
SCIENTIFIC PROGRAMS AND ACTIVITIES 

September 22, 2014  
OVERVIEW A special feature of this workshop are 5 minicourses (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 minicourses, we plan to have several lecture series (2 lectures each) on various topics in the direction of the program aimed at postdocs and researchers:
Schedule

May 13 (Fields Institute, Stewart Library)  
Wed.
May 1 
Thur
May 2 
Fri
May 3 

10:0011:30

A. Quéguiner

A. Quéguiner

A.Quéguiner


May 610 (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) 

9:3010:30


10:3011:00

Coffee

Coffee at Fields

Coffee

Coffee

Coffee

11:0012:00


12:0013:30

Lunch

Lunch

Lunch

Lunch

Lunch

13:3014:30


14:4015:10


15:1015:30

Coffee

Coffee

Coffee

Coffee


15:3016:30


May 1317 (Fields Institute, Room 230)  
Mon
May 13 
Tues
May 14 
Wed
May 15 
Thur
May 16 
Fri
May 17 

9:3010:30


10:3011:00

Coffee

Coffee

Coffee

Coffee

Coffee

11:0012:00


12:0014:00

Lunch

Lunch

Lunch

Lunch

Lunch

14:0015:00


15:0015:30

Coffee

Coffee

Coffee

Coffee


15:3016:30


16:4017:10


17:1517:45

Speaker & Affiliation  Title and Abstract 
Alexey Anayevski St.Petersburg University 
SLoriented 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 SLoriented 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 d’Artois 
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 wellknown 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 SeveriBrauer Varieties A SeveriBrauer variety is a twisted form of projective space. I consider twisted forms of toric varieties as a natural generalization of SeveriBrauer 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

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 (pullback, pushforward, homotopy invariance, localization). Characteristic classes. Basics on the Intersection theory. Chow motives. Motives of flag varieties (cellular decomposition, BruhatTits decomposition). Rost nilpotence and the KrullSchmidt Theorem. 
Christian Haesemeyer University of California at LosAngeles 
Rational points, zero cycles of degree one, and A^1 homotopy theory.

David Harari Université de ParisSud 
Duality theorems over a padic function field. Let K be the function field of a curve over a padic field. We prove PoitouTatelike duality theorems for Ktori and finite Galois modules over K, and give applications to the arithmetic of torsors under Ktori (joint work with Tamas Szamuely). 
Julia Hartmann RWTH Aachen 
Localglobal principles in the theory of linear algebraic groups

Olivier Haution University of Munich 
Singularities of codimension two and algebraic cycles Using Lipman's work on resolution of twodimensional 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 gammafiltration 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 gammarings of twisted flag varieties. In this project, we construct a nontrivial torsion element in the gammaring of a complete flag variety twisted by means of a PGOtorsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline. 
Daniel Krashen University of Georgia 
Minilecture on Patching and a localglobal principle

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 CalmesPetrov 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 hmotive 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 ladic 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 pseudooctonion 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. 

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

Anthony Ruozzi Emory University 
Degree 3 Cohomological Invariants of Split Semisimple Groups

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 Frational, i.e. birational to affine space over the field F. In recent work with Parimala and Suresh, we show that this localglobal principle may fail when the group G is not Frational. The obstruction we use comes from the BlochOgus 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. 

J.P. Tignol l'Université de Louvain 
The discriminant of symplectic involutions

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 BlochKato 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

MINI COURSES
May 17
May 13 at 10:0011:30 a.m.(Stewart library at Fields)
May 6 at 3:304:30 p.m. (Bahen Ctr,
Room 1130)
May 7 at 3:304:30 p.m. (Bahen Ctr,
Room 1240)
MiniCourse 4: Exceptional isomorphisms, triality, and applications to central
simple algebras with involution (Anne QuéguinerMathieu)
May 610Algebras 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 socalled exceptional isomorphisms and triality. We will also explain how they can be used to provide interesting structure theorems, as well as surprising examples.
Chow groups (pullback, pushforward, homotopy invariance, localization). Characteristic classes. Basics on the Intersection theory.
Chow motives. Motives of flag varieties (cellular decomposition, BruhatTits decomposition). Rost nilpotence and the KrullSchmidt Theorem.
May 1317
MiniCourse 6: Localglobal principles in the theory of linear algebraic groups
(Julia Hartmann)
May 1317 at 9:30 am
In this course, we consider localglobal 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 localglobal map with respect to discrete valuations. The proofs use patching methods.
May 610
MiniCourse 7: Motives and algebraic cycles on twisted flag varieties (Kirill
Zainoulline)
May 710 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 BlochKato 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 Jinvariant), explain relations to canonical dimensions, Ktheory, Chow groups, algebraic cobordism of twisted flag variaties and linear algebraic groups.
May 1317
MiniCourse 8: An introduction to the theory of essential dimension. (Zinovy
Reichstein)
May 1315 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  BarIlan University 
Chernousov, Vladimir  University of Alberta 
Chintala, Vineeth  Tata Institute of Fundamental Research 
ColliotThélène, JeanLouis  Universite de ParisSud 
Crooks, Peter  University of Toronto 
De Clercq, Charles  Université Paris 13 
Dolphin, Andrew  Université catholique de Louvain 
Duncan, Alexander  University of Michigan 
GarciaArmas, 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 ParisSud (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, TingYu  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 
QuéguinerMathieu, 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, JeanPierre  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(at)fields.utoronto.ca