\[\newcommand{\KO}{\mathrm{KO}} \newcommand{\TMF}{\mathrm{TMF}} \newcommand{\KU}{\mathrm{KU}} \newcommand{\Mfg}{M_\mathrm{fg}} \newcommand{\spec}{\mathrm{Spec}} \newcommand{\dX}{\mathcal{X}} \newcommand{\co}{\mathcal{O}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\top}{\mathrm{top}} \newcommand{\E}{\mathbf{E}} \newcommand{\GG}{\mathbf{G}} \newcommand{\DD}{\mathbf{D}} \newcommand{\Mell}{M_\mathrm{ell}} \newcommand{\ce}{\mathcal{E}} \newcommand{\H}{\mathrm{H}} \newcommand{\dR}{\mathrm{dR}} \newcommand{\cf}{\mathcal{F}} \newcommand{\FF}{\mathbf{F}} \newcommand{\spf}{\mathrm{Spf}\ } \newcommand{\cA}{\mathcal{A}} \newcommand{\Mcub}{M_\mathrm{cub}} \newcommand{\Mqd}{M_\mathrm{qd}} \newcommand{\Aut}{\mathrm{Aut}} \newcommand{\tmf}{\mathrm{tmf}} \newcommand{\Tmf}{\mathrm{Tmf}} \newcommand{\Map}{\mathrm{Map}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\End}{\mathrm{End}} \newcommand{\cg}{\mathcal{G}} \newcommand{\Hom}{\mathrm{Hom}} \newcommand{BP}[1]{\mathrm{BP}\langle #1\rangle} \newcommand{\QQ}{\mathbf{Q}} \newcommand{\PP}{\mathbf{P}} \newcommand{\Gal}{\mathrm{Gal}} \newcommand{\proj}{\mathrm{Proj}\ } \newcommand{\Ltwo}{L_{K(2)}} \newcommand{\Eoo}{\mathbf{E}_\infty} \newcommand{\ol}[1]{\overline #1} \newcommand{\pic}{\mathrm{Pic}} \newcommand{\RP}{\mathbf{R}P} \newcommand{\BO}{\mathrm{BO}} \newcommand{\cc}{\mathbf{C}} \newcommand{\CP}{\mathbf{CP}} \newcommand{\wt}[1]{\widetilde #1} \newcommand{\RR}{\mathbf{R}} \renewcommand{\S}{\mathbb{S}} \newcommand{\Lone}{L_{K(1)}} \newcommand{\Ltwo}{L_{K(2)}} \newcommand{\gl}{\mathrm{gl}} \newcommand{\Lk}{L_{K(n)}} \newcommand{\ku}{\mathrm{ku}} \newcommand{\ko}{\mathrm{ko}} \newcommand{\GL}{\mathrm{GL}} \newcommand{\Ln}{L_n} \newcommand{\Lt}{L_{K(t)}} \newcommand{\W}{\mathrm{W}} \newcommand{\BU}{\mathrm{BU}} \newcommand{\SU}{\mathrm{SU}} \newcommand{\MU}{\mathrm{MU}} \newcommand{\ul}[1]{\underline #1} \newcommand{\Sq}{\mathrm{Sq}} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\d}{\mathrm{d}} \newcommand{\HP}{\mathrm{HP}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cs}{\mathrm{cs}} \newcommand{\cd}{\mathcal{D}} \newcommand{\Bord}{\mathrm{Bord}} \newcommand{\SO}{\mathrm{SO}} \newcommand{\cC}{\mathcal{C}} \newcommand{\Sp}{\mathrm{Sp}} \newcommand{\Lkd}{L_{K(d)}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\cH}{\mathcal{H}} \newcommand{\ket}[1]{|#1 \rangle} \newcommand{\Bun}{\mathrm{Bun}} \newcommand{\cM}{\mathcal{M}} \newcommand{\MTH}{\mathrm{MT}H} \newcommand{\MSO}{\mathrm{MSO}} \newcommand{\twsm}{\ \widetilde{\wedge} \ } \newcommand{\BSpin}{\mathrm{BSpin}} \newcommand{\MSpin}{\mathrm{MSpin}} \newcommand{\BString}{\mathrm{BString}} \newcommand{\MString}{\mathrm{MString}} \newcommand{\B}{\mathrm{B}} \newcommand{\der}{\mathrm{der}} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\QCoh}{\mathrm{QCoh}} \newcommand{\LMod}{\mathrm{LMod}} \newcommand{\Rep}{\mathrm{Rep}} \newcommand{\HH}{\mathrm{HH}} \newcommand{\ZZ}{\mathfrak{Z}} \newcommand{\fr}{\mathrm{fr}} \newcommand{\BSO}{\mathrm{BSO}} \newcommand{\BSU}{\mathrm{BSU}} \newcommand{\id}{\mathrm{id}} \newcommand{\Top}{\mathrm{Top}} \newcommand{\cP}{\mathcal{P}} \newcommand{\U}{\mathrm{U}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\coker}{\mathrm{coker}} \newcommand{\Vect}{\mathrm{Vect}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\Res}{\mathrm{Res}} \newcommand{\Ind}{\mathrm{Ind}} \newcommand{\MO}{\mathrm{MO}} \newcommand{\MSO}{\mathrm{MSO}} \newcommand{\colim}{\mathrm{colim}} \newcommand{\hocolim}{\mathrm{hocolim}} \newcommand{\fin}{\mathrm{fin}} \newcommand{\Tot}{\mathrm{Tot}} \newcommand{\im}{\mathrm{im}} \newcommand{\FT}{\mathrm{FT}} \newcommand{\AA}{\mathbf{A}} \newcommand{\Frob}{\mathrm{Frob}} \newcommand{\ex}{\mathrm{ex,pr}} \newcommand{\fil}{\mathrm{fil}} \newcommand{\gr}{\mathrm{gr}} \newcommand{\un}{\mathrm{un}} \newcommand{\Syn}{\mathrm{Syn}} \newcommand{\Cart}{\mathrm{Cart}} \newcommand{\bo}{\mathrm{bo}} \newcommand{\mmod}{/\!