BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Si Ying Lee (Harvard University) DTSTART:20200408T140000Z DTEND:20200408T153000Z DTSTAMP:20260422T184930Z UID:STAGE/1 DESCRIPTION:Title: Mo dular forms on Hilbert modular varieties\nby Si Ying Lee (Harvard Univ ersity) as part of STAGE\n\n\nAbstract\nWe will give an overview of Katz's paper on the construction of $p$-adic L-functions for CM fields. A key in put in this paper is the consideration of modular forms on Hilbert modular varieties. We will discuss some key properties of Hilbert-Blumenthal abel ian varieties\, and the associated moduli spaces. We will also define modu lar forms on Hilbert modular varieties\, and prove a $q$-expansion princip le.\n LOCATION:https://researchseminars.org/talk/STAGE/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Petrov DTSTART:20200415T140000Z DTEND:20200415T153000Z DTSTAMP:20260422T184930Z UID:STAGE/2 DESCRIPTION:Title: $p $-adic modular forms on Hilbert modular varieties\nby Alexander Petrov as part of STAGE\n\n\nAbstract\nWe will define $p$-adic Hilbert modular f orms via level $p^{\\infty}$ formal Hilbert modular schemes and study the relative de Rham cohomology over that scheme using the Frobenius endomorph ism.\n LOCATION:https://researchseminars.org/talk/STAGE/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Feng DTSTART:20200422T140000Z DTEND:20200422T153000Z DTSTAMP:20260422T184930Z UID:STAGE/3 DESCRIPTION:Title: Di fferential operators on modular forms\nby Tony Feng as part of STAGE\n \n\nAbstract\nI will cover Section 2 of Katz's paper\, constructing (analy tic and p-adic) differential operators on modular forms.\n LOCATION:https://researchseminars.org/talk/STAGE/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Ziquan Yang (Harvard University) DTSTART:20200429T140000Z DTEND:20200429T153000Z DTSTAMP:20260422T184930Z UID:STAGE/4 DESCRIPTION:Title: $p $-adic Eisenstein series\nby Ziquan Yang (Harvard University) as part of STAGE\n\n\nAbstract\nI will cover Section 3 of Katz's paper on $p$-adic Eisenstein series.\n LOCATION:https://researchseminars.org/talk/STAGE/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Zijian Yao (Harvard) DTSTART:20200506T140000Z DTEND:20200506T153000Z DTSTAMP:20260422T184930Z UID:STAGE/5 DESCRIPTION:Title: CM Hilbert-Blumenthal abelian varieties\nby Zijian Yao (Harvard) as part of STAGE\n\n\nAbstract\nKatz 1978\, Sections 5.0 and 5.1\, and the statem ents of 5.2.26 and 5.2.29.\n LOCATION:https://researchseminars.org/talk/STAGE/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Kriz (Massachusetts Institute of Technology) DTSTART:20200513T140000Z DTEND:20200513T153000Z DTSTAMP:20260422T184930Z UID:STAGE/6 DESCRIPTION:Title: Co nstruction of Katz $p$-adic $L$-functions\nby Daniel Kriz (Massachuset ts Institute of Technology) as part of STAGE\n\n\nAbstract\nWe will descri be Katz's construction of a $p$-adic measure on the $p^{\\infty}$ ray clas s group of CM fields\, whose Mellin transform is a $p$-adic $L$-function i nterpolating critical values of Hecke $L$-functions. First\, we will recal l some basics of measures and the construction of the $p$-adic modular for m-valued Eisenstein measure. Next\, we will obtain Katz's measure by evalu ating the Eisenstein measure at CM points. Finally\, we will recover the a forementioned interpolation via Katz's insight that the values of the $p$- adic and complex differential operators at CM points coincide\, which foll ows from the moduli-theoretic definitions of these operators.\n LOCATION:https://researchseminars.org/talk/STAGE/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Danielle Wang (MIT) DTSTART:20200907T190000Z DTEND:20200907T203000Z DTSTAMP:20260422T184930Z UID:STAGE/7 DESCRIPTION:Title: St atements of the Weil conjectures\, proof for curves via the Hodge index th eorem\nby Danielle Wang (MIT) as part of STAGE\n\n\nAbstract\nReferenc es: Poonen\, Rat ional points on varieties\, Chapter 7 up to Section 7.5.1\; Milne\, The Riemann Hypothesis over Finite Fields: from Weil to the present day\, pages 8-10.\n\nThe Weil conjectures concern the zeta functions of varieties over a finite fie ld\, which for a smooth proper variety are rational functions that satisfy a functional equation and the Riemann hypothesis. The conjectures led to the development of étale cohomology by Grothendieck and Artin. In this ta lk\, we will state the Weil conjectures and prove the Riemann hypothesis f or curves using the Hodge index theorem.\n LOCATION:https://researchseminars.org/talk/STAGE/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20200914T190000Z DTEND:20200914T203000Z DTSTAMP:20260422T184930Z UID:STAGE/8 DESCRIPTION:Title: Sm ooth and étale morphisms\nby Niven Achenjang (MIT) as part of STAGE\n \n\nAbstract\nReferences: Mumford\, The red book of varieties and schemes\, III.5 and II I.10\; or Poonen \, Rational points on varieties\, Section 3.5.\n\nSmooth varieties giv e an algebraic analogue of (smooth) manifolds from differential geometry\, while smooth and étale morphisms give algebraic analogues of submersions and local isomorphisms. In addition to translating important notions from differential geometry into the algebraic setting\, maps of these types pl ay an important role in later development of étale cohomology. In this ta lk\, we will introduce the definitions and basic properties of smooth and étale morphisms with an emphasis on providing intuition for thinking abou t them.\n LOCATION:https://researchseminars.org/talk/STAGE/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Matthew Hase-Liu (Harvard) DTSTART:20200921T190000Z DTEND:20200921T203000Z DTSTAMP:20260422T184930Z UID:STAGE/9 DESCRIPTION:Title: In troduction to étale cohomology\nby Matthew Hase-Liu (Harvard) as part of STAGE\n\n\nAbstract\nReferences: Poonen\, Rational \npoints on varieties\, Chapter 6\; or Milne\, Lectures on é\;tale cohomology.\n\nA crash course on étale coho mology covering the following: sites and cohomology\, the étale site and operations on étale sheaves\, Frobenius action\, stalks of étale sheaves \, cohomology with compact support\, and important theorems/necessity of t orsion coefficients.\n LOCATION:https://researchseminars.org/talk/STAGE/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Samuel Marks (Harvard) DTSTART:20200928T190000Z DTEND:20200928T203000Z DTSTAMP:20260422T184930Z UID:STAGE/10 DESCRIPTION:Title: R ationality and functional equation of the zeta function\nby Samuel Mar ks (Harvard) as part of STAGE\n\n\nAbstract\nGiven a variety $X/\\mathbb{F }_q$\, the étale cohomology groups $H^i(X_{\\overline{\\mathbb{F}_q}}\,\\ mathbb{Q}_\\ell)$ come equipped with an action of $\\mathrm{Gal}(\\overlin e{\\mathbb{F}_q}/\\mathbb{F}_q)$\, and in particular with an action of the $q$-power Frobenius. This Frobenius action can also be described as comin g from the Frobenius morphism $\\mathrm{Fr}:X\\rightarrow X$. By using the se two perspectives on the Frobenius and some abstract cohomological input s\, we deduce the rationality and functional equation of $Z(X\,T)$ for nic e varieties $X$.\n\nReference: Jannsen\, Deligne's proof of the W eil-conjecture (course notes)\, Section 1.\n LOCATION:https://researchseminars.org/talk/STAGE/10/ END:VEVENT BEGIN:VEVENT SUMMARY:David Roe (MIT) DTSTART:20201005T190000Z DTEND:20201005T203000Z DTSTAMP:20260422T184930Z UID:STAGE/11 DESCRIPTION:Title: C onstructible sheaves\nby David Roe (MIT) as part of STAGE\n\n\nAbstrac t\nConstructible sheaves are built from locally constant sheaves and serve as the coefficients for étale cohomology. We will discuss the motivatio n behind their definition\, examples and some basic properties.\n\nReferen ce: Jannsen\, Deligne's proof of the Weil-conjecture (course note s)\, Section 2.\n LOCATION:https://researchseminars.org/talk/STAGE/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Sanath Devalapurkar (Harvard) DTSTART:20201012T190000Z DTEND:20201012T203000Z DTSTAMP:20260422T184930Z UID:STAGE/12 DESCRIPTION:Title: Étale fundamental groups\nby Sanath Devalapurkar (Harvard) as part of STAGE\n\n\nAbstract\nMotivated by topological considerations\, one can de fine an algebraic analogue of the fundamental group\, called the etale fun damental group. We will give a definition (via the abstract theory of Galo is categories from SGA)\, and review some basic calculations.\n\nReference s: Milne\, Lectu res on étale cohomology\, Chapter 3\; and/or Poonen\, Rational points on varieties\, Sections 3.5.9 and 3.5.11.\n LOCATION:https://researchseminars.org/talk/STAGE/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Francesc Fité DTSTART:20201019T190000Z DTEND:20201019T203000Z DTSTAMP:20260422T184930Z UID:STAGE/13 DESCRIPTION:Title: D eligne's version of the Rankin method\nby Francesc Fité as part of ST AGE\n\n\nAbstract\nWe will present a proof of the Riemann hypothesis for s mooth and projective curves defined over a finite field due to Katz. The p roof reduces the general case to the case of Fermat curves via a deformati on argument (the "connect by curves lemma") and the use of Deligne's versi on of the Rankin method. For the case of Fermat curves\, we will recall ho w the Riemann hypothesis amounts to a classical well-known result about th e size of Jacobi sums.\n\nReference: Katz\, A note on Riemann hypothesis for curves and hypersurfaces over finite fields\, Sections 1-4.\n LOCATION:https://researchseminars.org/talk/STAGE/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Ziquan Yang (Harvard University) DTSTART:20201026T190000Z DTEND:20201026T203000Z DTSTAMP:20260422T184930Z UID:STAGE/14 DESCRIPTION:Title: T he Riemann hypothesis for hypersurfaces\nby Ziquan Yang (Harvard Unive rsity) as part of STAGE\n\n\nAbstract\nI will talk about Katz' method of p roving the Riemann hypothesis (RH) for hypersurfaces. The basic idea is ve ry similar to what we saw last time: We reduce to showing RH for a particu lar hypersurface. Then we show RH for this particular hypersurface by a po int-counting argument. \n\nReference: Katz\, A note on Riemann hypothesis for curves and hypersurface s over finite fields\, Sections 5-8.\n LOCATION:https://researchseminars.org/talk/STAGE/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Hyuk Jun Kweon (MIT) DTSTART:20201102T200000Z DTEND:20201102T213000Z DTSTAMP:20260422T184930Z UID:STAGE/15 DESCRIPTION:Title: A lterations\nby Hyuk Jun Kweon (MIT) as part of STAGE\n\n\nAbstract\nIn 1964\, Hironaka proved that over a field of characteristic zero\, every a lgebraic variety admits a resolution of singularities. However\, the probl em of resolution of singularities is still open in positive characteristic . As a weaker result\, de Jong proved that every algebraic variety admits regular alterations. We will discuss background\, main statements and some applications for de Jong's result. If time allows\, we will discuss a ver y rough sketch of the proof.\n\n\nReference: Notes from Conrad's lectures on alternations\, Section 1. The goal is to understand the statement of the main theorem on alterations.\n LOCATION:https://researchseminars.org/talk/STAGE/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Petrov (Harvard University) DTSTART:20201109T200000Z DTEND:20201109T213000Z DTSTAMP:20260422T184930Z UID:STAGE/16 DESCRIPTION:Title: W eights and monodromy\nby Alexander Petrov (Harvard University) as part of STAGE\n\n\nAbstract\nReference: Scholl\, Hypersurfaces and the Weil conjectures\, Sections 1 a nd 2.\n\nWe will discuss the relationship between the action of local mono dromy around a singular fiber of a proper family and the Frobenius action\ , proving Deligne's weight monodromy theorem in equal characteristic.\n LOCATION:https://researchseminars.org/talk/STAGE/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Zhiyu Zhang (MIT) DTSTART:20201116T200000Z DTEND:20201116T213000Z DTSTAMP:20260422T184930Z UID:STAGE/17 DESCRIPTION:Title: V anishing cycles and deformation to hypersurfaces\nby Zhiyu Zhang (MIT) as part of STAGE\n\n\nAbstract\nFirstly\, we give a very brief review of Weil conjecture. Following works of Scholl and Katz\, we then outline a "1 0-line" proof of the Weil conjecture by deformation to smooth hypersurface s and induction on the dimension. In particular\, we will explain the last step i.e how to derive RH of the special fiber from the (equal characteri stic) weight-monodromy conjecture of the generic fiber\, using the weight spectral sequence as an input.\n LOCATION:https://researchseminars.org/talk/STAGE/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Raymond van Bommel (MIT) DTSTART:20201130T200000Z DTEND:20201130T213000Z DTSTAMP:20260422T184930Z UID:STAGE/18 DESCRIPTION:Title: T he Bombieri-Stepanov approach to the Riemann hypothesis for curves over fi nite fields\nby Raymond van Bommel (MIT) as part of STAGE\n\n\nAbstrac t\nIn this talk\, we will discuss an elementary proof for the Riemann hypo thesis for curves over finite fields due to Bombieri\, based on previous w ork by Stepanov and Schmidt. It uses a method which we would now call the polynomial method\, and the Riemann Roch theorem to prove an upper bound f or the number of rational points on a curve.\n\nThe slides for the talk wi ll be available on Monday 30 November.\n LOCATION:https://researchseminars.org/talk/STAGE/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Kriz (MIT) DTSTART:20201207T200000Z DTEND:20201207T213000Z DTSTAMP:20260422T184930Z UID:STAGE/19 DESCRIPTION:Title: D work's $p$-adic proof of rationality\nby Daniel Kriz (MIT) as part of STAGE\n\n\nAbstract\nIn 1959\, ex-electrical engineer Bernard Dwork shocke d the mathematical world by proving the first Weil conjecture on the ratio nality of the zeta function. Dwork's proof introduced striking new $p$-adi c methods\, and defied the expectation that the Weil conjectures could onl y be solved by developing a suitable Weil cohomology theory (later found t o be $l$-adic etale cohomology). In this talk we will outline Dwork's proo f and begin the initial part of the argument\, introducing Dwork's general notion of "splitting functions"\, the Artin-Hasse exponential and Dwork's lemma. \n\n\nReference: Koblitz\, p-adic numbers\, p-adic analysis\, and zeta-fu nctions\, pp. 92-95 and then Section V.2 to the end of the book\, some of which may be covered in a second lecture.\n LOCATION:https://researchseminars.org/talk/STAGE/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Kriz (MIT) DTSTART:20201214T200000Z DTEND:20201214T213000Z DTSTAMP:20260422T184930Z UID:STAGE/20 DESCRIPTION:Title: D work's $p$-adic proof of rationality\, continued\nby Daniel Kriz (MIT) as part of STAGE\n\n\nAbstract\nWe will go over the main steps of Dwork's argument in detail. First\, we will construct a splitting function for th e standard additive character and show it has good convergence properties using Dwork's lemma. Next we will establish the "analytic Lefschetz fixed point formula" by studying the trace of this splitting function acting on $p$-adic Banach spaces of power series. Finally\, we will show this analyt ic fixed point formula implies the zeta-function is the ratio of two entir e functions\, and conclude with a general rationality criterion for $p$-ad ic power series that implies the zeta-function is rational. \n\n\nReferenc e: Kobl itz\, p-adic numbers\, p-adic analysis\, and zeta-functions\, whatever remains of Chapter V after the first lecture.