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SCIENTIFIC PROGRAMS AND ACTIVITIES |
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OVERVIEW A special feature of this workshop are 5 mini-courses (5 lectures each) which provide an introduction to the workshop's theme from different perspectives. While introductory, these courses will lead the audience to a survey of the present state of the art:
Besides the mini-courses, we plan to have several lecture series (2 lectures each) on various topics in the direction of the program aimed at post-docs and researchers:
Schedule
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May 1-3 (Fields Institute, Stewart Library) | |||||
Wed.
May 1 |
Thur
May 2 |
Fri
May 3 |
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10:00-11:30
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A. Quéguiner
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A. Quéguiner
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A.Quéguiner
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May 6-10 (Bahen Centre Map to Bahen) | |||||
Mon May 6
(BA1130) |
Tues
May 7 (BA1240) |
Wed.
May 8 (BA1130) |
Thur
May 9 (BA1130) |
Fri
May 10 (BA1190) |
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9:30-10:30
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10:30-11:00
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Coffee
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Coffee at Fields
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Coffee
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Coffee
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Coffee
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11:00-12:00
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12:00-13:30
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Lunch
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Lunch
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Lunch
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Lunch
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Lunch
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13:30-14:30
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14:40-15:10
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15:10-15:30
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Coffee
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Coffee
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Coffee
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Coffee
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15:30-16:30
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May 13-17 (Fields Institute, Room 230) | |||||
Mon
May 13 |
Tues
May 14 |
Wed
May 15 |
Thur
May 16 |
Fri
May 17 |
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9:30-10:30
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10:30-11:00
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Coffee
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Coffee
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Coffee
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Coffee
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Coffee
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11:00-12:00
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12:00-14:00
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Lunch
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Lunch
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Lunch
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Lunch
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Lunch
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14:00-15:00
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15:00-15:30
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Coffee
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Coffee
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Coffee
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Coffee
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15:30-16:30
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16:40-17:10
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17:15-17:45
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Speaker & Affiliation | Title and Abstract |
Alexey Anayevski St.Petersburg University |
SL-oriented cohomology theories The basic and most fundamental computation for an oriented cohomology theory is thE projective bundle theorem claiming A(P^n_k) to be a truncated polynomial ring over A(k) with an explicit basis given by the powers of a Chern class. Having this result at hand one can introduce characteristic classes and carry out a variety of <<geometric>> computations. We establish analogous results for a representable SL-oriented cohomology theory A_\eta with the stable Hopf map inverted. A typical example of a cohomology theory with the prescribed properties is given by the derived Witt groups with the special linear orientation defined via Koszul complexes. It turns out that in this setting one should look at the varieties SL_{n+2}/(SL_2x SL_n) instead of the projective spaces P^n. |
Asher Auel Emory University |
Orthogonal group schemes with simple degeneration Over a base scheme, I will discuss a class of quadratic forms that have the simplest type of nontrivial degeneration along a divisor. Such forms naturally arise in number theory and algebraic geometry; I will give examples related to Gauss composition and to cubic fourfolds containing a plane. Quadratic forms with such simple degeneration turn out to be torsors for orthogonal group schemes that are smooth, yet not reductive, over the base. I will describe the local structure of these orthogonal group schemes, which are interesting objects in their own right. |
Sanghoon Baek KAIST, South Korea |
Semiorthogonal decomposition for twisted Grassmannians A basic way to study a derived category of coherent sheaves is to decompose it into simpler subcategories and this can be implemented by using the notion of semi orthogonal decomposition. Orlov gave the semiorthogonal decompositions for projective, grassmann, and flag bundles, which generalize the full exceptional collections on the corresponding varieties by Beilinson and Kapranov. In the case of projective bundles, Bernardara extended the semiorthogonal decomposition to the twisted forms. In this talk, we present, in a similar way, semiorthogonal decompositions for twisted forms of grassmannians. |
Baptiste Calmes University dArtois |
Torsors over general bases In these lectures, I will give concrete descriptions of categories of torsors under various classical reductive groups. The emphasis will be on working over a general base S rather than over a field. I will also explain how these torsors are mapped to each other using well-known exact sequences of algebraic groups between simply connected forms, adjoint forms, etc. The framework of Giraud's "Cohomologie non abélienne" will be used, but I will try to keep everything elementary, so that someone who is not familiar with stacks, gerbes, etc. should get a first idea of these concepts, without being lost in their generality. |
Alex Duncan University of Michigan |
Toric Varieties and Severi-Brauer Varieties A Severi-Brauer variety is a twisted form of projective space. I consider twisted forms of toric varieties as a natural generalization of Severi-Brauer varieties and discuss how many wellknown structural results have extensions to this more general setting. The main tool is a description of the automorphisms of the Cox ring of a toric variety (a notion closely related to universal torsors). |
Mathieu Florence Universite Paris 6 |
On the rationality of some
homogeneous spaces
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Skip Garibaldi Emory University |
Cohomological invariants of exceptional groups We survey what is known about the cohomological invariants of exceptional groups. For each of the groups, we discuss: Do we know all the cohomological invariants? What are the fibers of the known invariants? Do the values of the cohomological invariants determine the Tits index of the corresponding twisted group? |
Stefan Gille University of Alberta |
Introduction to Chow groups and Chow motives Chow groups (pull-back, push-forward, homotopy invariance, localization). Characteristic classes. Basics on the Intersection theory. Chow motives. Motives of flag varieties (cellular decomposition, Bruhat-Tits decomposition). Rost nilpotence and the Krull-Schmidt Theorem. |
Christian Haesemeyer University of California at Los-Angeles |
Rational points, zero cycles of degree one, and A^1 homotopy theory.