\!/} \newcommand{\bsp}{\mathrm{bsp}} \newcommand{\BPP}{\mathrm{BP}} \newcommand{\CAlg}{\mathrm{CAlg}} \newcommand{\TCart}{\mathrm{TCart}} \newcommand{\BW}{\mathrm{BW}} \newcommand{\Br}{\mathrm{Br}} \newcommand{\Pic}{\mathrm{Pic}} \newcommand{\ext}{\mathrm{ext}} \newcommand{\Cl}{\mathrm{Cl}} \newcommand{\disc}{\mathrm{disc}} \newcommand{\dir}{D\!\!\!\!/} \newcommand{\bb}{\mathrm{bb}} \newcommand{\Ran}{\mathrm{Ran}} \newcommand{\cu}{\mathcal{U}} \newcommand{\Gr}{\mathrm{Gr}} \newcommand{\Alg}{\mathrm{Alg}} \newcommand{\G}{\Gamma} \newcommand{\dY}{\mathcal{Y}} \newcommand{\Line}{\mathrm{Line}} \newcommand{\MUP}{\mathrm{MUP}} \newcommand{\Bl}{\mathrm{Bl}} \newcommand{\St}{\mathrm{St}} \newcommand{\cB}{\mathcal{B}} \newcommand{\SL}{\mathrm{SL}} \newcommand{\XX}{\mathrm{X}} \newcommand{\gp}{\mathrm{gp}} \newcommand{\Q}{\mathrm{Q}} \newcommand{\cI}{\mathcal{I}} \newcommand{\T}{\mathrm{T}} \newcommand{\D}{\mathrm{D}} \newcommand{\Aff}{\mathrm{Aff}} \newcommand{\BGL}{\mathrm{BGL}} \newcommand{\an}{\mathrm{an}} \newcommand{\Tel}{\mathrm{Tel}} \renewcommand{\|}{|} \newcommand{\MGL}{\mathrm{MGL}} \newcommand{\CH}{\mathrm{CH}} \newcommand{\fr}[1]{\mathfrak #1} \newcommand{\ch}{\mathrm{ch}} \newcommand{\Td}{\mathrm{Td}} \newcommand{\Spin}{\mathrm{Spin}} \newcommand{\G}{\mathrm{G}} \newcommand{\cHH}{\mathcal{H}} \newcommand{\F}{\mathrm{F}} \newcommand{\B}{\mathrm{B}} \newcommand{\HC}{\mathrm{HC}} \newcommand{\fib}{\mathrm{fib}} \newcommand{\tr}{\mathrm{tr}} \newcommand{\TR}{\mathrm{TR}} \newcommand{\THH}{\mathrm{THH}} \newcommand{\TP}{\mathrm{TP}} \newcommand{\crys}{\mathrm{crys}} \newcommand{\even}{\mathrm{even}} \newcommand{\odd}{\mathrm{odd}} \newcommand{\hodge}{\mathrm{Hodge}} \newcommand{\V}{\mathrm{V}} \newcommand{\C}{C} \newcommand{\K}{K} \newcommand{\M}{M} \newcommand{\X}{X} \newcommand{\h}{h} \newcommand{\TC}{\mathrm{TC}} \newcommand{\t}{t} \newcommand{\DC}{\mathrm{DC}} \newcommand{\str}{\mathrm{str}} \newcommand{\gen}{\mathrm{gen}} \newcommand{\Cyc}{\mathrm{CycSp}} \newcommand{\DA}{\mathrm{DA}} \newcommand{\AG}{\mathbf{A}^1/\mathbf{G}_m} \newcommand{\Crys}{\mathrm{Crys}} \newcommand{\cV}{\mathcal{V}} \newcommand{\can}{\mathrm{can}} \newcommand{\Coh}{\mathrm{Coh}} \newcommand{\SH}{\mathrm{SH}} \newcommand{\ld}[1]{\check{#1}} \newcommand{\PGL}{\mathrm{PGL}} \newcommand{\Sp}{\mathrm{Sp}} \newcommand{\SO}{\mathrm{SO}} \newcommand{\ls}[1]{(\!({#1})\!)} \newcommand{\pw}[1]{[\![{#1}]\!]} \newcommand{\Shv}{\mathrm{Shv}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\fr}[1]{\mathfrak{#1}}\]

I used to post relatively frequently on this blog as an undergraduate, and essentially stopped once I started grad school. It seems fun to try to restart it (and post with far less frequency than I used to). (I have archived the old posts; if, for some reason, you want to see them, let me know.) I will soon add to this post a discussion about a recent article that I finished. It is concerned with a few different, but related, topics, and I would like to illustrate some of the pretty linear algebra and geometry surrounding them.