\n LOCATION:https://researchseminars.org/talk/STAGE/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Ishan Levy (MIT) DTSTART:20210219T180000Z DTEND:20210219T193000Z DTSTAMP:20260422T184930Z UID:STAGE/21 DESCRIPTION:Title: T he infinitesimal site and algebraic de Rham cohomology\nby Ishan Levy (MIT) as part of STAGE\n\n\nAbstract\nThe de Rham cohomology of the analyt ification of a smooth projective\nvariety over $\\mathbb{C}$ can be comput ed via an algebraic de Rham complex.\nUnfortunately\, the algebraic de Rha m complex is somewhat poorly behaved in\npositive characteristic. To solv e this problem\, Grothendieck\nshowed first how to reinterpret de Rham coh omology in characteristic 0\nas cohomology on a site (the infinitesimal si te)\, and second\nhow to modify the infinitesimal site to obtain a site\nt hat works well also in characteristic p (the crystalline site).\n\nIn this talk\, we will explain algebraic de Rham cohomology\nand define the infin itesimal and stratifying sites. \nWe also will define the notion of a clas sical Weil cohomology theory\,\nwhich de Rham cohomology (char 0) and crys talline cohomology give examples of.\n LOCATION:https://researchseminars.org/talk/STAGE/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Naomi Sweeting (Harvard) DTSTART:20210226T180000Z DTEND:20210226T193000Z DTSTAMP:20260422T184930Z UID:STAGE/22 DESCRIPTION:Title: C rystalline cohomology\nby Naomi Sweeting (Harvard) as part of STAGE\n\ n\nAbstract\nThis talk will provide an overview of key concepts in crystal line cohomology. We will begin with Grothendieck's heuristic argument tha t\, because de Rham cohomology is independent of choice of smooth lift\, a n intrinsic characteristic zero-valued cohomology should exist for schemes in characteristic p. We will then discuss divided power structures and the crystalline site. After stating the key theorems\, we will describe a relative setup in which the general theory of topoi plays a more prominen t role. We will conclude with sketches of crucial ideas in the comparison isomorphisms\, and a glimpse of the relationship between crystals and con nections.\n LOCATION:https://researchseminars.org/talk/STAGE/22/ END:VEVENT BEGIN:VEVENT SUMMARY:James Myer (The CUNY Graduate Center) DTSTART:20210305T180000Z DTEND:20210305T193000Z DTSTAMP:20260422T184930Z UID:STAGE/23 DESCRIPTION:Title: I ntroduction to prismatic cohomology\nby James Myer (The CUNY Graduate Center) as part of STAGE\n\n\nAbstract\nThe study of the cohomology of alg ebraic varieties is depicted by Peter Scholze as a “plane worth” of pa irs of primes $(p\,\\ell)$\, each indexing a cohomology theory for varieti es over $\\mathbb{F}_p$ with coefficients in $\\mathbb{F}_{\\ell}$. The si ngular cohomology occupies a vertical line over $\\infty$\; the étale coh omology dances around\, avoiding the pairs $(p\,p)$\; the analytic de Rham cohomology occupies the top right corner\, intersecting the singular coho mology @ $(\\infty\,\\infty)$\, symbolizing the classical de Rham comparis on theorem\, while the diagonal is picked off by the algebraic de Rham coh omology. Zooming in on a point on the diagonal\, we begin to wonder whethe r there is a cohomology theory interpolating between the étale to the cry stalline (and de Rham). In fact\, the depiction of the plane of pairs of p rimes is striated by lines from each of the various cohomology theories\, but no cohomology theory seems to “wash over” any 2-dimensional part o f the picture and “phase in and out” between any one or the other. The prismatic cohomology theory is this “2-dimensional” theory interpolat ing between the étale and crystalline (and de Rham) theories.\n\nThe clas sical de Rham comparison theorem between the (dual of the) analytic de Rha m cohomology and the singular homology offers a geometric interpretation o f a (co)homology class as an obstruction to (globally) integrating a diffe rential form. This geometric interpretation loses steam when faced with to rsion classes because the integral over a torsion class is always zero. It is also worthwhile to note the relative ease with which we may calculate the de Rham cohomology of a variety (this can be done by machine\, e.g. Ma caulay2) as opposed to the singular cohomology of a variety. So\, how do w e detect these torsion cycles algebraically? We will see via a calculation applying the universal coefficients theorem that the hypothesis of equali ty of dimensions of the analytic and algebraic de Rham cohomology groups o f a variety implies lack of torsion in singular cohomology. Somewhat conve rsely\, we’ll see that the presence of torsion in the singular cohomolog y of the analytic space associated to a variety forces the algebraic de Rh am cohomology group to be larger than expected. This interplay between the various cohomology theories for varieties\, e.g. singular\, étale\, anal ytic de Rham\, algebraic de Rham\, crystalline\, is facilitated by a (spec ialization of a sequence of) remarkable theorem(s) whose proof depends on the existence of\, and motivates the construction of\, the prismatic cohom ology theory. \n\nFollowing this introduction\, we will venture into some detail\, set up some notation for the next speaker\, and elaborate a bit m ore on the story to come.\n LOCATION:https://researchseminars.org/talk/STAGE/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikayel Mkrtchyan (Harvard) DTSTART:20210312T180000Z DTEND:20210312T193000Z DTSTAMP:20260422T184930Z UID:STAGE/24 DESCRIPTION:Title: D elta rings\nby Mikayel Mkrtchyan (Harvard) as part of STAGE\n\n\nAbstr act\nThis talk will explain some basic properties of $\\delta$-rings\, fol lowing Bhatt-Scholze. This will include examples\, categorical properties of delta-rings\, Witt vector considerations\, and\, time permitting\, a co nnection with pd-envelopes.\n LOCATION:https://researchseminars.org/talk/STAGE/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Tobias Shin (Stony Brook) DTSTART:20210319T170000Z DTEND:20210319T183000Z DTSTAMP:20260422T184930Z UID:STAGE/25 DESCRIPTION:Title: D erived categories for the working graduate student\nby Tobias Shin (St ony Brook) as part of STAGE\n\n\nAbstract\nWe give a brief review of deriv ed categories\, then discuss derived tensor products and derived completio ns.\n\nReferences: The Stacks project section on derived completion and the references liste d there.\n LOCATION:https://researchseminars.org/talk/STAGE/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Samuel Marks (Harvard) DTSTART:20210326T170000Z DTEND:20210326T183000Z DTSTAMP:20260422T184930Z UID:STAGE/26 DESCRIPTION:Title: D istinguished elements and prisms\nby Samuel Marks (Harvard) as part of STAGE\n\n\nAbstract\nGiven a divided power ring $(A\,I)$\, the crystallin e site is defined using divided power thickenings over $(A\,I)$. Analogous ly\, given a prism $(A\,I)$\, the prismatic site is defined using " prismatic thickenings" over $(A\,I)$. The goal of this talk is to define p risms and develop their basic properties.\n\nReferences: Lecture III of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://researchseminars.org/talk/STAGE/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Sanath Devalapurkar DTSTART:20210402T170000Z DTEND:20210402T183000Z DTSTAMP:20260422T184930Z UID:STAGE/27 DESCRIPTION:Title: P erfect prisms and perfectoid rings\nby Sanath Devalapurkar as part of STAGE\n\n\nAbstract\nWe will show that the category of perfect prisms is e quivalent to the category of perfectoid rings\, and use this to prove some structural results about perfectoid rings.\n\nReferences: Lecture IV of < a href="http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/ ">Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://researchseminars.org/talk/STAGE/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Konrad Zou (Bonn) DTSTART:20210409T170000Z DTEND:20210409T183000Z DTSTAMP:20260422T184930Z UID:STAGE/28 DESCRIPTION:Title: T he prismatic site\nby Konrad Zou (Bonn) as part of STAGE\n\n\nAbstract \nWe will introduce the prismatic site and finally define the prismatic co mplex and the Hodge-Tate complex.\nWe define the Hodge-Tate comparison map \, which relates the Kähler differentials to the cohomology of the Hodge- Tate complex.\nFinally\, we will introduce the Čech-Alexander complex\, w hich computes the prismatic complex in the affine case.\n\nReferences: Lec ture V of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://researchseminars.org/talk/STAGE/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Avi Zeff (Columbia) DTSTART:20210416T170000Z DTEND:20210416T183000Z DTSTAMP:20260422T184930Z UID:STAGE/29 DESCRIPTION:Title: T he Hodge-Tate and crystalline comparison theorems\nby Avi Zeff (Columb ia) as part of STAGE\n\n\nAbstract\nWe will briefly review crystalline coh omology and its relationship to prismatic cohomology\, and sketch a proof of the crystalline comparison theorem and of the Hodge-Tate comparison the orem as a corollary.\n\nReferences: Lecture VI of Bhatt's notes. F or more details\, see the Bhatt -Scholze paper.\n LOCATION:https://researchseminars.org/talk/STAGE/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Danielle Wang DTSTART:20210423T170000Z DTEND:20210423T183000Z DTSTAMP:20260422T184930Z UID:STAGE/30 DESCRIPTION:Title: T he $q$-de Rham complex\nby Danielle Wang as part of STAGE\n\n\nAbstrac t\nReferences: Lecture X of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n\ nIn this talk\, we define the q-de Rham complex\, show that it is a q-defo rmation of the usual de Rham complex\, and state a conjecture about the co ordinate independence of this construction (to be proved next lecture usin g q-crystalline cohomology).\n LOCATION:https://researchseminars.org/talk/STAGE/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Feng DTSTART:20210430T170000Z DTEND:20210430T183000Z DTSTAMP:20260422T184930Z UID:STAGE/31 DESCRIPTION:Title: $ q$-crystalline cohomology\nby Tony Feng as part of STAGE\n\n\nAbstract \nReferences: Lecture XI of Bhatt's notes. For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://researchseminars.org/talk/STAGE/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Zijian Yao DTSTART:20210507T170000Z DTEND:20210507T183000Z DTSTAMP:20260422T184930Z UID:STAGE/32 DESCRIPTION:Title: E xtension to the singular case via derived prismatic cohomology\nby Zij ian Yao as part of STAGE\n\n\nAbstract\nReferences: Lecture VII of Bhat t's notes. For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://researchseminars.org/talk/STAGE/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Zhiyu Zhang DTSTART:20210514T170000Z DTEND:20210514T183000Z DTSTAMP:20260422T184930Z UID:STAGE/33 DESCRIPTION:Title: P erfections in mixed characteristic\nby Zhiyu Zhang as part of STAGE\n\ n\nAbstract\nReferences: Lecture VIII of Bhatt's notes. For more d etails\, see the Bhatt-Scholze paper.\n LOCATION:https://researchseminars.org/talk/STAGE/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Shizhang Li (University of Michigan) DTSTART:20210521T170000Z DTEND:20210521T183000Z DTSTAMP:20260422T184930Z UID:STAGE/34 DESCRIPTION:Title: T he étale comparison theorem\nby Shizhang Li (University of Michigan) as part of STAGE\n\n\nAbstract\nReferences: Lecture IX of Bhatt's notes . For more details\, see the Bhatt-Scholze paper.\n LOCATION:https://researchseminars.org/talk/STAGE/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Angus McAndrew (Boston University) DTSTART:20210922T150000Z DTEND:20210922T163000Z DTSTAMP:20260422T184930Z UID:STAGE/35 DESCRIPTION:Title: I ntersection theory with divisors\nby Angus McAndrew (Boston University ) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buildin g.\n\nAbstract\nReferences: Appendix B of Kleiman\, The Picard scheme\, Contemp. Math.\, 2005 a nd/or Appendix VI.2 in Kollár\, Rational curves on algebraic varieties.\n LOCATION:https://researchseminars.org/talk/STAGE/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Nathan Chen (Harvard) DTSTART:20210929T140000Z DTEND:20210929T153000Z DTSTAMP:20260422T184930Z UID:STAGE/36 DESCRIPTION:Title: B ig and nef line bundles\nby Nathan Chen (Harvard) as part of STAGE\n\n Lecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nWe wil l give a gentle introduction to big and nef line bundles\, with an emphasi s on their properties and examples. Reference: Sections 1.4 and 2.2 of Lazarsfeld \, Positivity in algebraic geometry I\, Springer\, 2004.\n\nNon -MIT participants must click here to get a "Tim Ticket" well in advance\; thi s is required for access to the seminar.\n LOCATION:https://researchseminars.org/talk/STAGE/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Katia Bogdanova (Harvard) DTSTART:20211006T150000Z DTEND:20211006T163000Z DTSTAMP:20260422T184930Z UID:STAGE/37 DESCRIPTION:Title: H eight machine\nby Katia Bogdanova (Harvard) as part of STAGE\n\nLectur e held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nReferences: Sections B1-B3 in Hindry and Silverman\, Diophantine geometry\, Springer\, 2000 and/or Chapter 2 of Serre\, Lectures on the Mordell-Weil theorem \, 3rd edition\, Springer\, 1997.\n LOCATION:https://researchseminars.org/talk/STAGE/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Alice Lin (Harvard) DTSTART:20211013T150000Z DTEND:20211013T163000Z DTSTAMP:20260422T184930Z UID:STAGE/38 DESCRIPTION:Title: C omparison of Weil height and canonical height\nby Alice Lin (Harvard) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building. \n\nAbstract\nReferences: Sections B4-B5 in Hindry and Silverman\, Diophantine geometry\, Springer\, 2000 and/or Chapter 3 of Serre\, Lectures on th e Mordell-Weil theorem\, 3rd edition\, Springer\, 1997. Also\, Th eorem A of Silverman\, Heights and the specialization map for families of abelian varieties\, J. Reine Ang ew. Math. 342 (1983)\, 197–211.\n LOCATION:https://researchseminars.org/talk/STAGE/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20211020T150000Z DTEND:20211020T163000Z DTSTAMP:20260422T184930Z UID:STAGE/39 DESCRIPTION:Title: V ojta's approach to the Mordell conjecture I\nby Niven Achenjang (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building. \n\nAbstract\nWe will sketch Bombieri's simplification of Vojta's proof.\n \nReferences: Chapter 11 of Bombieri and Gubler\, Heights in diophantine geometry\, Cambri dge University Press\, 2006.\nand/or Part E of Hindry and Silverman\, Diop hantine geometry\, Springer\, 2000.