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David Harari Université de Paris-Sud |
Duality theorems over a p-adic function field. Let K be the function field of a curve over a p-adic field. We prove Poitou-Tate-like duality theorems for K-tori and finite Galois modules over K, and give applications to the arithmetic of torsors under K-tori (joint work with Tamas Szamuely). |
Julia Hartmann RWTH Aachen |
Local-global principles in the theory of linear algebraic groups
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Olivier Haution University of Munich |
Singularities of codimension two and algebraic cycles Using Lipman's work on resolution of two-dimensional singularities, I will provide a form of resolution of singularities of codimension two for excellent schemes. I will then discuss applications to the study of algebraic cycles : integrality of the Chern character, Steenrod squares, operational Chow groups. |
Detlev Hoffmann University of Dortmund |
Witt kernels in characteristic 2 for algebraic extensions. A natural question in the algebraic theory of quadratic forms is the determination of Witt kernels, i.e. the kernel of the restriction map when passing from the Witt ring or Witt group of a field to that of a field extension. In general, this is a difficult problem. For odd degree field extensions, the Witt kernels are zero due to a theorem of Springer. For degree 2 extensions, Witt kernels have been known for quite some time (in any characteristic). For degree 4 extensions, these kernels have been determined completely by Sivatski in characteristic not 2. We determine Witt kernels for degree 4 extensions in characteristic 2, extending the partial results that have been known so far. In characteristic 2, there is an added difficulty because of possible inseparability of the extensions |
Rick Jardine University of Western Ontario |
Simplicial sheaves, cocycles and torsors This talk gives a rapid introduction to simplicial sheaves, their |
Caroline Junkins University of Ottawa |
The twisted gamma-filtration and algebras with orthogonal involution For the Grothendieck group of a split simple linear algebraic group, the twisted gamma filtration provides a useful tool for constructing torsion elements in gamma-rings of twisted flag varieties. In this project, we construct a non-trivial torsion element in the gamma-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline. |
Daniel Krashen University of Georgia |
Mini-lecture on Patching and a local-global principle
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Alexander Neshitov University of Ottawa/ Steklov Institute |
Oriented cohomology of algebraic groups and motives of flag varieties In this talk we will discuss the application of the technique developed by Calmes-Petrov Zainoulline to oriented cohomology of algebraic groups and motives of twisted flag varieties. In particular we will show how one can compare oriented cohomology of algebraic group to its chow ring. As an example, we will be able to compute algebraic cobordism of some groups of small ranks. Also we will discuss the relation between the Chow motive of a twisted flag variety and its h-motive for an oriented cohomology h. |
Alena Pirutka IRMA |
On the Tate conjecture for integral classes on cubic fourfolds. Let X be a smooth projective variety defined over a finite field. The Tate conjecture predicts that the cycle class map from the Chow groups of X with rational coefficients to the l-adic étale cohomology groups is surjective. The integral version, which is known not to be true in general, investigates the similar question for integral coefficients. In this talk we will explain how to prove this integral version for codimension two cycles on a cubic fourfold. The strategy is very much inspired by the approach of Claire Voisin used in the context of the integral Hodge conjecture. This is a joint work with F. Charles. |
Anne Queguiner Universite Paris 13 |
Exceptional isomorphisms, triality, valuations, and applications to central simple algebras with involution |
Okubo algebras in characteristic 3 and valuations Okubo algebras are forms of pseudo-octonion algebras, i.e. octonion algebras with a twisted product. An Okubo algebra in characteristic different from 3 and without nonzero idempotents is described as a subspace of a degree 3 central division algebra endowed with the Okubo product. Given an Okubo algebra S in characteristic 0 contained in a division algebra D which is endowed with a valuation with residue characteristic 3, I prove that the residue of S is an Okubo algebra (in characteristic 3) if and only if the residue division algebra has dimension 9 over the ground field and the height of D is maximal. Moreover Okubo algebras in characteristic 3 are always the residue of some Okubo algebra in characteristic 0. |
