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DTSTART;TZID=Europe/Berlin:20220707T171500
DTEND;TZID=Europe/Berlin:20220707T183000
DTSTAMP:20260510T045452
CREATED:20220609T081436Z
LAST-MODIFIED:20220707T120136Z
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SUMMARY:Modularity of Galois representations\, from Ramanujan to Serre's Conjecture
DESCRIPTION:CRC Colloquium\n9. Emil Artin Lecture \nProf. Dr. Chandrashekhar Khare\, Department of Mathematics\, UCLA \nRamanujan made a series of conjectures in his 1916 paper “On some arithmetical functions’’ on what is now called the Ramanujan $\tau$ function.  Part of these conjectures were proved soon after Ramanujan formulated them by Mordell\, while one of his conjectures (which is now the first of a vast web of conjectures in the theory of automorphic forms)  took almost 6 decades to be settled (in work of Deligne). A  congruence Ramanujan observed for $\tau(n)$ modulo 691 in the same paper\,  led to Serre and Swinnerton-Dyer developing a geometric theory of mod $p$ modular forms to explain some of Ramanujan’s observations. It was in the context of the theory of mod $p$ modular forms that Serre made his modularity conjecture\, which was initially formulated in a letter of Serre to Tate in 1973.\nI will narrate  this story\, starting from Ramanujan’s work  in 1916\, to the formulation of Serre’s conjecture in 1973\,  to its resolution  in 2009 by Jean-Pierre Wintenberger and myself (using as a key ingredient the modularity lifting method developed by Wiles in his proof of Fermat’s Last Theorem). I will also try to indicate why this subject is very much alive and in spite of all the  progress still in its infancy.
URL:https://crc326gaus.de/event/modularity-of-galois-representations-from-ramanujan-to-serres-conjecture/
LOCATION:Heidelberg\, MATHEMATIKON\, Hörsaal
CATEGORIES:GAUS-Event
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BEGIN:VEVENT
DTSTART;TZID=Europe/Berlin:20220713T121500
DTEND;TZID=Europe/Berlin:20220713T134500
DTSTAMP:20260510T045452
CREATED:20220701T110741Z
LAST-MODIFIED:20220704T090127Z
UID:3237-1657714500-1657719900@crc326gaus.de
SUMMARY:Lecture Series "Some recent developments in singularity theory in mixed and positive characteristic algebraic geometry"
DESCRIPTION:There will be a 4-talk Lecture series on mixed characteristic algebraic geometry by Kevin Tucker (UIC) during July here in Mainz. The first talk is a Kolloquium style talk and should be interesting for (und largely understandable by) most people; and you can decide if you continue with the other 3\, more demanding (but even more rewarding)\, talks. \nTalk 1: Wednesday July 13: 12:15–13:45\nTalk 2: Monday July 18: 12:15–14:45\nTalk 3: Monday July 18: 16:15–17:45\nTalk 4: Wednesday July 20: 12:15-13:35 \nAll talks are taking place in the Hilbertraum 05-432\, Staudingerweg 9\, 55099 Mainz\, or alternatively via Zoom:\nhttps://zoom.us/j/91070632898?pwd=ODM2a1RlZ1RwdVhxVkg2dEk1Vy9CZz09 Meeting ID: 910 7063 2898 Passcode: 123123 \nTitle: Some recent developments in singularity theory in mixed and positive characteristic algebraic geometry\nSpeaker: Kevin Tucker (University of Illinois at Chicago)\nAbstract: Standard “reduction to characteristic p” techniques have long been used to relate singularities defined via the Frobenius map in positive characteristic and those arising in complex algebraic geometry and the Minimal Model Program (MMP). For example\, log terminal and F-regular singularities are known to correspond to one another via reduction to characteristic p >> 0. Exciting developments have recently made it possible to exploit these connections in the mixed characteristic setting as well\, drawing on the (conjectured) characterization of F-regular rings as splinters in positive characteristic. A ring is a splinter if it is a direct summand of every finite cover\, and Hochster’s direct summand conjecture (now a Theorem) is the modest assertion that a regular ring of any characteristic is a splinter. This conjecture was settled affirmatively by André in 2018 who proved the mixed characteristic case more than three decades after Hochster’s verification of the conjecture in equal characteristic using Frobenius techniques. In these talks\, I will discuss some recent works on splinter rings in mixed and positive characteristics. In particular\, inspired by the result of Bhatt in 2020 on the Cohen-Macaulayness of the absolute integral closure\, I will describe a global notion of splinter in the mixed characteristic setting called global +-regularity with applications to birational geometry in mixed characteristic. This can be seen as a generalization of the theory of globally F-regular pairs from positive to mixed characteristic\, and led to the successful development of the three dimensional MMP in mixed characteristics (0; p > 5).