\n LOCATION:https://researchseminars.org/talk/STAGE/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Vijay Srinivasan (MIT) DTSTART:20211103T150000Z DTEND:20211103T163000Z DTSTAMP:20260422T184930Z UID:STAGE/40 DESCRIPTION:Title: L ine bundles on complex tori\nby Vijay Srinivasan (MIT) as part of STAG E\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nS ections I.1 and I.2 of Mumford\, Abelian varieties\, Oxford Univers ity Press\, 1970.\n LOCATION:https://researchseminars.org/talk/STAGE/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Weixiao Lu (MIT) DTSTART:20211110T160000Z DTEND:20211110T173000Z DTSTAMP:20260422T184930Z UID:STAGE/41 DESCRIPTION:Title: A lgebraization of complex tori\nby Weixiao Lu (MIT) as part of STAGE\n\ nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nSecti on I.3 of Mumford\, Abelian varieties\, Oxford University Press\, 1 970.\n LOCATION:https://researchseminars.org/talk/STAGE/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Ryan Chen (MIT) DTSTART:20211117T160000Z DTEND:20211117T173000Z DTSTAMP:20260422T184930Z UID:STAGE/42 DESCRIPTION:Title: M oduli spaces of curves and abelian varieties\nby Ryan Chen (MIT) as pa rt of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbs tract: TBA\n LOCATION:https://researchseminars.org/talk/STAGE/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Yujie Xu (Harvard) DTSTART:20211201T160000Z DTEND:20211201T173000Z DTSTAMP:20260422T184930Z UID:STAGE/43 DESCRIPTION:Title: B etti map and Betti form I\nby Yujie Xu (Harvard) as part of STAGE\n\nL ecture held in Room 2-449 in the MIT Simons Building.\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/STAGE/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Yujie Xu (Harvard) DTSTART:20211208T150000Z DTEND:20211208T163000Z DTSTAMP:20260422T184930Z UID:STAGE/45 DESCRIPTION:Title: B etti map and Betti form II\nby Yujie Xu (Harvard) as part of STAGE\n\n Lecture held in Room 2-449 in the MIT Simons Building.\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/STAGE/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Aashraya Jha (Boston University) DTSTART:20211215T150000Z DTEND:20211215T163000Z DTSTAMP:20260422T184930Z UID:STAGE/46 DESCRIPTION:Title: T he height inequality and applications\nby Aashraya Jha (Boston Univers ity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buil ding.\n\nAbstract\nWe shall look at section 7 of Ziyang Gao's summary "Rec ent Developments of the Uniform Mordell–Lang\nConjecture". We shall stat e the Height Inequality from the paper "Uniformity in Mordell-Lang for cur ves" by Dimitrov-Gao-Habegger and an application to show a statement simil ar to the New Gap Principle.\n LOCATION:https://researchseminars.org/talk/STAGE/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20211027T140000Z DTEND:20211027T153000Z DTSTAMP:20260422T184930Z UID:STAGE/47 DESCRIPTION:Title: V ojta's approach to the Mordell conjecture II\nby Niven Achenjang (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building .\n\nAbstract\nWe will sketch Bombieri's simplification of Vojta's proof.\ n\nReferences: Chapter 11 of Bombieri and Gubler\, Heights in diophantine geometry\, Cambr idge University Press\, 2006.\nand/or Part E of Hindry and Silverman\, Dio phantine geometry\, Springer\, 2000.\n LOCATION:https://researchseminars.org/talk/STAGE/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Feng (MIT) DTSTART:20220223T150000Z DTEND:20220223T163000Z DTSTAMP:20260422T184930Z UID:STAGE/48 DESCRIPTION:Title: U niform Mordell: review and preview 1\nby Tony Feng (MIT) as part of ST AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbstract: T BA\n LOCATION:https://researchseminars.org/talk/STAGE/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Feng (MIT) DTSTART:20220302T150000Z DTEND:20220302T163000Z DTSTAMP:20260422T184930Z UID:STAGE/49 DESCRIPTION:Title: U niform Mordell: review and preview 2\nby Tony Feng (MIT) as part of ST AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\nAbstract: T BA\n LOCATION:https://researchseminars.org/talk/STAGE/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Cong Wen (Boston University) DTSTART:20220316T140000Z DTEND:20220316T153000Z DTSTAMP:20260422T184930Z UID:STAGE/50 DESCRIPTION:Title: I ntersection theory and height inequality 1\nby Cong Wen (Boston Univer sity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Bui lding.\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/STAGE/50/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou (Boston University) DTSTART:20220330T140000Z DTEND:20220330T153000Z DTSTAMP:20260422T184930Z UID:STAGE/52 DESCRIPTION:Title: I ntersection theory and height inequality 2\nby Xinyu Zhou (Boston Univ ersity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons B uilding.\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/STAGE/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Alice Lin (Harvard) DTSTART:20220406T140000Z DTEND:20220406T153000Z DTSTAMP:20260422T184930Z UID:STAGE/53 DESCRIPTION:Title: H eight bounds for nondegenerate varieties\nby Alice Lin (Harvard) as pa rt of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nA bstract\nWe will prove the Silverman-Tate theorem in Appendix 5 of [DGH]\, which upper-bounds the difference between the Neron-Tate height and the W eil height of a point $P$ in an abelian scheme $\\pi: \\mathcal{A}\\to S$ in terms of the height of the point $\\pi(P)$ in the base scheme. Then\, w e'll apply this result\, together with last week's Proposition 4.1 of [DGH ]\, to prove Theorem 1.6 in [DGH]\, which gives a lower bound on the Neron -Tate height of $P$ in a nondegenerate subvariety $X$ of $\\mathcal{A}\\to S$ in terms of the height of $\\pi(P)$. For this application\, we follow Section 5 of [DGH].\n LOCATION:https://researchseminars.org/talk/STAGE/53/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20220413T140000Z DTEND:20220413T153000Z DTSTAMP:20260422T184930Z UID:STAGE/54 DESCRIPTION:Title: P roof of the new gap principle 1\nby Niven Achenjang (MIT) as part of S TAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract \nOver the next two talks to prove Proposition 7.1 of [DGH] which\, roughl y-speaking\, bounds the number of points on a curve within a fixed distanc e of a given point. In this talk we prepare for the proof of this proposit ion by proving a series of lemmas from section 6 of [DGH]. Specifically\, after stating Proposition 7.1 of [DGH]\, we will prove Theorem 6.2 (which shows non-degeneracy of a certain subvariety of the universal abelian vari ety) followed by Lemmas 6.3 and 6.1 (which will be used to obtain the boun d in Proposition 7.1).\n LOCATION:https://researchseminars.org/talk/STAGE/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Aashraya Jha (Boston University) DTSTART:20220420T140000Z DTEND:20220420T153000Z DTSTAMP:20260422T184930Z UID:STAGE/55 DESCRIPTION:Title: P roof of the new gap principle 2\nby Aashraya Jha (Boston University) a s part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\ n\nAbstract\nIn this talk\, we will prove Proposition 7.1 of [DGH]\, the s o called "New Gap Principle". We will first prove a couple of lemmas (Lemm a 6.3 and Lemma 6.4 of [DGH]) using techniques from enumerative geometry w hich bounds the number of points on a given curve lying in proper subsets of a certain product of varieties . We then use height bounds of points on non degenerate varieties (Theorem 1.6 and Theorem 6.2 of [DGH]) along wit h lemmas proven to use an inductive argument to prove the New Gap Principl e.\n LOCATION:https://researchseminars.org/talk/STAGE/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Fei Hu (Harvard) DTSTART:20220427T140000Z DTEND:20220427T153000Z DTSTAMP:20260422T184930Z UID:STAGE/56 DESCRIPTION:Title: U niformity for rational points\nby Fei Hu (Harvard) as part of STAGE\n\ nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nWe di scuss the proof of Proposition 8.1 in [DGH]\, which gives a uniform bound for the intersection of rational points $C(\\overline\\mathbb{Q})$ of a cu rve $C$ of large modular height in an abelian variety $A$ and a finite ran k subgroup $\\Gamma\\subseteq A(\\overline\\mathbb{Q})$.\nThe number of la rge points can be handled by a standard application of the Vojta and\nMumf ord inequalities.\nThe key of [DGH] is to bound the number of those small points using the so-called New Gap Principle.\n\nWe then deduce the unifor m boundedness of rational/torsion points of curves in [DGH]\, i.e.\, their Theorems 1.1\, 1.2\, and 1.4\, from the above Proposition 8.1 (for curves of large modular height) and some other classical results (taking care of curves of small modular height).\n LOCATION:https://researchseminars.org/talk/STAGE/56/ END:VEVENT BEGIN:VEVENT SUMMARY:Anlong Chua DTSTART:20220504T140000Z DTEND:20220504T153000Z DTSTAMP:20260422T184930Z UID:STAGE/57 DESCRIPTION:Title: U nlikely intersection theory and the Ax-Schanuel theorem\nby Anlong Chu a as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buildin g.\n\nAbstract\nCounting dimensions heuristically tells us whether geometr ic objects are "likely" or "unlikely" to intersect. For instance\, Bezout' s theorem tells us that two curves in $\\mathbb{P}^2$ always intersect. On the other hand\, two curves in $\\mathbb{P}^3$ are unlikely to intersect. In number theory\, one is often concerned with unlikely intersection prob lems — for example\, when does a subvariety of an abelian variety contai n many torsion points?\n\nIn this talk\, I will try to explain the connect ions between functional transcendence\, unlikely intersections\, and numbe r theory. Time permitting\, I will discuss the answer to the question pose d above and more. On our journey\, we will pass through the fascinating wo rld of o-minimality\, which I hope to describe in broad strokes.\n LOCATION:https://researchseminars.org/talk/STAGE/57/ END:VEVENT BEGIN:VEVENT SUMMARY:Raymond van Bommel (Massachusetts Institute of Technology) DTSTART:20220511T140000Z DTEND:20220511T153000Z DTSTAMP:20260422T184930Z UID:STAGE/58 DESCRIPTION:Title: P roof of the amplification principle 1\nby Raymond van Bommel (Massachu setts Institute of Technology) as part of STAGE\n\nLecture held in Room 2- 449 in the MIT Simons Building.\n\nAbstract\nWe will recall the definition s of the Betti map and Betti rank\, and look at the degeneration locus of abelian schemes. We will see how these notions are related to each other\, and the bi-algebraic structure that we saw in the previous talk.\n\nAll p articipants should abide by MIT's COVID policies https://now.mit.edu/polic ies/events/\n LOCATION:https://researchseminars.org/talk/STAGE/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Hyuk Jun Kweon (Massachusetts Institute of Technology) DTSTART:20220518T140000Z DTEND:20220518T153000Z DTSTAMP:20260422T184930Z UID:STAGE/59 DESCRIPTION:Title: P roof of the amplification principle 2\nby Hyuk Jun Kweon (Massachusett s Institute of Technology) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nIn the previous talk\, we proved several results on the Betti rank. In this talk\, we will prove more gener alized versions of these results. Then we will prove that the rank of Bett i become maximal if we take enough iterated fibered products\, under some mild conditions.\n LOCATION:https://researchseminars.org/talk/STAGE/59/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam Schiavone (MIT) DTSTART:20220913T150000Z DTEND:20220913T163000Z DTSTAMP:20260422T184930Z UID:STAGE/60 DESCRIPTION:Title: B rauer groups of fields\nby Sam Schiavone (MIT) as part of STAGE\n\nLec ture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: D efinition of Brauer group in terms of central simple algebras (also known as Azumaya algebras over a field)\; definition of Brauer group in terms of Galois cohomology\; cyclic algebras\; Brauer groups of finite fields\, lo cal fields\, and global fields (without proofs).\n\nReferences: Poonen\, Rational \npoi nts on varieties\, Section 1.5. See also Gille and Szamuely\, Cen tral simple algebras and Galois cohomology\, Sections 2.4-2.6\, for some o f the topics. Also see Milne\, Class field theory\, Chapter IV and Theorem VII I.4.2.\n LOCATION:https://researchseminars.org/talk/STAGE/60/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20220920T150000Z DTEND:20220920T163000Z DTSTAMP:20260422T184930Z UID:STAGE/61 DESCRIPTION:Title: R eview of étale cohomology\nby Kenta Suzuki (MIT) as part of STAGE\n\n Lecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopic: A crash course on étale cohomology covering étale morphisms\, sites and cohomology\, and the étale site.\n\nReferences: Poonen\, Rational \npoints on varieti es\, Sections 3.5 (just enough to define étale morphism) and 6.1- 6.4\; or Milne\, Lectures on é\;tale cohomology.\n LOCATION:https://researchseminars.org/talk/STAGE/61/ END:VEVENT BEGIN:VEVENT SUMMARY:Hao Peng (MIT) DTSTART:20220927T150000Z DTEND:20220927T163000Z DTSTAMP:20260422T184930Z UID:STAGE/62 DESCRIPTION:Title: B rauer groups of schemes\nby Hao Peng (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: Étal e cohomology of $\\mathbb{G}_m$\; definition of cohomological Brauer group of a scheme\; Azumaya algebras\; definition of Azumaya Brauer group\; com parison (without proof).\n\nReference: Poonen\, Rational \npoints on varieties\ , Section 6.6. See also Colliot-Thé\;lè\;ne and Skorobogatov\, Th e Brauer-Grothendieck group\, Sections 3.1-3.3 and Chapter 4.\n LOCATION:https://researchseminars.org/talk/STAGE/62/ END:VEVENT BEGIN:VEVENT SUMMARY:Haoshuo Fu (MIT) DTSTART:20221004T150000Z DTEND:20221004T163000Z DTSTAMP:20260422T184930Z UID:STAGE/63 DESCRIPTION:Title: T he Hochschild-Serre spectral sequence\nby Haoshuo Fu (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstrac t\nTopics: Spectral sequences\; spectral sequence from a composition of fu nctors\; the Hochschild-Serre spectral sequence in group cohomology and é tale cohomology.\n\nReference: Poonen\, Rational \npoints on varieties\, Sectio n 6.7.\n LOCATION:https://researchseminars.org/talk/STAGE/63/ END:VEVENT BEGIN:VEVENT SUMMARY:Weixiao Lu (MIT) DTSTART:20221011T150000Z DTEND:20221011T163000Z DTSTAMP:20260422T184930Z UID:STAGE/64 DESCRIPTION:Title: R esidue homomorphisms and examples of Brauer groups\nby Weixiao Lu (MIT ) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buildin g.\n\nAbstract\nTopics: Residue homomorphisms\; purity\; examples of Braue r groups of schemes.\n\nReferences: Poonen\, Rational \npoints on varieties\, S ections 6.8-6.9\; Colliot-Thé\;lè\;ne and Skorobogatov\, The Braue r-Grothendieck group\, Section 3.7.\n LOCATION:https://researchseminars.org/talk/STAGE/64/ END:VEVENT BEGIN:VEVENT SUMMARY:Vijay Srinivasan (MIT) DTSTART:20221018T150000Z DTEND:20221018T163000Z DTSTAMP:20260422T184930Z UID:STAGE/65 DESCRIPTION:Title: T he Brauer-Manin obstruction\nby Vijay Srinivasan (MIT) as part of STAG E\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nT opics: Brauer evaluation\; Brauer set\; Brauer-Manin obstruction to the lo cal-global principle or to weak approximation\; Brauer evaluation is local ly constant.