu es such as a highly homotopy-coherent chain rule for functors of structure d ring spectra.\n LOCATION:https://researchseminars.org/talk/GOATS/4/ END:VEVENT END:VCALENDAR