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Andrei Rapinchuk University of Virginia |
On division algebras having the same maximal subfields. The talk will address the following question: Let D and T be central division algebras over a field K. When does the fact that D and T have the same maximal subfields imply that D and T are actually isomorphic over K? I will discuss various motivations for this question and some recent results. Time permitting, I will also indicate some variations of this question and its generalizations to algebraic groups. This is a joint work with V. Chernousov and I. Rapinchuk. |
Zinovy Reichstein University of British Columbia Lecture Notes |
An introduction to the theory of essential dimension
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Anthony Ruozzi Emory University |
Degree 3 Cohomological Invariants of Split Semisimple Groups
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Espaces homogènes sur les corps de fonctions de courbes sur
un corps local Over such a function field F, D. Harbater, J. Hartmann and D. Krashen have proved a localglobal principle for the existence of rational points on principal homogeneous spaces under a connected linear algebraic group G over F when the underlying variety of G is F-rational, i.e. birational to affine space over the field F. In recent work with Parimala and Suresh, we show that this local-global principle may fail when the group G is not F-rational. The obstruction we use comes from the Bloch-Ogus complex for étale cohomology over an arithmetic surface extending the curve. One may then ask when this new obstruction is the only obstruction to the existence of rational points. |
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J.-P. Tignol l'Université de Louvain |
The discriminant of symplectic involutions
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Kirill Zainoulline |
Motives and algebraic cycles on twisted flag varieties Motives of twisted flag varieties is an important tool in the study of splitting properties of torsors, in the geometric theory of quadratic forms and central simple algebras with involutions. For instance, motives of Pfister quadrics played a key role in the proofs of the Milnor conjecture on quadratic forms and the Bloch-Kato conjecture. Another applications include cohomological invariants, the theory of canonical and essential dimensions of linear algebraic groups. |
Changlong Zhong
University of Ottawa |
On the gamma filtration of oriented cohomology of flag varieties
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MINI- COURSES Algebras with involution are pretty well understood in small degree. As
some essential dimension computation shows, the theory is far less complicated
up to degree 14. Moreover, their automorphism groups are algebraic groups
of small rank that have specific properties. After recalling the basic definitions
and theorems on algebras with involution, we will introduce those tools,
in particular the so-called exceptional isomorphisms and triality. We will
also explain how they can be used to provide interesting structure theorems,
as well as surprising examples.
May 1-7
May 1-3 at 10:00-11:30 a.m.(Stewart library at Fields)
May 6 at 3:30-4:30 p.m. (Bahen Ctr,
Room 1130)
May 7 at 3:30-4:30 p.m. (Bahen Ctr,
Room 1240)
Mini-Course 4: Exceptional isomorphisms, triality, and applications to central
simple algebras with involution (Anne Quéguiner-Mathieu)
May 6-10
Mini-Course 5: Introduction to Chow groups and Chow motives (Stefan Gille)
May 6 at 9:30 and 11 a.m.
May 8, 9, 10 at 9:30 a.m.
Chow groups (pull-back, push-forward, homotopy invariance, localization). Characteristic classes. Basics on the Intersection theory.
Chow motives. Motives of flag varieties (cellular decomposition, Bruhat-Tits decomposition). Rost nilpotence and the Krull-Schmidt Theorem.
May 13-17
Mini-Course 6: Local-global principles in the theory of linear algebraic groups
(Julia Hartmann)
May 13-17 at 9:30 am
In this course, we consider local-global principles for torsors when the base field is an algebraic function field over a complete discretely valued field. We compute the obstructions to these principles with respect to certain other families of overfields. The results then give insight about the original local-global map with respect to discrete valuations. The proofs use patching methods.
May 6-10
Mini-Course 7: Motives and algebraic cycles on twisted flag varieties (Kirill
Zainoulline)
May 7-10 at 11:00 a.m.
