{ "145710-21897721": { "datetime_modified":"20260417T111735", "datetime_start":"20260508T143000", "datetime_end":"20260508T153000", "has_end_time":1, "date_start":"2026-05-08", "date_end":"2026-05-08", "time_start":"14:30:00", "time_end":"15:30:00", "time_zone":"America\/Detroit", "event_title":"Growth of the length function for finite-dimensional algebras (Combinatorics Seminar)", "occurrence_title":"", "combined_title":"Growth of the length function for finite-dimensional algebras (Combinatorics Seminar): Alexander Guterman (Bar-Ilan University)", "event_subtitle":"Alexander Guterman (Bar-Ilan University)", "event_type":"Workshop \/ Seminar", "event_type_id":"21", "description":"Let *A* be a finite-dimensional algebra over a field, and *S* be its generating set. The smallest positive integer *k* such that all products of the elements from *S* of degree not exceeding *k* span *A* is called the *length of S*. The maximal length of *S* among all sets *S* that generate *A* is called the *length of A*. The length function is an important invariant widely used to study finite-dimensional algebras since 1959. Due to its numerous applications, it was thoroughly investigated as a purely algebraic problem. It is straightforward to check that the length of an associative finite-dimensional algebra is less than its dimension *d*, and can be equal to *d-1* only for commutative algebras. \n\nWe show that the length of a *d*-dimensional non-associative algebra is bounded by *2^{d-2}*, and this bound is sharp. The investigations of the length function in the non-associative case is closely related with the combinatorial properties of addition chains, i.e., the sequences of natural numbers in which each term is a sum of two previous terms. These sequences are known since ancient times and are useful in the number of applications. In particular, Fibonacci sequence is a classical example of an addition chain without doublings. We prove that the length of a quadratic algebra of dimension *n* is bounded by the Fibonacci number *F_{n-1}*, and this bound is sharp. \n\nThe precise length evaluation is a difficult problem even in the associative case. For example, the length of the full matrix algebra is unknown. It was conjectured by Paz in 1984 to be a linear function of the matrix size; this conjecture is still open. We investigate different algebraic properties of the length function for associative and non-associative algebras and estimate length for different classes of non-associative algebras. \n\nThe talk is based on a series of joint works with Dmitry Kudryavtsev, Olga Markova and Svetlana Zhilina.", "occurrence_notes":null, "guid":"145710-21897721@events.umich.edu", "permalink":"http:\/\/events.umich.edu\/event\/145710", "building_id":"1000166", "building_name":"East Hall", "campus_maps_id":"53", "room":"3866 East Hall", "location_name":"East Hall", "has_livestream":0, "cost":"", "tags":["Mathematics"], "website":"", "sponsors":[ { "group_name":"Combinatorics Seminar - Department of Mathematics", "group_id":"4885", "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":21897721 } , "148085-21902932": { "datetime_modified":"20260505T110722", "datetime_start":"20260508T143000", "datetime_end":"20260508T150000", "has_end_time":1, "date_start":"2026-05-08", "date_end":"2026-05-08", "time_start":"14:30:00", "time_end":"15:00:00", "time_zone":"America\/Detroit", "event_title":"Recent advances in proboscidean paleontology", "occurrence_title":"", "combined_title":"Recent advances in proboscidean paleontology: Advait M. Jukar, Assistant Curator of Vertebrate Paleontology, University of Florida", "event_subtitle":"Advait M. Jukar, Assistant Curator of Vertebrate Paleontology, University of Florida", "event_type":"Lecture \/ Discussion", "event_type_id":"13", "description":"Recent advances in proboscidean paleontology will be discussed. Please join us.", "occurrence_notes":null, "guid":"148085-21902932@events.umich.edu", "permalink":"http:\/\/events.umich.edu\/event\/148085", "building_id":"1005169", "building_name":"Biological Sciences Building", "campus_maps_id":"631", "room":"3150", "location_name":"Biological Sciences Building", "has_livestream":0, "cost":"", "tags":["Museum Of Paleontology","Paleontology","Research Museums Center"], "website":"", "sponsors":[ { "group_name":"Museum Paleontology", "group_id":"3535", "website":"" } ], "image_url":"https:\/\/events.umich.edu\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "styled_images":{ "event_thumb":"https:\/\/events.umich.edu\/media\/cache\/resolve\/event_thumb\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "event_large":"https:\/\/events.umich.edu\/media\/cache\/event_large\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "event_large_2x":"https:\/\/events.umich.edu\/media\/cache\/event_large_2x\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "event_large_lightbox":"https:\/\/events.umich.edu\/media\/cache\/event_large_lightbox\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "group_thumb":"https:\/\/events.umich.edu\/media\/cache\/resolve\/group_thumb\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "group_thumb_square":"https:\/\/events.umich.edu\/media\/cache\/resolve\/group_thumb_square\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "group_large":"https:\/\/events.umich.edu\/media\/cache\/resolve\/group_large\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "group_large_lightbox":"https:\/\/events.umich.edu\/media\/cache\/resolve\/group_large_lightbox\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "event_large_crop":"https:\/\/events.umich.edu\/media\/cache\/resolve\/event_large_crop\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "event_list":"https:\/\/events.umich.edu\/media\/cache\/resolve\/event_list\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "event_list_2x":"https:\/\/events.umich.edu\/media\/cache\/event_list_2x\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "event_grid":"https:\/\/events.umich.edu\/media\/cache\/event_grid\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "event_grid_2x":"https:\/\/events.umich.edu\/media\/cache\/resolve\/event_grid_2x\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "event_feature_large":"https:\/\/events.umich.edu\/media\/cache\/resolve\/event_feature_large\/media\/attachments\/2026\/05\/event_148085_original-2.jpg", "event_feature_thumb":"https:\/\/events.umich.edu\/media\/cache\/resolve\/event_feature_thumb\/media\/attachments\/2026\/05\/event_148085_original-2.jpg" }, "occurrence_count":1, "first_occurrence":21902932 } }