\n\nReference: Poonen\, Rational \npoints on varieties\, Sections 8 .2.1-8.2.4.\n LOCATION:https://researchseminars.org/talk/STAGE/65/ END:VEVENT BEGIN:VEVENT SUMMARY:Aashraya Jha (Boston University) DTSTART:20221025T150000Z DTEND:20221025T163000Z DTSTAMP:20260422T184930Z UID:STAGE/66 DESCRIPTION:Title: T he Brauer-Manin obstruction for conic bundles\nby Aashraya Jha (Boston University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Sim ons Building.\n\nAbstract\nTopics: Iskovskikh's example\; Brauer groups of conic bundles.\n\nReference: Poonen\, Rational \npoints on varieties\, Section 8.2.5\; and Skorobogatov\, Torsors and rational points\, Section 7 .1.\n LOCATION:https://researchseminars.org/talk/STAGE/66/ END:VEVENT BEGIN:VEVENT SUMMARY:Anne Larsen (MIT) DTSTART:20221101T150000Z DTEND:20221101T163000Z DTSTAMP:20260422T184930Z UID:STAGE/67 DESCRIPTION:Title: T orsors of algebraic groups over a field\nby Anne Larsen (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbst ract\nTopics: Torsors of groups\; torsors of algebraic groups over a field \; examples\; classification by $H^1$\; operations on torsors.\n\nReferenc e: Poonen\, R ational \npoints on varieties\, Sections 5.12.1-5.12.5.\n LOCATION:https://researchseminars.org/talk/STAGE/67/ END:VEVENT BEGIN:VEVENT SUMMARY:Thomas Rüd (MIT) DTSTART:20221108T160000Z DTEND:20221108T173000Z DTSTAMP:20260422T184930Z UID:STAGE/68 DESCRIPTION:Title: T orsors over finite fields\, local fields\, and global fields\nby Thoma s Rüd (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Sim ons Building.\n\nAbstract\nTopics: Torsors over fields of dimension $\\le 1$\; torsors over local fields\; local-global principle for torsors over g lobal fields.\n\nReference: Poonen\, Rational \npoints on varieties\, Sections 5.12.6-5.12.8.\n LOCATION:https://researchseminars.org/talk/STAGE/68/ END:VEVENT BEGIN:VEVENT SUMMARY:Alice Lin (Harvard) DTSTART:20221115T160000Z DTEND:20221115T173000Z DTSTAMP:20260422T184930Z UID:STAGE/69 DESCRIPTION:Title: T orsors over a scheme\nby Alice Lin (Harvard) as part of STAGE\n\nLectu re held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: Tor sors over a scheme\; torsor sheaves\; torsors and $H^1$\; geometric operat ions on torsors.\n\nReference: Poonen\, Rational \npoints on varieties\, Sectio ns 6.5.1-6.5.6.\n LOCATION:https://researchseminars.org/talk/STAGE/69/ END:VEVENT BEGIN:VEVENT SUMMARY:Daishi Kiyohara (MIT) DTSTART:20221122T160000Z DTEND:20221122T173000Z DTSTAMP:20260422T184930Z UID:STAGE/70 DESCRIPTION:Title: U nramified torsors\nby Daishi Kiyohara (MIT) as part of STAGE\n\nLectur e held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopic: Unram ified torsors.\n\nReference: Poonen\, Rational \npoints on varieties\, Section 6.5.7.\n LOCATION:https://researchseminars.org/talk/STAGE/70/ END:VEVENT BEGIN:VEVENT SUMMARY:Kush Singhal (Harvard) DTSTART:20221129T160000Z DTEND:20221129T173000Z DTSTAMP:20260422T184930Z UID:STAGE/71 DESCRIPTION:Title: A n example of descent\nby Kush Singhal (Harvard) as part of STAGE\n\nLe cture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: Example of descent on a genus 2 curve\; explanation in terms of twists of a Galois covering.\n\nReference: Poonen\, Rational \npoints on varieties\, Sect ion 8.3.\n LOCATION:https://researchseminars.org/talk/STAGE/71/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20221206T160000Z DTEND:20221206T173000Z DTSTAMP:20260422T184930Z UID:STAGE/72 DESCRIPTION:Title: T he descent obstruction\nby Niven Achenjang (MIT) as part of STAGE\n\nL ecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nTopics: Evaluation of torsors\; Selmer set\; weak Mordell-Weil theorem\; descent obstruction.\n\nReference: Poonen\, Rational \npoints on varieties\, Sections 8 .4.1-8.4.5 and 8.4.7.\n LOCATION:https://researchseminars.org/talk/STAGE/72/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou (Boston University) DTSTART:20221213T160000Z DTEND:20221213T173000Z DTSTAMP:20260422T184930Z UID:STAGE/73 DESCRIPTION:Title: T he étale-Brauer obstruction and insufficiency of the obstructions\nby Xinyu Zhou (Boston University) as part of STAGE\n\nLecture held in Room 2 -449 in the MIT Simons Building.\n\nAbstract\nTopics: The étale-Brauer se t\; comparison with the descent set\; insufficiency of the obstructions fo r a quadric bundle over a curve.\n\nReference: Poonen\, Rational \npoints on varieties< /i>\, Sections 8.5.2-8.5.3 and 8.6.2.\n LOCATION:https://researchseminars.org/talk/STAGE/73/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20230213T210000Z DTEND:20230213T223000Z DTSTAMP:20260422T184930Z UID:STAGE/74 DESCRIPTION:Title: C omplex tori and abelian varieties\nby Kenta Suzuki (MIT) as part of ST AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\ nWe will define abelian varieties and discuss polarization. We then discus s Riemann's criterion for when a period matrix gives rise to an abelian va riety\, and if we have time\, will see how the Siegel upper-half space par ametrizes abelian varieties.\n\nReference: Section 1.1 (and maybe 1.2) of Genestier and Ngo\ , Lectures on Shimura varieties.\n LOCATION:https://researchseminars.org/talk/STAGE/74/ END:VEVENT BEGIN:VEVENT SUMMARY:Meng Yan (Brandeis University) DTSTART:20230227T210000Z DTEND:20230227T223000Z DTSTAMP:20260422T184930Z UID:STAGE/75 DESCRIPTION:Title: Q uotients of the Siegel upper half space\nby Meng Yan (Brandeis Univers ity) as part of STAGE\n\nLecture held in Room 2-135 in the MIT Simons Buil ding.\n\nAbstract\nWe will first talk about Riemann's theorem of polarizat ion of complex tori and then give canonical bijections between polarized a belian varieties and Siegel upper half-spaces. If time permits\, we will a lso define principal level structures on abelian varieties to build isomor phisms to smooth complex analytic spaces.\n\nReference: The end of Section 1.1\, and Section 1.2 of Genestier and Ngo\, Lectures on Shimura varieties.\n LOCATION:https://researchseminars.org/talk/STAGE/75/ END:VEVENT BEGIN:VEVENT SUMMARY:Hao Peng (MIT) DTSTART:20230306T213000Z DTEND:20230306T230000Z DTSTAMP:20260422T184930Z UID:STAGE/76 DESCRIPTION:Title: M oduli space of abelian varieties I\nby Hao Peng (MIT) as part of STAGE \n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nSe ctions 2.1-2.3 of Genestier and Ngo\, Lectures on Shimura varieties. We will first finis h the part on classifying isomorphism of polarized Abelian varieties over \\mathbb C\, then introduce dual Abelian schemes\, calculate cohomology of line bundles on Abelian varieties\, and verify the representability of th e moduli problem \\mathcal A classifying Abelian varieties with polarizati ons and Level strictures.\n LOCATION:https://researchseminars.org/talk/STAGE/76/ END:VEVENT BEGIN:VEVENT SUMMARY:Zifan Wang (MIT) DTSTART:20230313T203000Z DTEND:20230313T220000Z DTSTAMP:20260422T184930Z UID:STAGE/77 DESCRIPTION:Title: M oduli space of abelian varieties II\nby Zifan Wang (MIT) as part of ST AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\ nSections 2.4.-2.6 of Genestier and Ngo\, Lectures on Shimura varieties. We will finish the proof that the functor $\\mathcal{A}$ is represented by a smooth quasi projective scheme. In particular\, to show the smoothness of $\\mathcal{A} $\, we review Grothendieck and Messing's theorem on deformations of abelia n schemes. Finally\, if we have time\, we will give an adelic description of $\\mathcal{A}$ and define Hecke operators.\n LOCATION:https://researchseminars.org/talk/STAGE/77/ END:VEVENT BEGIN:VEVENT SUMMARY:Kush Singhal (Harvard University) DTSTART:20230320T203000Z DTEND:20230320T220000Z DTSTAMP:20260422T184930Z UID:STAGE/78 DESCRIPTION:Title: R eview of reductive algebraic groups\nby Kush Singhal (Harvard Universi ty) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Build ing.\n\nAbstract\nThis will be a crash course on the theory of reductive a lgebraic groups. We will run through basic definitions and results on affi ne algebraic groups\, reductive groups\, and tori. This will be followed b y a discussion on the Lie algebra and the adjoint representation of a redu ctive group. Finally\, if time allows\, we will briefly discuss Borel and parabolic subgroups and their relation to (generalized) flag varieties. No proofs will be given due to time constraints. We will mostly follow parts of Milne's book on Algebraic Groups (available at https://math.ucr.edu/ho me/baez/qg-fall2016/Milne_iAG.pdf) specifically various subsections of cha pters 1-4\, 8\, 9\, 12\, 14\, 18\, & 19. I will thus be covering the prere quisites for Milne's notes on Shimura Varieties (https://www.jmilne.org/ma th/xnotes/svi.pdf)\, as well as the beginning few subsections of Chapters 2 and 5 of these notes.\n LOCATION:https://researchseminars.org/talk/STAGE/78/ END:VEVENT BEGIN:VEVENT SUMMARY:Gefei Dang (MIT) DTSTART:20230403T203000Z DTEND:20230403T220000Z DTSTAMP:20260422T184930Z UID:STAGE/79 DESCRIPTION:Title: H odge structures and variations\nby Gefei Dang (MIT) as part of STAGE\n \nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nWe w ill first introduce Hodge structures\, give some examples\, and rephrase t hem as representations of the Deligne torus $\\mathbb{S}$. Then we will ta lk about Hodge tensors\, polarizations\, and variations of Hodge structure s. Finally\, we will briefly introduce hermitian symmetric domains and rea lize them as parameter spaces for variations of Hodge structures.\n\nRefer ence: Milne\, Introdu ction to Shimura varieties\, Chapter 2.\n LOCATION:https://researchseminars.org/talk/STAGE/79/ END:VEVENT BEGIN:VEVENT SUMMARY:Anne Larsen (MIT) DTSTART:20230410T203000Z DTEND:20230410T220000Z DTSTAMP:20260422T184930Z UID:STAGE/80 DESCRIPTION:Title: H ermitian symmetric domains and locally symmetric varieties\nby Anne La rsen (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simon s Building.\n\nAbstract\nThe goal of this week's talk is to give the neces sary background for the definition of a Shimura variety\, to be given next week. In the first part of the talk\, we will discuss hermitian symmetric domains and their groups of automorphisms (including the homomorphism fro m U_1 associated with each point and Cartan involutions on the associated real adjoint group). In the second part\, we will define arithmetic groups and state some of the main theorems about the algebraic variety structure and group of automorphisms of the quotients of hermitian symmetric domain s by torsion-free arithmetic groups.\n\nReference: Milne\, Introduction to Shimura varieties\, Chapter 1 and 3.\n LOCATION:https://researchseminars.org/talk/STAGE/80/ END:VEVENT BEGIN:VEVENT SUMMARY:Eunsu Hur (MIT) DTSTART:20230424T203000Z DTEND:20230424T220000Z DTSTAMP:20260422T184930Z UID:STAGE/81 DESCRIPTION:Title: S himura data and Shimura varieties\nby Eunsu Hur (MIT) as part of STAGE \n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nPr imarily cover Chapter 4-5 of Milne.\n\nDefine congruence subgroup and rela te to compact open subgroups of $G(\\mathbb{A}_f)$\, no proofs necessary. Define connected Shimura datum\, equivalence via Prop. 4.8. Proposition 4. 9. Define connected Shimura variety. Cover Example 4.14 on Hilbert modula r varieties. Give the adelic description in Prop 4.18 and Prop 4.19.\n\nRe mind us of $G^{\\mathrm{der}}$ and $G^{\\mathrm{ad}}$. Define Shimura datu m\, compare to connected Shimura datum. Give Ex 5.6. Cover Prop 5.7\, Cor 5.8\, Prop 5.9. Define Shimura varieties. Define a morphism of Shimura var ieties.\n LOCATION:https://researchseminars.org/talk/STAGE/81/ END:VEVENT BEGIN:VEVENT SUMMARY:Dylan Pentland (Harvard University) DTSTART:20230501T203000Z DTEND:20230501T220000Z DTSTAMP:20260422T184930Z UID:STAGE/82 DESCRIPTION:Title: C lassification of Shimura varieties\nby Dylan Pentland (Harvard Univers ity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buil ding.\n\nAbstract\nPrimarily cover Chapters 6-8 of Milne.\n\nRemind us of the definition of a Shimura datum\, and maybe give SV2*-SV6 on p.63. Sketc h the construction of the Siegel modular variety in Chapter 6 and why it s atisfies SV1-SV6. Show that the Siegel modular variety parametrizes polari zed abelian varieties over $\\mathbb{C}$ with symplectic level structure.\ n\nSummarize Hodge type Shimura varieties as in Chapter 7.\n\nIf you have time\, sketch what changes to go from Siegel modular varieties to PEL Shim ura varieties (Chapter 8). It would be great to cover some idea of Shimura varieties of abelian type (Chapter 9).\n LOCATION:https://researchseminars.org/talk/STAGE/82/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20230508T203000Z DTEND:20230508T220000Z DTSTAMP:20260422T184930Z UID:STAGE/83 DESCRIPTION:Title: C omplex Multiplication\, Shimura-Taniyama formula\nby Niven Achenjang ( MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Buil ding.\n\nAbstract\nMotivated by the desire to construct canonical models o f Shimura curves (in the final talk)\, we introduce the theory of complex multiplication (CM) of abelian varieties. After briefly discussing the con nection between CM and canonical models\, we will cover the basic properti es of CM abelian varieties\, state the Shimura-Taniyama formula (without p roof)\, and then give the main theorem of complex multiplication. Our main reference for all of this will be chapters 10 and 11 of Milne.\n LOCATION:https://researchseminars.org/talk/STAGE/83/ END:VEVENT BEGIN:VEVENT SUMMARY:Aaron Landesman (MIT) DTSTART:20230515T203000Z DTEND:20230515T220000Z DTSTAMP:20260422T184930Z UID:STAGE/84 DESCRIPTION:Title: C anonical models of Shimura varieties\nby Aaron Landesman (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbs tract\nDefinition of canonical model in Chapter 12\, uniqueness in Chapter 13\, existence in Chapter 14.\n LOCATION:https://researchseminars.org/talk/STAGE/84/ END:VEVENT BEGIN:VEVENT SUMMARY:Daishi Kiyohara (Harvard) DTSTART:20231130T210000Z DTEND:20231130T223000Z DTSTAMP:20260422T184930Z UID:STAGE/85 DESCRIPTION:Title: T he Eisenstein quotient\nby Daishi Kiyohara (Harvard) as part of STAGE\ n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nDef ine the Eisenstein quotient and show it has the properties necessary to de duce the nonexistence of rational $p$-torsion in elliptic curves over $\\m athbb{Q}$ for $p \\ge 11\, p\\ne 13$.\n LOCATION:https://researchseminars.org/talk/STAGE/85/ END:VEVENT BEGIN:VEVENT SUMMARY:Barinder Banwait (Boston University) DTSTART:20231207T210000Z DTEND:20231207T223000Z DTSTAMP:20260422T184930Z UID:STAGE/86 DESCRIPTION:Title: P oints of order 13\nby Barinder Banwait (Boston University) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstrac t\nProve that an elliptic curve over $\\mathbb{Q}$ cannot have a rational point of order $13$\, following the paper of Mazur and Tate.