{ "144446-21895363": { "datetime_modified":"20260125T172305", "datetime_start":"20260127T150000", "datetime_end":"20260127T160000", "has_end_time":1, "date_start":"2026-01-27", "date_end":"2026-01-27", "time_start":"15:00:00", "time_end":"16:00:00", "time_zone":"America\/Detroit", "event_title":"Derived Fun: Towards Homotopical Algebra", "occurrence_title":"", "combined_title":"Derived Fun: Towards Homotopical Algebra: Ajay Srinivasan", "event_subtitle":"Ajay Srinivasan", "event_type":"Workshop \/ Seminar", "event_type_id":"21", "description":"Derived functors play a central role in commutative algebra, organizing algebraic computations through exact sequences and homological methods. This talk begins with a brief conceptual revisit of derived functors in homological algebra, and then examines K\u00e4hler differentials of commutative rings as a motivating example for why we want derived functors beyond abelian categories. \n\nTo address this problem, we trace Dan Quillen\u2019s insight that homotopical methods provide an ideal framework for defining derived functors in nonabelian settings. Along the way, we encounter his definition of model categories, which have since become central tools in modern homotopy theory. From this perspective, the cotangent complex arises as a derived replacement for K\u00e4hler differentials. We will outline the construction of the cotangent complex and discuss some of its powerful applications to deformation theory. Time permitting, I will say something about how the homotopical viewpoint reconciles with homological algebra.", "occurrence_notes":null, "guid":"144446-21895363@events.umich.edu", "permalink":"http:\/\/events.umich.edu\/event\/144446", "building_id":"1000166", "building_name":"East Hall", "campus_maps_id":"53", "room":"3088", "location_name":"East Hall", "has_livestream":0, "cost":"", "tags":["Mathematics"], "website":"", "sponsors":[ { "group_name":"Student Commutative Algebra Seminar - Department of Mathematics", "group_id":"4908", "website":"" }, { "group_name":"Department of Mathematics", "group_id":"3791", "website":"" } ], "image_url":"", "styled_images":{ "event_thumb":"", "event_large":"", "event_large_2x":"", "event_large_lightbox":"", "group_thumb":"", "group_thumb_square":"", "group_large":"", "group_large_lightbox":"", "event_large_crop":"", "event_list":"", "event_list_2x":"", "event_grid":"", "event_grid_2x":"", "event_feature_large":"", "event_feature_thumb":"" }, "occurrence_count":1, "first_occurrence":21895363 } , "143123-21892177": { "datetime_modified":"20251226T220321", "datetime_start":"20260127T160000", "datetime_end":"20260127T170000", "has_end_time":1, "date_start":"2026-01-27", "date_end":"2026-01-27", "time_start":"16:00:00", "time_end":"17:00:00", "time_zone":"America\/Detroit", "event_title":"Colloquium: Topology, graphs, and data", "occurrence_title":"", "combined_title":"Colloquium: Topology, graphs, and data: Chris Kapulkin, University of Western Ontario", "event_subtitle":"Chris Kapulkin, University of Western Ontario", "event_type":"Lecture \/ Discussion", "event_type_id":"13", "description":"This talk will be an introduction to the emerging area of discrete\r\nhomotopy theory, which applies intuitions and techniques from the\r\ncontinuous setting to discrete objects such as graphs. It has found a\r\nbroad range of applications, both within and outside mathematics,\r\nincluding to matroid theory, hyperplane arrangements, and data analysis.\r\n\r\nI will discuss two of my own contributions to discrete homotopy theory,\r\none more theoretical and one more applied. The first is a proof, joint\r\nwith D. Carranza (Compos. Math., 2024), of the conjecture by E. Babson,\r\nH. Barcelo, M. de Longueville, and R. Laubenbacher that discrete\r\nhomotopy groups can be topologically realized. The second, joint with N.\r\nKershaw (arXiv:2506.15020), builds on this result and introduces a new\r\nmethod of data analysis, which we call persistent discrete homology. We\r\nshow that in addition to its utility for clustering, it can detect other\r\ngeometric features of a data set. It is furthermore highly noise\r\nresistant, and as such provides a powerful alternative to the usual\r\nmethods of (unsupervised) machine learning, especially in areas subject\r\nto high uncertainty, such as seismology or crime linkage.", "occurrence_notes":null, "guid":"143123-21892177@events.umich.edu", "permalink":"http:\/\/events.umich.edu\/event\/143123", "building_id":"1000166", "building_name":"East Hall", "campus_maps_id":"53", "room":"1360", "location_name":"East Hall", "has_livestream":0, "cost":"", "tags":["Mathematics"], "website":"https:\/\/www.math.uwo.ca\/faculty\/kapulkin\/index.html", "sponsors":[ { "group_name":"Colloquium Series - Department of Mathematics", "group_id":"4884", "website":"" }, { "group_name":"Department of Mathematics", "group_id":"3791", "website":"" } ], "image_url":"https:\/\/events.umich.edu\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "styled_images":{ "event_thumb":"https:\/\/events.umich.edu\/media\/cache\/event_thumb\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "event_large":"https:\/\/events.umich.edu\/media\/cache\/event_large\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "event_large_2x":"https:\/\/events.umich.edu\/media\/cache\/event_large_2x\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "event_large_lightbox":"https:\/\/events.umich.edu\/media\/cache\/event_large_lightbox\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "group_thumb":"https:\/\/events.umich.edu\/media\/cache\/group_thumb\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "group_thumb_square":"https:\/\/events.umich.edu\/media\/cache\/group_thumb_square\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "group_large":"https:\/\/events.umich.edu\/media\/cache\/group_large\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "group_large_lightbox":"https:\/\/events.umich.edu\/media\/cache\/group_large_lightbox\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "event_large_crop":"https:\/\/events.umich.edu\/media\/cache\/event_large_crop\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "event_list":"https:\/\/events.umich.edu\/media\/cache\/event_list\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "event_list_2x":"https:\/\/events.umich.edu\/media\/cache\/event_list_2x\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "event_grid":"https:\/\/events.umich.edu\/media\/cache\/event_grid\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "event_grid_2x":"https:\/\/events.umich.edu\/media\/cache\/event_grid_2x\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "event_feature_large":"https:\/\/events.umich.edu\/media\/cache\/event_feature_large\/media\/attachments\/2025\/12\/event_143123_original-1.jpg", "event_feature_thumb":"https:\/\/events.umich.edu\/media\/cache\/event_feature_thumb\/media\/attachments\/2025\/12\/event_143123_original-1.jpg" }, "occurrence_count":1, "first_occurrence":21892177 } }