URL:https://crc326gaus.de/event/lecture-series-on-mixed-characteristic-birational-geometry/
LOCATION:Mainz\, Hilbertraum (05-432) and Zoom
CATEGORIES:GAUS-Event
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Europe/Berlin:20220718T121500
DTEND;TZID=Europe/Berlin:20220718T144500
DTSTAMP:20260510T045452
CREATED:20220701T110952Z
LAST-MODIFIED:20220704T090227Z
UID:3239-1658146500-1658155500@crc326gaus.de
SUMMARY:Lecture Series "Some recent developments in singularity theory in mixed and positive characteristic algebraic geometry"
DESCRIPTION:There will be a 4-talk Lecture series on mixed characteristic algebraic geometry by Kevin Tucker (UIC) during July here in Mainz. The first talk is a Kolloquium style talk and should be interesting for (und largely understandable by) most people; and you can decide if you continue with the other 3\, more demanding (but even more rewarding)\, talks. \nTalk 1: Wednesday July 13: 12:15–13:45\nTalk 2: Monday July 18: 12:15–14:45\nTalk 3: Monday July 18: 16:15–17:45\nTalk 4: Wednesday July 20: 12:15-13:35 \nAll talks are taking place in the Hilbertraum 05-432\, Staudingerweg 9\, 55099 Mainz\, or alternatively via Zoom:\nhttps://zoom.us/j/91070632898?pwd=ODM2a1RlZ1RwdVhxVkg2dEk1Vy9CZz09 Meeting ID: 910 7063 2898 Passcode: 123123 \nTitle: Some recent developments in singularity theory in mixed and positive characteristic algebraic geometry\nSpeaker: Kevin Tucker (University of Illinois at Chicago)\nAbstract: Standard “reduction to characteristic p” techniques have long been used to relate singularities defined via the Frobenius map in positive characteristic and those arising in complex algebraic geometry and the Minimal Model Program (MMP). For example\, log terminal and F-regular singularities are known to correspond to one another via reduction to characteristic p >> 0. Exciting developments have recently made it possible to exploit these connections in the mixed characteristic setting as well\, drawing on the (conjectured) characterization of F-regular rings as splinters in positive characteristic. A ring is a splinter if it is a direct summand of every finite cover\, and Hochster’s direct summand conjecture (now a Theorem) is the modest assertion that a regular ring of any characteristic is a splinter. This conjecture was settled affirmatively by André in 2018 who proved the mixed characteristic case more than three decades after Hochster’s verification of the conjecture in equal characteristic using Frobenius techniques. In these talks\, I will discuss some recent works on splinter rings in mixed and positive characteristics. In particular\, inspired by the result of Bhatt in 2020 on the Cohen-Macaulayness of the absolute integral closure\, I will describe a global notion of splinter in the mixed characteristic setting called global +-regularity with applications to birational geometry in mixed characteristic. This can be seen as a generalization of the theory of globally F-regular pairs from positive to mixed characteristic\, and led to the successful development of the three dimensional MMP in mixed characteristics (0; p > 5).
URL:https://crc326gaus.de/event/lecture-series-on-mixed-characteristic-birational-geometry-2/
LOCATION:Mainz\, Hilbertraum (05-432) and Zoom
CATEGORIES:GAUS-Event
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Europe/Berlin:20220718T161500
DTEND;TZID=Europe/Berlin:20220718T174500
DTSTAMP:20260510T045452
CREATED:20220701T111115Z
LAST-MODIFIED:20220704T090302Z
UID:3241-1658160900-1658166300@crc326gaus.de
SUMMARY:Lecture Series "Some recent developments in singularity theory in mixed and positive characteristic algebraic geometry"
DESCRIPTION:There will be a 4-talk Lecture series on mixed characteristic algebraic geometry by Kevin Tucker (UIC) during July here in Mainz. The first talk is a Kolloquium style talk and should be interesting for (und largely understandable by) most people; and you can decide if you continue with the other 3\, more demanding (but even more rewarding)\, talks. \nTalk 1: Wednesday July 13: 12:15–13:45\nTalk 2: Monday July 18: 12:15–14:45\nTalk 3: Monday July 18: 16:15–17:45\nTalk 4: Wednesday July 20: 12:15-13:35 \nAll talks are taking place in the Hilbertraum 05-432\, Staudingerweg 9\, 55099 Mainz\, or alternatively via Zoom:\nhttps://zoom.us/j/91070632898?pwd=ODM2a1RlZ1RwdVhxVkg2dEk1Vy9CZz09 Meeting ID: 910 7063 2898 Passcode: 123123 \nTitle: Some recent developments in singularity theory in mixed and positive characteristic algebraic geometry\nSpeaker: Kevin Tucker (University of Illinois at Chicago)\nAbstract: Standard “reduction to characteristic p” techniques have long been used to relate singularities defined via the Frobenius map in positive characteristic and those arising in complex algebraic geometry and