\n\nReferences : [MT]\n LOCATION:https://researchseminars.org/talk/STAGE/86/ END:VEVENT BEGIN:VEVENT SUMMARY:Vijay Srinivasan (MIT) DTSTART:20230907T200000Z DTEND:20230907T213000Z DTSTAMP:20260422T184930Z UID:STAGE/87 DESCRIPTION:Title: O verview of the proof\nby Vijay Srinivasan (MIT) as part of STAGE\n\nLe cture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nState th e main theorem and give a summary of the ingredients of the proof.\n LOCATION:https://researchseminars.org/talk/STAGE/87/ END:VEVENT BEGIN:VEVENT SUMMARY:Zhao Yu Ma (MIT) DTSTART:20230914T200000Z DTEND:20230914T213000Z DTSTAMP:20260422T184930Z UID:STAGE/88 DESCRIPTION:Title: A belian varieties\nby Zhao Yu Ma (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nBasic definitions regarding abelian varieties and abelian schemes (isogenies\, dual abelian variety\, polarizations)\, Poincaré reducibility theorem\, weak Mordell-W eil theorem\n LOCATION:https://researchseminars.org/talk/STAGE/88/ END:VEVENT BEGIN:VEVENT SUMMARY:Alice Lin (Harvard) DTSTART:20230921T200000Z DTEND:20230921T213000Z DTSTAMP:20260422T184930Z UID:STAGE/89 DESCRIPTION:Title: G roup schemes\nby Alice Lin (Harvard) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nPreliminaries on th e theory of group schemes with emphasis on finite flat group schemes (conn ected-étale sequence\, Cartier duality\, Frobenius/Verschiebung\, Raynaud 's theorem)\n LOCATION:https://researchseminars.org/talk/STAGE/89/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam Schiavone (MIT) DTSTART:20230928T200000Z DTEND:20230928T213000Z DTSTAMP:20260422T184930Z UID:STAGE/90 DESCRIPTION:Title: E lliptic curves over a local field\nby Sam Schiavone (MIT) as part of S TAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract \nBasic theory of Weierstrass equations over a DVR (including semistable r eduction theorem\, Néron-Ogg-Shafarevich criterion)\n LOCATION:https://researchseminars.org/talk/STAGE/90/ END:VEVENT BEGIN:VEVENT SUMMARY:Frank Lu (Harvard) DTSTART:20231005T200000Z DTEND:20231005T213000Z DTSTAMP:20260422T184930Z UID:STAGE/91 DESCRIPTION:Title: N éron models\nby Frank Lu (Harvard) as part of STAGE\n\nLecture held i n Room 2-131 in the MIT Simons Building.\n\nAbstract\nNéron models of ell iptic curves\, Néron models of abelian varieties (omitting proof of exist ence)\, reduction types of Néron models\n LOCATION:https://researchseminars.org/talk/STAGE/91/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Hu (Harvard) DTSTART:20231012T200000Z DTEND:20231012T213000Z DTSTAMP:20260422T184930Z UID:STAGE/92 DESCRIPTION:Title: R elative Picard functor\nby Daniel Hu (Harvard) as part of STAGE\n\nLec ture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nDefine th e relative Picard functor and discuss representability\, discuss the case of curves and abelian schemes.\n LOCATION:https://researchseminars.org/talk/STAGE/92/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikayel Mkrtchyan (MIT) DTSTART:20231019T200000Z DTEND:20231019T213000Z DTSTAMP:20260422T184930Z UID:STAGE/93 DESCRIPTION:Title: J acobians\nby Mikayel Mkrtchyan (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nDiscuss Jacobians o f smooth curves\, reduced proper curves\, and families of semistable curve s.\n LOCATION:https://researchseminars.org/talk/STAGE/93/ END:VEVENT BEGIN:VEVENT SUMMARY:Eunsu Hur (MIT) DTSTART:20231026T200000Z DTEND:20231026T213000Z DTSTAMP:20260422T184930Z UID:STAGE/94 DESCRIPTION:Title: M odular forms and Hecke operators\nby Eunsu Hur (MIT) as part of STAGE\ n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nThe ory of modular forms and Hecke operators over $\\mathbb{C}$\, modular curv es over $\\mathbb{C}$ and geometric interpretation of modular forms\, the divisor $[0]-[\\infty]$ on $X_0(p)$\n LOCATION:https://researchseminars.org/talk/STAGE/94/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20231102T200000Z DTEND:20231102T213000Z DTSTAMP:20260422T184930Z UID:STAGE/95 DESCRIPTION:Title: I ntegral models of modular curves\nby Niven Achenjang (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstrac t\nSmooth models of $X_0(N)\, X_1(N)$ over $\\mathbb{Z}[1/N]$ and models o ver $\\mathbb{Z}$ à la Deligne-Rapoport and Katz-Mazur\, consequences for the structure of the Néron model of $J_0(p)$\n LOCATION:https://researchseminars.org/talk/STAGE/95/ END:VEVENT BEGIN:VEVENT SUMMARY:Hari Iyer (Harvard) DTSTART:20231109T210000Z DTEND:20231109T223000Z DTSTAMP:20260422T184930Z UID:STAGE/96 DESCRIPTION:Title: G alois representations and modular forms\nby Hari Iyer (Harvard) as par t of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAb stract\nEichler-Shimura relation on the special fiber of $J_0(N)$\, associ ating Galois representations and abelian varieties to weight 2 cusp forms. \n LOCATION:https://researchseminars.org/talk/STAGE/96/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20231116T210000Z DTEND:20231116T223000Z DTSTAMP:20260422T184930Z UID:STAGE/97 DESCRIPTION:Title: C riterion for the nonexistence of rational $p$-torsion\nby Kenta Suzuki (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Bu ilding.\n\nAbstract\nReduce the proof of the main theorem to showing that there exists a rank-$0$ quotient $A$ of $J_0(p)$ such that $[0]\\ne [\\inf ty]$ in $A$.\n\nReference: [Ma1] Section III.5\n LOCATION:https://researchseminars.org/talk/STAGE/97/ END:VEVENT BEGIN:VEVENT SUMMARY:Raymond van Bommel (Massachusetts Institute of Technology) DTSTART:20240208T210000Z DTEND:20240208T223000Z DTSTAMP:20260422T184930Z UID:STAGE/98 DESCRIPTION:Title: H eight functions\nby Raymond van Bommel (Massachusetts Institute of Tec hnology) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nThis talk will be a survey of the theory of heights . We will consider heights for projective varieties over number fields and function fields. We will not only consider finiteness of points of bounde d degree and height\, but also the number of such points. If time allows\, we will consider how the heights associated to different line bundles on a projective variety are related.\n\nThe contents of this talk are based o n Chapter 2 of the book. Another good source is Lang's book.\n LOCATION:https://researchseminars.org/talk/STAGE/98/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou DTSTART:20240222T210000Z DTEND:20240222T223000Z DTSTAMP:20260422T184930Z UID:STAGE/99 DESCRIPTION:Title: N ormalized heights\nby Xinyu Zhou as part of STAGE\n\nLecture held in R oom 2-131 in the MIT Simons Building.\n\nAbstract\nChapter 3 of Serre\, Le ctures on the Mordell-Weil theorem\n LOCATION:https://researchseminars.org/talk/STAGE/99/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven T. Achenjang (MIT) DTSTART:20240229T210000Z DTEND:20240229T223000Z DTSTAMP:20260422T184930Z UID:STAGE/100 DESCRIPTION:Title: The Mordell-Weil theorem and Chabauty's theorem\nby Niven T. Achenjang (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Bu ilding.\n\nAbstract\nChapter 4 and Section 5.1 of Serre\, Lectures on the Mordell-Weil theorem.\n\nThis talk will be split into two parts. In the fi rst part\, we will discuss the Mordell-Weil Theorem\, which states that th e abelian group of rational points on an abelian variety $A$ defined over a global field $K$ is finitely generated. We will show that this theorem f ollows from some classical finiteness results in algebraic number theory a long with the theory of heights built up in previous talks. Time permittin g\, we will conclude the first part by proving a theorem of Neron which gi ves an asymptotic count for the number of points of bounded height on an a belian variety of rank $\\rho$. In the second part\, we will turn our atte ntion towards curves of genus $g\\ge2$. For such curves $C/K$\, we will pr ove Chabauty's Theorem that $C(K)$ is finite if $\\operatorname{rank}\\ope ratorname{Jac}(C)(K) < g$ (finiteness of $C(K)$ is now known even when $C$ 's Jacobian has large rank).\n LOCATION:https://researchseminars.org/talk/STAGE/100/ END:VEVENT BEGIN:VEVENT SUMMARY:Frank Lu (Harvard) DTSTART:20240307T210000Z DTEND:20240307T223000Z DTSTAMP:20260422T184930Z UID:STAGE/101 DESCRIPTION:Title: The Demyanenko-Manin method and Mumford's inequality\nby Frank Lu (Har vard) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Bui lding.\n\nAbstract\nIn this talk\, we will discuss two theorems regarding the number of rational points on curves of genus $g \\geq 2:$ the Demyanen ko-Manin theorem and Mumford's inequality. We will begin with the Demyanen ko-Manin theorem\, which tells us how the existence of enough functions $f _i: C \\rightarrow A\,$ for some abelian variety $A\,$ allows us to show t he number of rational points on $C$ is finite. After outlining the proof o f this theorem and discussing an application to modular curves\, we will t hen sketch a proof of Mumford's inequality\, which gives an asymptotic bou nd on the number of points of bounded height without knowing Falting's the orem.\n LOCATION:https://researchseminars.org/talk/STAGE/101/ END:VEVENT BEGIN:VEVENT SUMMARY:Bjorn Poonen (MIT) DTSTART:20240321T200000Z DTEND:20240321T213000Z DTSTAMP:20260422T184930Z UID:STAGE/103 DESCRIPTION:Title: Siegel's method\nby Bjorn Poonen (MIT) as part of STAGE\n\nLecture hel d in Room 2-131 in the MIT Simons Building.\n\nAbstract\nWe will show how theorems about diophantine approximation (e.g.\, Roth's theorem that irrat ional algebraic numbers cannot be approximated too well by rational number s) can be used to prove one of the most famous theorems of 20th century ar ithmetic geometry\, Siegel's theorem that a hyperbolic affine curve can ha ve only finitely many integral points. The proof is ineffective\, however : 95 years later it is still not known if there is an algorithm that takes as input the equation of a curve and returns the list of its integral poi nts.\n\nReference: Chapter 7 of Serre\, Lectures on the Mordell-Weil theor em.\n LOCATION:https://researchseminars.org/talk/STAGE/103/ END:VEVENT BEGIN:VEVENT SUMMARY:Shiva Chidambaram (MIT) DTSTART:20240404T200000Z DTEND:20240404T213000Z DTSTAMP:20260422T184930Z UID:STAGE/104 DESCRIPTION:Title: Baker's method\nby Shiva Chidambaram (MIT) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nIn this talk\ , we will discuss Baker's theorem on lower bounds for linear forms in loga rithms\, and how it gives effective bounds for quasi-integral points on $\ \mathbb{P}^1 \\setminus \\{0\,1\,\\infty\\}$. Using coverings\, this furth er yields effective bounds for quasi-integral points on elliptic\, superel liptic and certain hyperelliptic affine curves. We will also discuss an ap plication towards finding elliptic curves with good reduction outside a gi ven finite set of places.\n\nReference: Chapter 8 of Serre\, Lectures on t he Mordell-Weil theorem.\n LOCATION:https://researchseminars.org/talk/STAGE/104/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Hu (Harvard) DTSTART:20240411T200000Z DTEND:20240411T213000Z DTSTAMP:20260422T184930Z UID:STAGE/105 DESCRIPTION:Title: The Hilbert irreducibility theorem\nby Daniel Hu (Harvard) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstrac t\nI will introduce the notion of thin sets and discuss some applications to Galois groups of polynomials. Then\, I will state and prove Hilbert's i rreducibility theorem following Serre's account of Lang's proof.\n\nRefere nce: Chapter 9 of Serre\, Lectures on the Mordell-Weil theorem.\n LOCATION:https://researchseminars.org/talk/STAGE/105/ END:VEVENT BEGIN:VEVENT SUMMARY:Vijay Srinivasan DTSTART:20240418T200000Z DTEND:20240418T213000Z DTSTAMP:20260422T184930Z UID:STAGE/106 DESCRIPTION:Title: Construction of Galois extensions\nby Vijay Srinivasan as part of STAG E\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstract\nT he inverse Galois problem asks whether every finite group can be realized as the Galois group of a finite extension of $\\mathbb{Q}$. In this talk\, we will discuss the cases of $S_n$ and $\\text{PGL}_2(\\mathbb{F}_p)$. Th ese methods can also be adapted to apply to the simple groups $A_n$ and $\ \text{PSL}_2(\\mathbb{F}_p)$ (for many $p$).\n LOCATION:https://researchseminars.org/talk/STAGE/106/ END:VEVENT BEGIN:VEVENT SUMMARY:Jane Shi DTSTART:20240502T200000Z DTEND:20240502T213000Z DTSTAMP:20260422T184930Z UID:STAGE/107 DESCRIPTION:Title: Construction of elliptic curves of large rank\nby Jane Shi as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAbstrac t\nIn this talk\, I will first prove Néron's theorem and explain how we c an use it as a basis to construct elliptic curves of large rank. Then\, I will discuss two methods for constructing elliptic curves of rank at least 9 and one method for constructing elliptic curves of rank at least 10. If there is more time\, I will discuss approaches for generating elliptic cu rves of rank at least $ 11$.\n LOCATION:https://researchseminars.org/talk/STAGE/107/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Larsen (Massachusetts Institute of Technology) DTSTART:20240425T200000Z DTEND:20240425T213000Z DTSTAMP:20260422T184930Z UID:STAGE/108 DESCRIPTION:Title: The large sieve\nby Daniel Larsen (Massachusetts Institute of Technolo gy) as part of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Build ing.\n\nAbstract\nIn this talk\, we will prove a version of the large siev e inequality\, a result from analytic number theory that will eventually b e used to give bounds on thin sets. Along the way\, we will prove the Dave nport-Halberstam theorem and generally try to understand how the support o f a function's Fourier transform influences the function's behavior.\n\nRe ference: Chapter 12 of Serre\, Lectures on the Mordell-Weil theorem.\nWear ing a mask is welcomed\, but optional.\n LOCATION:https://researchseminars.org/talk/STAGE/108/ END:VEVENT BEGIN:VEVENT SUMMARY:Hao Peng (MIT) DTSTART:20240509T200000Z DTEND:20240509T213000Z DTSTAMP:20260422T184930Z UID:STAGE/109 DESCRIPTION:Title: Applications of the large sieve to thin sets\nby Hao Peng (MIT) as par t of STAGE\n\nLecture held in Room 2-131 in the MIT Simons Building.\n\nAb stract\nWe review the proof of Cohen-Serre bound on the number of rational points on projective and affine varieties using the large sieve method an d Lang-Weil bound on rational points on varieties over finite fields. Noti ce that stronger bounds are known now by work of Browning\, Heath-Brown an d Salberger.\n LOCATION:https://researchseminars.org/talk/STAGE/109/ END:VEVENT BEGIN:VEVENT SUMMARY:Elia Gorokhovsky (Harvard) DTSTART:20240909T143000Z DTEND:20240909T160000Z DTSTAMP:20260422T184930Z UID:STAGE/110 DESCRIPTION:Title: Complex analytic spaces\, vector bundles\, and GAGA\nby Elia Gorokhovs ky (Harvard) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Sim ons Building.\n\nAbstract\nDefinition of the category of complex analytic spaces\, and statements of GAGA.\n\nReferences: \n

Hartshorne\, < i>Algebraic geometry\, 1977. Appendix B.1 and B.2.
Gunning and Rossi\, Analytic functions of several complex variables\, Prent ice-Hall (1965).
Serre\, Gé\;ometrie algé\;brique et gé\;ometrie analytique\, Ann. Inst. Fourier 6 (1956)\ , 1-42.