Motives of twisted flag varieties is an important tool in the study of splitting properties of torsors, in the geometric theory of quadratic forms and central simple algebras with involutions. For instance, motives of Pfister quadrics played a key role in the proofs of the Milnor conjecture on quadratic forms and the Bloch-Kato conjecture. Another applications include cohomological invariants, the theory of canonical and essential dimensions of linear algebraic groups.
The course will survey some of this research. It will be started with an introduction to the theory motives of twisted flag varieties and conclude with a discussion of open problems. We will introduce and study the discrete motivic invariant of a torsor (the J-invariant), explain relations to canonical dimensions, K-theory, Chow groups, algebraic cobordism of twisted flag variaties and linear algebraic groups.
May 13-17
Mini-Course 8: An introduction to the theory of essential dimension. (Zinovy
Reichstein)
May 13-15 at 11:00 a.m.
The essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. This notion was initially introduced in the context where the objects in question are finite field extensions. Essential dimension has since been investigated in several broader contexts, by a range of techniques, and has been found to have interesting and surprising connections to many problems in algebra and algebraic geometry.
The course will survey some of this research. It will be started with the definition of essential dimension and conclude with a discussion of open problems.
Participant List as of May 3, 2013
Full Name | University/Affiliation |
Ananyevskiy, Alexey | St.Petersburg State University |
Asok, Aravind | University of Southern California |
Auel, Asher | New York University |
Bacard, Hugo | Western University |
Baek, Sanghoon | KAIST |
Bermudez, Hernando | Emory University |
Bhaskhar, Nivedita | Emory University |
Black, Rebecca | University of Maryland |
Burda, Yuri | University of British Columbia |
Calmès, Baptiste | Université d'Artois |
Cely, Jorge | University of Pittsburgh |
Cernele, Shane | University of British Columbia |
Chang, Zhihua | University of Alberta |
Chapman, Adam | Bar-Ilan University |
Chernousov, Vladimir | University of Alberta |
Chintala, Vineeth | Tata Institute of Fundamental Research |
Colliot-Thélène, Jean-Louis | Universite de Paris-Sud |
Crooks, Peter | University of Toronto |
De Clercq, Charles | Université Paris 13 |
Dolphin, Andrew | Université catholique de Louvain |
Duncan, Alexander | University of Michigan |
Garcia-Armas, Mario | University of British Columbia |
Garibaldi, Skip | Emory University |
Gille, Stefan | University of Alberta |
Haesemeyer, Christian | University of California at Los Angeles |
Halacheva, Iva | University of Toronto |
Harari, David | Université de Paris-Sud (Orsay) |
Hartmann, Julia | Rwthaachen University |
Haution, Olivier | University of Munich |
Hoffmann, Detlev | Technische Universität Dortmund |
Jacobson, Jeremy | The Fields Institute |
Jardine, Rick | University of Western Ontario |
Junkins, Caroline | University of Ottawa |
Krashen, Daniel | University of Georgia |
Ledet, Arne | Texas Tech University |
Lee, Ting-Yu | The Fields Institute |
Lefebvre, Jerome | University of British Columbia |
Martel, Justin | University of British Columbia |
Mathieu, Florence | Institut de Mathematiques de Jussieu |
McFaddin, Patrick | University of Georgia |
Monson, Nathaniel | University of Maryland |
Nenashev, Alexander | York University, Glendon College |
Neshitov, Alexander | University of Ottawa |
Opara, Innocent | Central Institute of Mangement |
Parimala, Raman | Emory University |
Pirutka, Alena | Université de Strasbourg |
Pollio, Timothy | University of Virginia |
Prasad, Gopal | University of Michigan |
Quadrelli, Claudio | Western University |
Quéguiner-Mathieu, Anne | Université Paris 13 |
Raczek, Mélanie | Université catholique de Louvain |
Rapinchuk, Andrei | University of Virginia |
Rapinchuk, Igor | Yale University |
Reichstein, Zinovy | University of British Columbia |
Ruozzi, Anthony | Emory University |
Srimathy, Srinivasan | University of Maryland |
Stavrova, Anastasia | The Fields Institute |
Tignol, Jean-Pierre | Université catholique de Louvain |
Vavilov, Nikolai | St. Petersburg State University |
Weekes, Alex | University of Toronto |
Wong, Wanshun | The Fields Institute |
Yagita, Nobuaki | Ibaraki University |
Yahorau, Uladzimir | University of Alberta |
Zainoulline, Kirill | University of Ottawa |
Zhong, Changlong | The Fields Institute |
For additional information contact thematic(PUT_AT_SIGN_HERE)fields.utoronto.ca