the Minimal Model Program (MMP). For example\, log terminal and F-regular singularities are known to correspond to one another via reduction to characteristic p >> 0. Exciting developments have recently made it possible to exploit these connections in the mixed characteristic setting as well\, drawing on the (conjectured) characterization of F-regular rings as splinters in positive characteristic. A ring is a splinter if it is a direct summand of every finite cover\, and Hochster’s direct summand conjecture (now a Theorem) is the modest assertion that a regular ring of any characteristic is a splinter. This conjecture was settled affirmatively by André in 2018 who proved the mixed characteristic case more than three decades after Hochster’s verification of the conjecture in equal characteristic using Frobenius techniques. In these talks\, I will discuss some recent works on splinter rings in mixed and positive characteristics. In particular\, inspired by the result of Bhatt in 2020 on the Cohen-Macaulayness of the absolute integral closure\, I will describe a global notion of splinter in the mixed characteristic setting called global +-regularity with applications to birational geometry in mixed characteristic. This can be seen as a generalization of the theory of globally F-regular pairs from positive to mixed characteristic\, and led to the successful development of the three dimensional MMP in mixed characteristics (0; p > 5).
URL:https://crc326gaus.de/event/lecture-series-on-mixed-characteristic-birational-geometry-3/
LOCATION:Mainz\, Hilbertraum (05-432) and Zoom
CATEGORIES:GAUS-Event
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Europe/Berlin:20220720T121500
DTEND;TZID=Europe/Berlin:20220720T131500
DTSTAMP:20260510T045452
CREATED:20220701T111243Z
LAST-MODIFIED:20220704T090354Z
UID:3243-1658319300-1658322900@crc326gaus.de
SUMMARY:Lecture Series "Some recent developments in singularity theory in mixed and positive characteristic algebraic geometry"
DESCRIPTION:There will be a 4-talk Lecture series on mixed characteristic algebraic geometry by Kevin Tucker (UIC) during July here in Mainz. The first talk is a Kolloquium style talk and should be interesting for (und largely understandable by) most people; and you can decide if you continue with the other 3\, more demanding (but even more rewarding)\, talks. \nTalk 1: Wednesday July 13: 12:15–13:45\nTalk 2: Monday July 18: 12:15–14:45\nTalk 3: Monday July 18: 16:15–17:45\nTalk 4: Wednesday July 20: 12:15-13:35 \nAll talks are taking place in the Hilbertraum 05-432\, Staudingerweg 9\, 55099 Mainz\, or alternatively via Zoom:\nhttps://zoom.us/j/91070632898?pwd=ODM2a1RlZ1RwdVhxVkg2dEk1Vy9CZz09 Meeting ID: 910 7063 2898 Passcode: 123123 \nTitle: Some recent developments in singularity theory in mixed and positive characteristic algebraic geometry\nSpeaker: Kevin Tucker (University of Illinois at Chicago)\nAbstract: Standard “reduction to characteristic p” techniques have long been used to relate singularities defined via the Frobenius map in positive characteristic and those arising in complex algebraic geometry and the Minimal Model Program (MMP). For example\, log terminal and F-regular singularities are known to correspond to one another via reduction to characteristic p >> 0. Exciting developments have recently made it possible to exploit these connections in the mixed characteristic setting as well\, drawing on the (conjectured) characterization of F-regular rings as splinters in positive characteristic. A ring is a splinter if it is a direct summand of every finite cover\, and Hochster’s direct summand conjecture (now a Theorem) is the modest assertion that a regular ring of any characteristic is a splinter. This conjecture was settled affirmatively by André in 2018 who proved the mixed characteristic case more than three decades after Hochster’s verification of the conjecture in equal characteristic using Frobenius techniques. In these talks\, I will discuss some recent works on splinter rings in mixed and positive characteristics. In particular\, inspired by the result of Bhatt in 2020 on the Cohen-Macaulayness of the absolute integral closure\, I will describe a global notion of splinter in the mixed characteristic setting called global +-regularity with applications to birational geometry in mixed characteristic. This can be seen as a generalization of the theory of globally F-regular pairs from positive to mixed characteristic\, and led to the successful development of the three dimensional MMP in mixed characteristics (0; p > 5).
URL:https://crc326gaus.de/event/lecture-series-on-mixed-characteristic-birational-geometry-4/
LOCATION:Mainz\, Hilbertraum (05-432) and Zoom
CATEGORIES:GAUS-Event
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