Grothendieck\, SGA I\, Exp. XII

\n LOCATION:https://researchseminars.org/talk/STAGE/110/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard) DTSTART:20240916T143000Z DTEND:20240916T160000Z DTSTAMP:20260422T184930Z UID:STAGE/111 DESCRIPTION:Title: Local systems\, fundamental groupoid\, definition of connection\nby Xi nyu Fang (Harvard) as part of STAGE\n\nLecture held in Room 2-449 in the M IT Simons Building.\n\nAbstract\nThe definition of local systems vs. vecto r bundles. Definition of the fundamental group and fundamental groupoid o f a nice topological space. Statement of equivalence between the category of local systems and the category of finite-dimensional representations o f the fundamental group. Definition of connection on a vector bundle.\n\n Reference: Deligne\, up to Section 2.9.\n LOCATION:https://researchseminars.org/talk/STAGE/111/ END:VEVENT BEGIN:VEVENT SUMMARY:Atticus Wang (MIT) DTSTART:20240923T143000Z DTEND:20240923T160000Z DTSTAMP:20260422T184930Z UID:STAGE/112 DESCRIPTION:Title: Integrable connections\nby Atticus Wang (MIT) as part of STAGE\n\nLect ure held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nDefinition of curvature and integrable connection. Statement of the equivalence bet ween the category of local systems and the category of vector bundles with integrable connection. Variants: schemes\, relative setting.\n\nReferenc e: Deligne\, 2.10 to the end of Section 2\; Conrad\, Classical motivation for the Riemann-Hilbert correspondence. Notes from the talk are attached u nder "slides".\n LOCATION:https://researchseminars.org/talk/STAGE/112/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20240930T143000Z DTEND:20240930T160000Z DTSTAMP:20260422T184930Z UID:STAGE/113 DESCRIPTION:Title: Relationship to PDEs and $n$th order differential equations\nby Kenta Suzuki (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Sim ons Building.\n\nAbstract\nVia local trivializations of vector bundles and connections\, we translate the conditions of horizontal sections and flat connections in terms of classical differential equations. We then associa te vector bundles with connections to higher order differential equations and finally prove an equivalence between the categories of these objects ( with additional data).\n\nReference: Deligne\, Sections 3 and 4.\n LOCATION:https://researchseminars.org/talk/STAGE/113/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenz Kallal (Princeton University) DTSTART:20241007T143000Z DTEND:20241007T160000Z DTSTAMP:20260422T184930Z UID:STAGE/114 DESCRIPTION:Title: Second order differential equations and projective connections\nby Ken z Kallal (Princeton University) as part of STAGE\n\nLecture held in Room 2 -449 in the MIT Simons Building.\n\nAbstract\nReference: Deligne\, Section 5.\n\nIn the previous section\, Deligne sets up an equivalence of categor ies between order-n differential equations on line bundles on curves and r ank-n vector bundles with connection plus the extra data of a certain cycl ic morphism. \n\nIn section 5\, Deligne reinterprets the special case n = 2 in terms of a connection on a certain bundle and another uniformization datum called a projective connection. I will prove this alternative equiva lence of categories\, focusing on the different ways of viewing and comput ing with projective connections.\n LOCATION:https://researchseminars.org/talk/STAGE/114/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Barz (Princeton University) DTSTART:20241209T153000Z DTEND:20241209T170000Z DTSTAMP:20260422T184930Z UID:STAGE/115 DESCRIPTION:Title: Irregular connections and the Stokes phenomena\nby Michael Barz (Princ eton University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nDeligne's book focuses mostly on connection s with regular singularities -- in the 1970s\, Deligne found connec tions with irregular singularities to be pathological (see his article "Po urquoi un géomètre algébriste s'intéresse-t-il aux connexions irrégul ières?"). But since then\, Deligne\, Malgrange\, Sibuya\, and many others have noticed that irregular connections are home to many interesting phen omena which seem to mirror things occurring for ell-adic sheaves.\n\nRegul ar connections are the simplest to understand since\, by Riemann-Hilbert\, they are completely determined by the monodromy of their solutions. Unfor tunately\, this fails for irregular connections -- there are nontrivial ir regular connections whose solutions have no monodromy. In this talk we des cribe the Stokes data which one can use to help understand irregular conne ctions.\n\nReference: Malgrange\, Équations Différentielles à Coeffi cients Polynomiaux\, chapters 3 and 4\nBabbitt and Varadarajan\, Lo cal moduli for meromorphic differential equations\nDeligne\, Malgrange \, and Ramis\, Singularités Irrégulières: Correspondance et document s\, particularly the 19.4.78 letter from Deligne to Malgrange.\n LOCATION:https://researchseminars.org/talk/STAGE/115/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou (Boston University) DTSTART:20241021T143000Z DTEND:20241021T160000Z DTSTAMP:20260422T184930Z UID:STAGE/116 DESCRIPTION:Title: Multivalued functions\, and abstract derivations and connections\nby X inyu Zhou (Boston University) as part of STAGE\n\nLecture held in Room 2-4 49 in the MIT Simons Building.\n\nAbstract\nOur next goal is to study exte nsions of vector bundles and connections on an open (non-compact) surfaces . In this talk\, we introduce relevant concepts and their basic properties . We first generalize the idea of multivalued functions to sheaves and sho w its relations to monodromy representations. We also introduce and study the valuations of a function under the action of a connection.\n\nDeligne\ , Sections I.6 and the beginning of II.1.\n LOCATION:https://researchseminars.org/talk/STAGE/116/ END:VEVENT BEGIN:VEVENT SUMMARY:Julia Meng (MIT) DTSTART:20241028T143000Z DTEND:20241028T160000Z DTSTAMP:20260422T184930Z UID:STAGE/117 DESCRIPTION:Title: Regular connections in dimension 1\nby Julia Meng (MIT) as part of STA GE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\n Definition of meromorphic vector bundles with regular connections in dimen sion one. Meromorphic vector bundles on the punctured disc and monodromy t ransformations. Statement that two meromorphic vector bundles on the punct ured disc with regular connections are isomorphic if and only if they have the same monodromy.\n\nReference: Deligne\, section II.1.\n LOCATION:https://researchseminars.org/talk/STAGE/117/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikayel Mkrtchyan (MIT) DTSTART:20241104T153000Z DTEND:20241104T170000Z DTSTAMP:20260422T184930Z UID:STAGE/118 DESCRIPTION:Title: Connections with regular singularities in higher dimension\nby Mikayel Mkrtchyan (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nFollowing chapter 2 of Deligne\, we will in troduce aspects of regularity for connections on smooth varieties. This in cludes moderate growth conditions and\, in the normal crossing divisor cas e\, connections with regular singularities. Time-permitting\, we will disc uss the relations between the various characterizations of regular singula rities.\n LOCATION:https://researchseminars.org/talk/STAGE/118/ END:VEVENT BEGIN:VEVENT SUMMARY:Oakley Edens (Harvard) DTSTART:20241118T153000Z DTEND:20241118T170000Z DTSTAMP:20260422T184930Z UID:STAGE/119 DESCRIPTION:Title: Regular connections in dimension $n$\nby Oakley Edens (Harvard) as par t of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAb stract\nFollowing Deligne chapter $2$\, we discuss the relationship betwee n several characterizations of regular connections in higher dimensions. I n particular\, we show that the regularity is "local in codimension $1$ at $\\infty$". Time permitting\, we will prove the Riemann-Hilbert correspon dence for regular connections.\n LOCATION:https://researchseminars.org/talk/STAGE/119/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard) DTSTART:20241125T153000Z DTEND:20241125T170000Z DTSTAMP:20260422T184930Z UID:STAGE/120 DESCRIPTION:Title: Proof of the Riemann-Hilbert Correspondence\nby Xinyu Fang (Harvard) a s part of STAGE\n\nLecture held in Room 2-255 in the MIT Simons Building.\ n\nAbstract\nAfter a quick review of the key concepts we learned before\, I will present the statement and a sketch of the proof of the Riemann-Hilb ert correspondence\; namely\, the equivalence of categories between algebr aic vector bundles with regular integrable connections and holomorphic vec tor bundles with an integrable connection on a complex algebraic variety ( and its analytification\, respectively). After that\, I will also present a simple example to illustrate why we should have the regularity condition imposed on the algebraic side. We will follow Chapter II Section 5 of Del igne.\n LOCATION:https://researchseminars.org/talk/STAGE/120/ END:VEVENT BEGIN:VEVENT SUMMARY:Aashraya Jha (Boston University) DTSTART:20241202T153000Z DTEND:20241202T170000Z DTSTAMP:20260422T184930Z UID:STAGE/121 DESCRIPTION:Title: de Rham cohomology and Gauss-Manin connections\nby Aashraya Jha (Bosto n University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Si mons Building.\n\nAbstract\nWe state the equivalence of algebraic and anal ytic de Rham cohomologies for vector bundles with regular integrable conne ctions and discuss a relative version. We then discuss the Gauss-Manin con nection\, which is obtained on the derived pushforward sheaves of an integ rable connection and is a prominent example of connections considered in p ractice. We show that the Gauss-Manin connection is regular if the integra ble connection we start with is regular. We will try to provide examples a long the way to elucidate the theory.\n LOCATION:https://researchseminars.org/talk/STAGE/121/ END:VEVENT BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20250206T220000Z DTEND:20250206T233000Z DTSTAMP:20260422T184930Z UID:STAGE/122 DESCRIPTION:Title: Overview of the Lawrence-Venkatesh proof\nby Niven Achenjang (MIT) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\ nAbstract\nThe Mordell Conjecture states that a curve of genus $g\\ge2$ ov er a number field can only have finitely many rational points. This was fi rst proved by Faltings in his famous 1983 paper\, but more recently\, a ne w proof was given by Brian Lawrence and Akshay Venkatesh using $p$-adic me thods. In this talk\, after briefly setting up the context of Mordell's co njecture\, we will discuss\, in broad strokes\, the various ideas and resu lts which go into the Lawrence-Venkatesh proof.\n\nReferences: \n\n$\\bull et$ Poonen\, A $p$-adic approach to ratio nal points on curves\n\n$\\bullet$ Poonen\, $p$-adic approaches to rational an d integral points on curves\n\n$\\bullet$ Lawrence and Venkatesh\, Diopha ntine problems and $p$-adic period mappings\n LOCATION:https://researchseminars.org/talk/STAGE/122/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard University) DTSTART:20250213T220000Z DTEND:20250213T233000Z DTSTAMP:20260422T184930Z UID:STAGE/123 DESCRIPTION:Title: A tour of the Betti\, étale\, de Rham\, and crystalline cohomology\nb y Xinyu Fang (Harvard University) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nThis talk is a crash cours e on the properties of various cohomology theories that will be used in th e Lawrence-Venkatesh proof. These include the Betti cohomology\, étale co homology\, de Rham cohomology and crystalline cohomology. We review releva nt structures on each of these cohomology groups of a smooth proper variet y\, state some comparison theorems\, and explain how they come together to form the "big diagram" in the Lawrence-Venkatesh argument.\n\n[Update] I uploaded my outline for the talk to (slides) below. It is only a skeleton of the talk instead of a complete write-up\, so if you want to review the material\, I recommend reading the first reference (it's only 2 pages)\, a nd if you would like more detail\, look into the other references.\n\nRefe rence: \n\n$\\bullet$ Poonen\, $p$-adic approaches to rational and integral points on curves\, Sections 5 and 6.\n\nFor more details:\n\n$\\bullet$ Delign e\, Hodge cycles on abelian varieties\, Section 1 (for Betti and de Rh am cohomology).\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine problems a nd $p$-adic period mappings\, Section 3 (to see how cohomology theorie s are going to be used).\n\n$\\bullet$ Brinon and Conrad\, CMI summer school notes on $ p$-adic Hodge theory\, Section 9.1 (for details on the ring $B_{cris}$ and the functor $D_{cris}$).\n\n$\\bullet$ Nicole\, Cris is for Crystalline (notes for a seminar talk on crystalline cohomology) (for the definition and motivations for crystalline cohomology).\n\n$\\bul let$ Voisin\, Hodge th eory and complex algebraic geometry I\, Chapter II (for de Rham cohomo logy and the Hodge decomposition).\n LOCATION:https://researchseminars.org/talk/STAGE/123/ END:VEVENT BEGIN:VEVENT SUMMARY:Mohit Hulse (MIT) DTSTART:20250220T220000Z DTEND:20250220T233000Z DTSTAMP:20260422T184930Z UID:STAGE/124 DESCRIPTION:Title: Period maps and the Gauss-Manin connection\nby Mohit Hulse (MIT) as pa rt of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nA bstract\nFor a family of smooth projective varieties over a number field\, we have a complex period map and a $p$-adic period map\, and they are bot h governed by the (algebraic) Gauss-Manin connection. After some prelimina ries\, we introduce these objects and prove some bounds on the dimensions of their images.\n\nReference:\n\n$\\bullet$ Lawrence and Venkatesh\, Diophan tine problems and $p$-adic period mappings\, Section 3.\n\nFor more de tails:\n\n$\\bullet$ Deligne\, Hodge cycles on abelian varieties\, Secti on 2 (for Gauss-Manin connection).\n\n$\\bullet$ Katz and Oda\, On the differentiation of De Rham c ohomology classes with respect to parameters\n\n$\\bullet$ Voisin\, Hodge Theory and Complex Algebraic Geometry I\, Part III.\n\n$\\bullet$ Hotta\, Takeuchi and Tanisak i\, D-Modules\, Perverse Sheaves\, and Representation Theory (for mor e on Riemann-Hilbert).\n LOCATION:https://researchseminars.org/talk/STAGE/124/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou (Boston University) DTSTART:20250227T220000Z DTEND:20250227T233000Z DTSTAMP:20260422T184930Z UID:STAGE/125 DESCRIPTION:Title: Families of varieties with good reduction\nby Xinyu Zhou (Boston Unive rsity) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Bu ilding.\n\nAbstract\nWe review some constructions on crystalline represent ations and cohomology. Then we present a result in Lawrence-Venkatesh that shows the points in a residue disk that define semisimple representations are contained in a proper analytic subset. The proof illustrates the basi c strategy in Lawrence-Venkatesh: to show the finiteness of a set of point s\, one only need to show its image in the period domain is contained in a Zariski-closed subset with dimension smaller than that of the orbit of a point under the complex monodromy group.\n\nReference:\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine problems and $p$-adic period mappings\, S ection 3.\n\n$\\bullet$ Faltings\, Crystalline cohomology and p-adic Galois-repre sentations. Algebraic analysis\, geometry\, and number theory (Baltimo re\, MD\, 1988)\, 25–80.\n\n$\\bullet$ Illusie\, Crystalline cohomology. Section 3.Motives (S eattle\, WA\, 1991)\, 43–70.