BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Ken Ribet (UC Berkeley) DTSTART:20210105T000000Z DTEND:20210105T005000Z DTSTAMP:20260422T212606Z UID:CHAT/1 DESCRIPTION:Title: Lan glands correspondence and geometry\nby Ken Ribet (UC Berkeley) as part of CHAT (Career\, History And Thoughts) series\n\n\nAbstract\nThe Langlan ds program suggests innumerable problems in\narithmetic geometry. One cla ss of problems concerns geometric and\ncohomological relations between alg ebraic varieties in case such\nrelations are predicted by a Langlands corr espondence between spaces\nof automorphic forms for different algebraic gr oups. I will describe\nhow my quest for such a relation led me to realize that there can be a\nlink between the behavior of one Shimura variety in one characteristic\nand the behavior of a second Shimura variety in a seco nd\ncharacteristic.\n LOCATION:https://researchseminars.org/talk/CHAT/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Benedict Gross (UCSD) DTSTART:20210112T000000Z DTEND:20210112T005000Z DTSTAMP:20260422T212606Z UID:CHAT/2 DESCRIPTION:Title: The conjectures of Gan\, Gross\, and Prasad\nby Benedict Gross (UCSD) as part of CHAT (Career\, History And Thoughts) series\n\n\nAbstract\nI will review the conjectures I made with Wee Teck Gan and Dipendra Prasad\, whic h provide a bridge between number theory and representation theory. Beside s stating the various conjectures and reviewing the main results that have been obtained in this direction\, I'll make some historical remarks on ho w we came to formulate them.\n LOCATION:https://researchseminars.org/talk/CHAT/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Harris (Columbia) DTSTART:20210202T000000Z DTEND:20210202T005000Z DTSTAMP:20260422T212606Z UID:CHAT/3 DESCRIPTION:Title: Gal ois representations and torsion cohomology: a series of misunderstandings< /a>\nby Michael Harris (Columbia) as part of CHAT (Career\, History And Th oughts) series\n\n\nAbstract\nIn 2013\, Peter Scholze announced his proof that Galois representations with finite coefficients could be associated t o \ntorsion classes in the cohomology of certain locally symmetric spaces. The existence of such a\ncorrespondence had been predicted by a number o f mathematicians but for a long time no one had the slightest idea\nhow to construct the Galois representations. In this talk I will review some of the history of the problem\, with\nemphasis on the many false starts and occasional successes\, and on my own intermittent involvement\nwith this a nd related problems.\n LOCATION:https://researchseminars.org/talk/CHAT/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Barry Mazur (Harvard) DTSTART:20210405T230000Z DTEND:20210405T235000Z DTSTAMP:20260422T212606Z UID:CHAT/4 DESCRIPTION:Title: Tho ughts about Primes and Knots\nby Barry Mazur (Harvard) as part of CHAT (Career\, History And Thoughts) series\n\n\nAbstract\nKnots and their exq uisitely idiosyncratic properties\, are the vital essence of three-dimensi onal\ntopology. Primes and their exquisitely idiosyncratic properties\, ar e the vital essence of number\ntheory. A striking (and extremely useful) a nalogy between Knots and Primes helped me as I\nbecame as passionate about number theory as I was (and still am) about knots. I’m delighted\nto ha ve been asked by Shekhar and Chi-Yun to be part of the ‘experiment’ in this (experimental)\nseries of talks: CHAT: Career\, History and Thoughts to think again about this\, and take part\nin a Q&A with people in the se minar.\n LOCATION:https://researchseminars.org/talk/CHAT/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Peter Sarnak (Princeton) DTSTART:20210517T230000Z DTEND:20210517T235000Z DTSTAMP:20260422T212606Z UID:CHAT/5 DESCRIPTION:Title: Aut omorphic Cuspidal Representations and Maass Forms\nby Peter Sarnak (Pr inceton) as part of CHAT (Career\, History And Thoughts) series\n\n\nAbstr act\nThe building blocks for automorphic representations on $\\mathrm{GL}_ n$ are the cusp forms. Even the existence of Maass cusp forms is subtle an d tied to arithmetic. I will describe some of my many encounters with thes e trascendental objects and speculate about their role in number theory.