\n LOCATION:https://researchseminars.org/talk/STAGE/125/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20250306T220000Z DTEND:20250306T233000Z DTSTAMP:20260422T184930Z UID:STAGE/126 DESCRIPTION:Title: The $S$-unit equation\nby Kenta Suzuki (MIT) as part of STAGE\n\nLectu re held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nWe prove th at the $S$-unit equation\, $x+y=1$ where $x\,y\\in\\mathcal O_S^\\times$\, has finitely many solutions by using $p$-adic period mappings. To do so w e analyze the monodromy and period mapping for (a small modification of) t he Legendre family. Although not logically necessary for the proof of Falt ing's theorem\, many of the key ideas are already present in this special case.\n\nReference:\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine proble ms and $p$-adic period mappings\, Section 4.\n LOCATION:https://researchseminars.org/talk/STAGE/126/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard) DTSTART:20250403T210000Z DTEND:20250403T223000Z DTSTAMP:20260422T184930Z UID:STAGE/128 DESCRIPTION:Title: Outline of the argument for Mordell's conjecture\nby Xinyu Fang (Harva rd) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Build ing.\n\nAbstract\nI will present an outline of the argument for the proof of Mordell's conjecture\, following Section 5 of Lawrence-Venkatesh. Speci fically\, I will give an overview of two key inputs: the existence of a go od abelian-by-finite family (the Kodaira-Parshin family) and the finitenes s of rational points whose fiber along the finite map has large Galois orb its (proven using p-adic Hodge theory). Then\, I will explain how to reduc e Mordell's conjecture to these key inputs.\n\nReference:\n\n$\\bullet$ Lawre nce and Venkatesh\, Diophantine problems and $p$-adic period mappings\ , Section 5.\n LOCATION:https://researchseminars.org/talk/STAGE/128/ END:VEVENT BEGIN:VEVENT SUMMARY:Frank Lu (Harvard) DTSTART:20250410T210000Z DTEND:20250410T223000Z DTSTAMP:20260422T184930Z UID:STAGE/129 DESCRIPTION:Title: Abelian-by-finite families I\nby Frank Lu (Harvard) as part of STAGE\n \nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\nIn t his talk\, we will begin the proof that there are only finitely many ratio nal points whose pre-images along the finite map have large Galois orbits\ , introduced in the previous talk. The proof of this statement requires tw o lemmas: a generic simplicity statement\, and a finiteness statement if w e consider only the rational points corresponding to a given simple Galois representation. We will begin by presenting the proof\, assuming these tw o lemmas\, before proving the finiteness lemma. \n\nWe will follow part of La wrence and Venkatesh\, Diophantine problems and $p$-adic period mappings\, Section 6.\n LOCATION:https://researchseminars.org/talk/STAGE/129/ END:VEVENT BEGIN:VEVENT SUMMARY:Vijay Srinivasan (MIT) DTSTART:20250417T210000Z DTEND:20250417T223000Z DTSTAMP:20260422T184930Z UID:STAGE/130 DESCRIPTION:Title: Abelian-by-finite families II\nby Vijay Srinivasan (MIT) as part of ST AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\ nReference:\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine problems and $ p$-adic period mappings\, second half of Section 6.\n LOCATION:https://researchseminars.org/talk/STAGE/130/ END:VEVENT BEGIN:VEVENT SUMMARY:Elia Gorokhovsky (Harvard) DTSTART:20250424T210000Z DTEND:20250424T223000Z DTSTAMP:20260422T184930Z UID:STAGE/131 DESCRIPTION:Title: The Kodaira--Parshin family\nby Elia Gorokhovsky (Harvard) as part of STAGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstrac t\nIn this talk\, we describe the construction of an abelian-by-finite fam ily parametrized by a prime $q$ which is amenable to direct monodromy comp utations. This family\, the Kodaira-Parshin family\, serves as input to th e arguments in Section 6 which use an abelian-by-finite family with full m onodromy to prove finiteness of rational points. The bulk of the talk focu ses on the ``finite'' part of abelian-by-finite: we will describe the cons truction of an \\'etale cover of a curve $Y$ parametrizing $G$-covers of $ Y$ branched at a single point\, together with a universal curve.\n\nRefere nce:\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine problems and $p$-adic period mappings\, Section 7.\n LOCATION:https://researchseminars.org/talk/STAGE/131/ END:VEVENT BEGIN:VEVENT SUMMARY:Joye Chen (MIT) DTSTART:20250501T214500Z DTEND:20250501T231500Z DTSTAMP:20260422T184930Z UID:STAGE/132 DESCRIPTION:Title: Mapping class groups and Dehn twists\nby Joye Chen (MIT) as part of ST AGE\n\nLecture held in Room 2-449 in the MIT Simons Building.\n\nAbstract\ nReference:\n\n$\\bullet$ Farb and Margalit\, A Primer on Mapping Class Groups .\n LOCATION:https://researchseminars.org/talk/STAGE/132/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (MIT) DTSTART:20250508T193000Z DTEND:20250508T210000Z DTSTAMP:20260422T184930Z UID:STAGE/133 DESCRIPTION:Title: Monodromy of the Kodaira--Parshin family\nby Kenta Suzuki (MIT) as par t of STAGE\n\nLecture held in Room 4-149\, in the Maclaurin Buildings.\n\n Abstract\nWe complete the final step of the proof\, proving that the monod romy of the Kodaira-Parshin family is large. Let $Y$ be a compact orientab le surface or genus $g\\ge 2$ and let $Y^0$ be the puncture at a point. Le t $(Z_1\,\\pi_1)\,\\dots\,(Z_N\,\\pi_N)$ be the $\\mathrm{Aff}(q)$-covers of $Y^0$\, and let $\\mathrm{MCG}(Y^0)_0$ be those mapping classes of $Y^0 $ who induce trivial mapping classes on every $Z_i$. Then $\\mathrm{MCG}(Y ^0)_0$ acts on the primitive homomology $H_1^{\\mathrm{Pr}}(Z_i\,Y^0)$ of $Z_i$\, i.e.\, the orthogonal complement of $\\pi_i^*H_1(Z_i)\\subset H_1( Y^0)$. We prove that $\\mathrm{MCG}(Y^0)_0\\to\\prod_{i=1}^N\\mathrm{Sp}(H _1^{\\mathrm{Pr}}(Z_i\,Y^0))$ has Zariski dense image.\n\nBy Goursat's lem ma it suffices to prove each $\\mathrm{MCG}(Y^0)_0\\to\\mathrm{Sp}(H_1^{\\ mathrm{Pr}}(Z_i\,Y^0))$ has Zariski dense image\, which we prove by produc ing many Dehn twists on $Y^0$ inducing trivial mapping classes on $Z_i$.\n \nReference:\n\n$\\bullet$ Lawrence and Venkatesh\, Diophantine problems and $p$-adic period mappings\, Section 8.\n LOCATION:https://researchseminars.org/talk/STAGE/133/ END:VEVENT BEGIN:VEVENT SUMMARY:Ari Krishna + Sophie Zhu (Harvard) DTSTART:20250911T203000Z DTEND:20250911T220000Z DTSTAMP:20260422T184930Z UID:STAGE/134 DESCRIPTION:Title: Statements of the Weil conjectures\, proof for curves via the Hodge index theorem\nby Ari Krishna + Sophie Zhu (Harvard) as part of STAGE\n\nLec ture held in Room 4-163.\n\nAbstract\nState the Weil conjectures for smoot h proper varieties over finite fields. Explain the proof for curves via in tersection theory on surfaces\, in particular the Hodge index theorem.\n\n References: Poonen\, Rational points on varieties\, Chapter 7 up to Section 7.5.1\; Mi lne\, The Riemann Hyp othesis over Finite Fields: from Weil to the present day\, pages 8-10. \n LOCATION:https://researchseminars.org/talk/STAGE/134/ END:VEVENT BEGIN:VEVENT SUMMARY:Yutong Chen (MIT) DTSTART:20250918T203000Z DTEND:20250918T220000Z DTSTAMP:20260422T184930Z UID:STAGE/135 DESCRIPTION:Title: The étale site\nby Yutong Chen (MIT) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Simons Building.\n\nAbstract\nDefine étale morp hisms\, Grothendieck topologies\, the étale site\, and sheaves thereon.\n \nReference: Poonen\, Rational points on varieties\, Section 3.5\, 6.2\, 6.3.\n LOCATION:https://researchseminars.org/talk/STAGE/135/ END:VEVENT BEGIN:VEVENT SUMMARY:Nir Elber (MIT) DTSTART:20250925T203000Z DTEND:20250925T220000Z DTSTAMP:20260422T184930Z UID:STAGE/136 DESCRIPTION:Title: Étale cohomology\nby Nir Elber (MIT) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Simons Building.\n\nAbstract\nIn the first half of the talk\, we will define and explain some properties of $\\ell$-adic c ohomology\, more or less following SGA $4\\frac12$. In the second half of the talk\, we will explain what a Weil cohomology theory is and state that $\\ell$-adic cohomology is an example of a Weil cohomology theory. If we have any time remaining\, we may gesture towards other Weil cohomology the ories.\n LOCATION:https://researchseminars.org/talk/STAGE/136/ END:VEVENT BEGIN:VEVENT SUMMARY:Arav Karighattam (MIT) DTSTART:20251002T203000Z DTEND:20251002T220000Z DTSTAMP:20260422T184930Z UID:STAGE/137 DESCRIPTION:Title: Lefschetz trace formula and proof of rationality and functional equation\nby Arav Karighattam (MIT) as part of STAGE\n\nLecture held in Room 2-1 05 in the MIT Simons Building.\n\nAbstract\nState the Lefschetz trace form ula for étale cohomology\, and explain how to apply it to the Frobenius m orphism to deduce the Weil conjectures\, excluding the Riemann hypothesis. \n\nReference: Milne\, Lectures on étale cohomology\, Section 25 and 27.\n LOCATION:https://researchseminars.org/talk/STAGE/137/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikayel Mkrtchyan (MIT) DTSTART:20251009T203000Z DTEND:20251009T220000Z DTSTAMP:20260422T184930Z UID:STAGE/138 DESCRIPTION:Title: Constructible sheaves\, L-functions and base change theorems\nby Mikay el Mkrtchyan (MIT) as part of STAGE\n\nLecture held in Room 2-105 in the M IT Simons Building.\n\nAbstract\nIn this talk\, we will give an overview o f various foundational tools used for computations in etale cohomology. Ti me permitting\, this may include constructible sheaves\, their L-functions and the Grothendieck-Lefschetz trace formula\, and the proper base change theorem.\n LOCATION:https://researchseminars.org/talk/STAGE/138/ END:VEVENT BEGIN:VEVENT SUMMARY:Jane Shi (MIT) DTSTART:20251016T203000Z DTEND:20251016T220000Z DTSTAMP:20260422T184930Z UID:STAGE/139 DESCRIPTION:Title: Katz's proof of the Riemann hypothesis for curves\nby Jane Shi (MIT) a s part of STAGE\n\nLecture held in Room 2-105 in the MIT Simons Building.\ n\nAbstract\nIn this talk\, we'll study a proof of the Riemann Hypothesis for (projective\, smooth and geometrically connected) curves over finite fields by Katz. We'll study Deligne's version of Rankin's method and the " connect by curves" lemma\, and how they reduce a proof of RH on genus $g$ curves to a proof of RH on Fermat curves. Finally\, we'll introduce the "p ersistence of purity theorem"\, which will be useful for the next talk.\n\ n(Reference: section 1-4 of A Note on Riemann Hypothesis for Curves and Hypersurfaces Over Finite Fiel ds)\n LOCATION:https://researchseminars.org/talk/STAGE/139/ END:VEVENT BEGIN:VEVENT SUMMARY:Leonid Gorodetskii (MIT) DTSTART:20251023T203000Z DTEND:20251023T220000Z DTSTAMP:20260422T184930Z UID:STAGE/140 DESCRIPTION:Title: Katz's proof of the Riemann hypothesis for hypersurfaces\nby Leonid Go rodetskii (MIT) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Simons Building.\n\nAbstract\nIn this talk\, we will discuss Katz’s proo f of the Riemann Hypothesis for hypersurfaces in projective space. Buildin g on techniques developed last time\, we will see how the persistence of p urity theorem reduces the problem to explicit cases --- the Fermat and Gab ber hypersurfaces --- and we will complete the verification using Gauss su ms.\n\nReference: Katz\, A N ote on Riemann Hypothesis for Curves and Hypersurfaces Over Finite Fields< /a>\, Sections 5-8.\n LOCATION:https://researchseminars.org/talk/STAGE/140/ END:VEVENT BEGIN:VEVENT SUMMARY:Mohit Hulse (MIT) DTSTART:20251030T203000Z DTEND:20251030T220000Z DTSTAMP:20260422T184930Z UID:STAGE/141 DESCRIPTION:Title: Deligne's proof in Weil I (Main lemma)\nby Mohit Hulse (MIT) as part o f STAGE\n\nLecture held in Room 2-105 in the MIT Simons Building.\n\nAbstr act\nAfter a quick review of $\\ell$-adic local systems and the étale fun damental group\, I will state and prove Deligne's "main lemma." \nI will t hen derive some consequences to be used in the next few talks\, and if tim e permits\, explain a key fact about $\\operatorname{Sp}$-invariants used in the proof.\n\nReferences:
\n$\\bullet$ Milne\, Lectures on Étale Cohomology\ , Section 30.
\n$\\bullet$ Deligne\, La Conjecture de Weil. I \, Sections 1-3.
\n$\\bullet$ Fulton and Harris\, Representation Theory Appendix F. \n LOCATION:https://researchseminars.org/talk/STAGE/141/ END:VEVENT BEGIN:VEVENT SUMMARY:Jack Miller (Harvard) DTSTART:20251106T213000Z DTEND:20251106T230000Z DTSTAMP:20260422T184930Z UID:STAGE/142 DESCRIPTION:Title: Deligne's proof in Weil I (Lefschetz pencil)\nby Jack Miller (Harvard) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Simons Building .\n\nAbstract\nThe main player in this talk is the notion of a Lefsche tz pencil\, a special kind of 1-parameter family of varieties with ni ce degeneration properties. Because we have discussed how the Riemann Hypo thesis for varieties over finite fields reduces to studying the middle coh omology of an even dimensional variety\, we will produce a Lefschetz fibra tion with odd dimensional fibers whose middle cohomology contains an "inte resting piece\," which we will show has big symplectic monodromy.\n\nRefer ence: Milne\, Lectures on Étale Cohomology\, Section 31-32.\n LOCATION:https://researchseminars.org/talk/STAGE/142/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Zhou (Boston University) DTSTART:20251113T213000Z DTEND:20251113T230000Z DTSTAMP:20260422T184930Z UID:STAGE/143 DESCRIPTION:Title: Deligne's proof in Weil I (Completing the proof)\nby Xinyu Zhou (Bosto n University) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Si mons Building.\n\nAbstract\nIn this talk\, I will finish Deligne's proof o f the Riemann Hypothesis by applying all the tools developed in the previo us talks. Crucially\, the theory of Lefschetz pencils reduces the problem to a study of certain higher direct images on P^1. We then use the Lefsche tz-Picard formula and the Main Lemma to prove an estimate of the Frobenius -eigenvalues on the cohomology of the higher direct images\, which is suff icient to deduce the Riemann Hypothesis.\n\nReference: Milne\, Lectures on Étale Cohom ology\, Section 28 and 33.\n LOCATION:https://researchseminars.org/talk/STAGE/143/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki (Princeton University) DTSTART:20251120T213000Z DTEND:20251120T230000Z DTSTAMP:20260422T184930Z UID:STAGE/144 DESCRIPTION:Title: Weights and the statement of Weil II\nby Kenta Suzuki (Princeton Unive rsity) as part of STAGE\n\nLecture held in Room 2-105 in the MIT Simons Bu ilding.\n\nAbstract\nThe Weil conjecture states that given a smooth projec tive variety over a finite field\, the Frobenius eigenvalues on the étale cohomology have specific absolute values. As is usual in algebraic geomet ry\, we may ask for a relative analog: what happens when there is a morphi sm of schemes? We will introduce weights for étale sheaves on schemes and formulate Weil II\, which gives a relation between the weights of a sheaf to its pushforward. We will see how this recovers the Weil conjecture\, a nd record other consequences such as semisimplicity.\n\nReference:\n\n1. S zamuely\, Section 7.1-7.2.\n\n2. Kiehl-Weissauer\, Weil Conjectures\, Perverse Sheaves and $l$-adic Fourier Transform\, Section I.2\, I.7.\n\n3. Del igne\, Weil II.\n LOCATION:https://researchseminars.org/talk/STAGE/144/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Hu (Harvard) DTSTART:20251204T213000Z DTEND:20251204T230000Z DTSTAMP:20260422T184930Z UID:STAGE/145 DESCRIPTION:Title: Local Monodromy\nby Daniel Hu (Harvard) as part of STAGE\n\nLecture he ld in Room 2-105 in the MIT Simons Building.