\n LOCATION:https://researchseminars.org/talk/CHAT/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Henri Darmon (McGill) DTSTART:20210524T230000Z DTEND:20210524T235000Z DTSTAMP:20260422T212606Z UID:CHAT/6 DESCRIPTION:Title: Mod ular functions and explicit class field theory: private reminiscences and public confessions\nby Henri Darmon (McGill) as part of CHAT (Career\, History And Thoughts) series\n\n\nAbstract\nThe problem of constructing c lass fields of number fields from explicit values of modular functions ha s its roots in the theory of cyclotomic fields and the theory of complex m ultiplication. The latter theory acquired a renewed currency in the second half of the 20th century through its connections to the arithmetic of ell iptic curves\, manifested in the work of Coates--Wiles\, Rubin\, Gross--Za gier\, and Kolyvagin. \n\nI will give a personal account of my path towar ds a (slightly) better understanding of explicit class field theory for re al quadratic fields and its applications to elliptic curves\, taking advan tage of the CHAT format to focus on the misconceptions\, false starts\, an d dead ends that have marked my roundabout and tortuous\, but also very en joyable\, mathematical journey so far.\n LOCATION:https://researchseminars.org/talk/CHAT/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Hélène Esnault (FU Berlin/Harvard/Copenhagen) DTSTART:20231204T230000Z DTEND:20231204T235000Z DTSTAMP:20260422T212606Z UID:CHAT/7 DESCRIPTION:Title: Cod imension one in Algebraic and Arithmetic Geometry\nby Hélène Esnault (FU Berlin/Harvard/Copenhagen) as part of CHAT (Career\, History And Thou ghts) series\n\n\nAbstract\nThe notions of $\\textit{weight}$ in complex g eometry and in $\\ell$-adic theory in geometry over a finite field\nhave b een developed by Deligne and by the Grothendieck school. The analogy betw een the theories is foundational \nand led to predictions and theorems o n both sides. \nOn the complex Hodge theory side\, not only do we have the weight filtration\, but we also have the Hodge filtration. \nThe analogy on the $\\ell$-adic side over a finite field hasn’t really been documen ted by Deligne. \nThinking of this gave the way to understand the Lang--Ma nin conjecture according to which smooth projective\n rationally connected varieties over a finite field possess a rational point. \n$\\url{http://p age.mi.fu-berlin.de/esnault/preprints/helene/62-chowgroup.pdf}$\n\nOn the other hand\, we know the formulation in complex geometry of the Hodge conj ecture: on a smooth projective complex variety $X$\, \na sub-Hodge structu re of $H^{2j}(X)$ of Hodge type $(j\,j)$ should be supported on a codi mension $j$ cycle. The analog $\\ell$-adic conjecture \nhas been formulat ed by Tate\, even over a number field. Grothendieck’s generalized Hodge conjecture is straightforwardly formulated: \na sub-Hodge structure $H$ of $H^i(X)$ of Hodge type $(i-1\,1)\, (i-2\,2)\, \\ldots\, (1\,i-1)$ should be supported on a codimension $1$ cycle. \nEquivalently it should die at t he generic point of the variety. \nThis is difficult to formulate because Hodge structures are complicated to describe. But there is one instance fo r which we can bypass the Hodge formulation:\n$H=H^i(X)$ and $H^{0\,i}=H^i (X\, \\mathcal O)(=H^{i\,0}=H^0(X\, \\Omega^i))=0$. Then the conjecture de scends to the field of definition of $X$ and becomes purely algebraic. \nI t is on the one hand related to the (quite bold) motivic conjectures predi cting that $H^i(X\,\\mathcal O)=0$ for all $i\\neq 0$ should be equivalent to the triviality of the Chow group of $0$-cycles over a large field (thi s brings us back to the proof of the Lang--Manin conjecture). On the other hand\, as it is purely algebraic\, one can try to think of it in the fram ework of today’s $p$-adic Hodge theory.\n LOCATION:https://researchseminars.org/talk/CHAT/7/ END:VEVENT END:VCALENDAR