\n\nAbstract\nWe'll continue o ur discussion of Deligne's generalization of the Weil conjectures to the r elative setting. A theorem of Grothendieck allows us to define the monodro my operator on an ell-adic Galois representation that comes from cohomolog y of smooth proper varieties over local fields. This allows us to define a new filtration on H^i\, called the monodromy filtration. The monodromy-we ight conjecture states that it coincides with the weight filtration\, up t o a shift. We'll apply the case of function fields of curves to deduce the statement of Weil II.\n\nReference:\n\n1. Szamuely\, A course on the Weil conjectures\, Section 7.6.\n\n2. Kiehl-Weissauer\, Weil Conjectures\, Perverse Sheaves and $ l$-adic Fourier Transform\, Section I.3\, I.9.\n\n3. Deligne\, La conj ecture de Weil. II\, 1.3\, 1.7\, 1.8?\n LOCATION:https://researchseminars.org/talk/STAGE/145/ END:VEVENT BEGIN:VEVENT SUMMARY:Elia Gorokhovsky (Harvard) DTSTART:20251211T213000Z DTEND:20251211T230000Z DTSTAMP:20260422T184930Z UID:STAGE/146 DESCRIPTION:Title: Applications of Weil II\nby Elia Gorokhovsky (Harvard) as part of STAG E\n\nLecture held in Room 2-105 in the MIT Simons Building.\n\nAbstract\nW e state two applications of Deligne's theory of weights. The first is the semisimplicity theorem\, which states that a lisse\, pure $\\overline{\\Q_ \\ell}$-sheaf on a normal base over $\\mathbb F_q$ decomposes as a direct sum of irreducible subsheaves over $\\overline{\\mathbb{F}_q}$. The second is a very general theorem about equidistribution of Frobenius elements in the monodromy group\, which enables proofs of several important results i n arithmetic statistics\, such as the Sato-Tate conjecture over function f ields and a version of the Cohen-Lenstra heuristics.\n\nReference:\n\n1. S zamuely. A Course on the Weil Conjectures\, Section 7.2.\n\n2. Katz. Gauss Sums\, Kloosterman Sums\, and Monodromy Groups\, Chapter 3.\n\n3. Deligne . Weil II\, Sections 3.4\, 3.5\n\nSee also:\n\n4. Katz\, Sarnak. Random Ma trices\, Frobenius Eigenvalues\, and Monodromy.\n LOCATION:https://researchseminars.org/talk/STAGE/146/ END:VEVENT BEGIN:VEVENT SUMMARY:Aloysius Ng (MIT) DTSTART:20260205T220000Z DTEND:20260205T233000Z DTSTAMP:20260422T184930Z UID:STAGE/147 DESCRIPTION:Title: Torsors of algebraic groups over fields\nby Aloysius Ng (MIT) as part of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbst ract\nTo build towards the descent obstruction\, we begin by defining tors ors. In this talk\, we introduce $G$-torsors under smooth algebraic group $G/k$ as a generalization of simply transitive $G$-sets of group $G$. We d iscuss its classification and some results that arise from it\, laying the foundation for understanding them over arbitrary bases.\n\nReference: Poo nen\, Rationa l points on varieties\, Section 5.12.\n LOCATION:https://researchseminars.org/talk/STAGE/147/ END:VEVENT BEGIN:VEVENT SUMMARY:Sophie Zhu (Harvard) DTSTART:20260212T220000Z DTEND:20260212T233000Z DTSTAMP:20260422T184930Z UID:STAGE/148 DESCRIPTION:Title: Torsors over an arbitrary base\nby Sophie Zhu (Harvard) as part of STA GE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbstract\n Reference: Poonen\, Rational points on varieties\, Section 6.5 (up to 6.5.5 or 6.5.6) and review of fppf cohomology.\n LOCATION:https://researchseminars.org/talk/STAGE/148/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard) DTSTART:20260219T220000Z DTEND:20260219T233000Z DTSTAMP:20260422T184930Z UID:STAGE/149 DESCRIPTION:Title: Descent obstruction and the local-global principle for torsors\nby Xin yu Fang (Harvard) as part of STAGE\n\nLecture held in Room 2-139 in the MI T Simons Building.\n\nAbstract\nWe start with a quick overview of the desc ent obstruction for the Hasse principle\, indicating how torsors and $H^1( k\,G)$ show up in this context. \nNext\, we introduce contracted products and twisted torsors\, which will be important for us later. Finally\, we d iscuss finiteness results for torsors over local fields and the local-glob al principle for torsors. These will be useful when we discuss unramified torsors and the descent obstruction in more detail later.\n\nReference: \n \n1) Poonen\, Rational points on varieties\, Sections 8.4.7\, 5.12.5-5.12.8\, a nd 6.5.6.\n\n2) Skorobogatov\, Torsors and rational points\, Section 2.2.\ n LOCATION:https://researchseminars.org/talk/STAGE/149/ END:VEVENT BEGIN:VEVENT SUMMARY:Ari Krishna DTSTART:20260226T220000Z DTEND:20260226T233000Z DTSTAMP:20260422T184930Z UID:STAGE/150 DESCRIPTION:Title: Unramified torsors\nby Ari Krishna as part of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbstract\nWe introduce unramifi ed torsors as a tool for further studying local-global questions. A class $\\tau \\in H^1(k\,G)$ is unramified at a place v if it extends over $\\ma thcal O_{k\,v}$\, i.e. if it lies in the image of $H^1(\\mathcal{O}_{k\,v} \,\\mathcal G)\\to H^1(k\,G).$ For a finite set of places $S$ and an exact sequence $$1\\to \\mathcal G^0 \\to \\mathcal G \\to \\mathcal F \\to 1$$ with $\\mathcal{G}^0$ having connected fibers and $\\mathcal{F}$ finite étale\, we prove that the maps $$H^1_S(k\,\\mathcal G) \\to H^1_S(k\,\\m athcal F) \\to \\prod_{v\\in S} H^1(k_v\,F)$$ have finite fibers\, and tha t $H^{1}_S(k\,\\mathcal{G})$ is finite when $k$ is a number field. These f initeness results are key to proving the finiteness of Selmer sets\, which \, e.g.\, offers one route to weak Mordell-Weil in the case of abelian var ieties. Along the way\, we analyze torsors over finite fields using Lang ’s theorem.\n\n\nReference: Poonen\, Rational points on varieties\, 6.5.7.\n LOCATION:https://researchseminars.org/talk/STAGE/150/ END:VEVENT BEGIN:VEVENT SUMMARY:Pitchayut (Mark) Saengrungkongka (MIT) DTSTART:20260305T220000Z DTEND:20260305T233000Z DTSTAMP:20260422T184930Z UID:STAGE/151 DESCRIPTION:Title: Examples of descent\nby Pitchayut (Mark) Saengrungkongka (MIT) as part of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAbs tract\nThe main theorem of descent states that if $X$ is a smooth proper v ariety over global field $k$\, $G$ is a smooth affine algebraic group over $k$\, and $f : Z\\to X$ is a $G$-torsor over $X$\, then the $k$-rational points of $X$ correspond to a union of $k$-rational points of finitely man y twists of $Z$. This reduces the problem of finding rational points of $X $ to finding rational points of finitely many twists of $Z$. We illustrate this theorem through several examples\, including the Weak Mordell-Weil T heorem.\n\nReference: Poonen\, Rational points on varieties\, Section 8.3.\, 8. 4.1-2\, 8.4.4-5.\n LOCATION:https://researchseminars.org/talk/STAGE/151/ END:VEVENT BEGIN:VEVENT SUMMARY:Yutong Chen (MIT) DTSTART:20260312T210000Z DTEND:20260312T223000Z DTSTAMP:20260422T184930Z UID:STAGE/152 DESCRIPTION:Title: The descent obstruction\nby Yutong Chen (MIT) as part of STAGE\n\nLect ure held in Room 2-139 in the MIT Simons Building.\n\nAbstract\nIn this ta lk\, we will briefly review the notion of Selmer sets and how they provide an obstruction to the existence of rational points. We will then prove a theorem showing that\, the Selmer set is finite in interested cases. Next\ , we will define the descent obstruction to the local-global principle and compare it with the Brauer-Manin obstruction. Finally\, we will construct explicit torsors over Iskovskikh's surface to demonstrate that it exhibit s a descent obstruction and\, in particular\, possesses no rational points . We may also apply descent obstruction to see the failure of strong appro ximation if time permits.\n\nReference: Poonen\, Rational points on varieties\, Section 8.1\,8.4\, 8.5.1.\n LOCATION:https://researchseminars.org/talk/STAGE/152/ END:VEVENT BEGIN:VEVENT SUMMARY:Arav Karighattam (MIT) DTSTART:20260319T200000Z DTEND:20260319T213000Z DTSTAMP:20260422T184930Z UID:STAGE/153 DESCRIPTION:Title: $x^2+y^3=z^7$\nby Arav Karighattam (MIT) as part of STAGE\n\nLecture h eld in Room 2-143 in the MIT Simons Building.\n\nAbstract\nThe generalized Fermat equation $ax^p + by^q = cz^r$ admits a fascinating trichotomy in t he theory of its primitive integer solutions: if the invariant $\\chi=1/p+ 1/q+1/r-1$ is positive\, there are either zero or infinitely many solution s (Beukers)\, if $\\chi=0$ it reduces to solving certain elliptic curves\, and if $\\chi<0$ there are only finitely many solutions (Darmon-Granville ). The striking similarity between this result and the finiteness of rati onal points on algebraic curves over $\\Q$ of genus $g\\ge2$ (Faltings' th eorem) is not a coincidence. In fact\, primitive integer solutions to the generalized Fermat equations correspond to rational points on a "stacky c urve" whose Euler characteristic is $\\chi$. The Riemann existence theore m guarantees us a finite étale covering of this stacky curve by an ordina ry curve (which is in our case a branched covering of $\\mathbb{P}^1$ with prescribed ramification). In this talk\, I will explain this general the ory in the case $\\chi<0$ and focus on the explicit computations due to Po onen-Schaefer-Stoll\, using twists of the triply branched covering $\\pi\\ colon X(7)\\to\\mathbb{P}^1$ and their pullbacks to the punctured affine s urface $S=\\text{Spec }\\Z[x\,y\,z]/(x^2+y^3-z^7)\\setminus0$ to determine the primitive integer solutions to $x^2+y^3=z^7$. This is an example of the descent obstruction applied to $G$-torsors over $S$\, where $G=\\text{ Aut }\\pi=\\text{PSL}_2(\\mathbb{F}_7)$!\n\n*Note different time and lo cation.\n\nReference: Poo nen-Schaefer-Stoll\, Twists of $X(7)$ and primitive solutions to $x^ 2+y^3=z^7$\; Darmon\, Faltings plus epsilon\, Wiles plus epsilon\, and the Generalized Fermat Equation\; Darmon-Granville\, On the equations $z^m=F(x\,y)$ and $Ax^p+By^q=Cz^r$ .\n LOCATION:https://researchseminars.org/talk/STAGE/153/ END:VEVENT BEGIN:VEVENT SUMMARY:Leonid Gorodetskii (MIT) DTSTART:20260402T210000Z DTEND:20260402T223000Z DTSTAMP:20260422T184930Z UID:STAGE/154 DESCRIPTION:Title: Universal torsors\nby Leonid Gorodetskii (MIT) as part of STAGE\n\nLec ture held in Room 2-139 in the MIT Simons Building.\n\nAbstract\nIn this t alk\, we focus on torsors under algebraic tori. For an algebraic variety $ X$ with discrete $\\operatorname{Pic}(X_{\\bar k})$ (for instance\, when $ X_{\\bar k}$ is rationally connected)\, the dual group $T = \\operatorname {Hom}(\\operatorname{Pic}(X_{\\bar k})\, \\mathbb{G}_m)$ is a natural alge braic torus associated to $X$. Universal torsors are a class of torsors ov er $X$ under $T$ which\, on the one hand\, can capture the set of all rati onal points on $X$ and\, on the other\, often admit an explicit descriptio n. After developing the theory\, we will see how universal torsors can be used to prove the existence of rational points.\n\nReferences:\n
\n- Wittenberg\, Rational points and zero-cycles on rationally connected varieties over number fields\, Section 3.3. \n
\n- Skorobogatov\, Torsors and Rational Points\, Section 2.3\, p.25 and the references there.\n LOCATION:https://researchseminars.org/talk/STAGE/154/ END:VEVENT BEGIN:VEVENT SUMMARY:Ari Krishna (Harvard) DTSTART:20260409T210000Z DTEND:20260409T223000Z DTSTAMP:20260422T184930Z UID:STAGE/155 DESCRIPTION:Title: Torsors over a diagonal cubic surface\nby Ari Krishna (Harvard) as par t of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n\nAb stract\nDiagonal cubic surfaces are an interesting class of varieties in t he context of the Hasse principle\, since they have a very simple form and the Brauer-Manin obstruction is conjectured to be the only obstruction to the Hasse principle. \n\nIn this and the next talk\, we will construct to rsors over diagonal cubic surfaces under certain tori that play the role o f the universal torsors. We will show that checking whether the Brauer-Man in obstruction is the only one amounts to understanding the Hasse principl e on these torsors. \n\nThe first talk will focus on the construction of s uch a torsor.\n\nReference: Colliot-Thélène\, Kanevsky\, Sansuc\, Arithmétique des surfaces cubiques diagonales\, Section 10(a)(b).\n(English translation)\n LOCATION:https://researchseminars.org/talk/STAGE/155/ END:VEVENT BEGIN:VEVENT SUMMARY:Xinyu Fang (Harvard) DTSTART:20260416T210000Z DTEND:20260416T223000Z DTSTAMP:20260422T184930Z UID:STAGE/156 DESCRIPTION:Title: Descent varieties and Brauer-Manin obstruction on diagonal cubic surfaces< /a>\nby Xinyu Fang (Harvard) as part of STAGE\n\nLecture held in Room 2-13 9 in the MIT Simons Building.\n\nAbstract\nDiagonal cubic surfaces are an interesting class of varieties in the context of the Hasse principle\, sin ce they have a very simple form and the Brauer-Manin obstruction is conjec tured to be the only obstruction to the Hasse principle.\n\nIn this talk\, we construct torsors over diagonal cubic surfaces under a torus\, which p lay the role of the universal torsor. We define the "type" of a torsor\, a nd the obstruction defined by a given type. The main theorem is the equiva lence between the Brauer-Manin obstruction and the obstruction defined by torsors of type $i$ that we constructed earlier. This reduces the problem of whether "the Brauer-Manin obstruction is the only one" to the validity of the Hasse principle for these torsors.\n\nReference: \n\n1. Colliot-Th élène\, Kanevsky\, and Sansuc\, Arithmétique des surfaces cubiques diagonales\ , Section 10(c) + Proposition 10 from (d).\n(English tr anslation)\n\n2. Colliot-Thélène and Sansuc. La descente sur les var iétés rationnelles\, II. (1987)\n LOCATION:https://researchseminars.org/talk/STAGE/156/ END:VEVENT BEGIN:VEVENT SUMMARY:Cedric Xiao (MIT) DTSTART:20260423T210000Z DTEND:20260423T223000Z DTSTAMP:20260422T184930Z UID:STAGE/157 DESCRIPTION:Title: Finite descent on curves I\nby Cedric Xiao (MIT) as part of STAGE\n\nL ecture held in Room 2-139 in the MIT Simons Building.\n\nAbstract\nIn this and the next talk\, we're focusing on the descents of finite étale group schemes\, and similar properties over finite solvable and abelian group s chemes. We discuss on the subset of adelic points cut by these covering co nditions\, and about their relation with the rational points on the variet y. We would further relate these obstruction with the Brauer-Manin obstruc tion\, and develop results over curves. \n\nReference: Stoll\, Finite Descent Obstruc tions and Rational Points on Curves\, Section 5-6 (note the erratum)\n LOCATION:https://researchseminars.org/talk/STAGE/157/ END:VEVENT BEGIN:VEVENT SUMMARY:Kenta Suzuki DTSTART:20260430T210000Z DTEND:20260430T223000Z DTSTAMP:20260422T184930Z UID:STAGE/158 DESCRIPTION:Title: Finite descent on curves II\nby Kenta Suzuki as part of STAGE\n\nLectu re held in Room 2-139 in the MIT Simons Building.\n\nAbstract\nReference: Stoll\, Fin ite descent obstructions and rational points on curves\, Sections 7-9 (not e the erratum)\n LOCATION:https://researchseminars.org/talk/STAGE/158/ END:VEVENT BEGIN:VEVENT SUMMARY:Mohit Hulse DTSTART:20260507T210000Z DTEND:20260507T223000Z DTSTAMP:20260422T184930Z UID:STAGE/159 DESCRIPTION:Title: Finite descent obstruction on curves and modularity\nby Mohit Hulse as part of STAGE\n\nLecture held in Room 2-139 in the MIT Simons Building.\n \nAbstract\nReference: Helm\, Voloch\, Finite descent obstruction on curve s and modularity.\n LOCATION:https://researchseminars.org/talk/STAGE/159/ END:VEVENT BEGIN:VEVENT SUMMARY:TBA DTSTART:20260514T210000Z DTEND:20260514T223000Z DTSTAMP:20260422T184930Z UID:STAGE/160 DESCRIPTION:by TBA as part of STAGE\n\nLecture held in Room 2-139 in the M IT Simons Building.\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/STAGE/160/ END:VEVENT END:VCALENDAR