
THEMATIC PROGRAMS 

January 15, 2021  
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Times and dates of courses will be announced when finalized.
ncategories with duals
and TQFT

Tuesday 16 Jan. 3:305 p.m.  room 230 
Wednesday 17 Jan. 3:305 p.m.  room 230 
Thursday 18 Jan. 3:305 p.m  room 230 
Friday 19 Jan. 1:303 p.m.  room 230 
Reading list:
1) J.E. Bergner, "A survey of $(\infty,1)$categories. In preparation.
2) A. Joyal, "Quasicategories and Kan complexes" JPAA
vol 175 (2002), 207222.
3) A. Joyal, "The theory of quasicategories I". In preparation.
4) A. Joyal and M. Tierney, "Quasicategories vs Segal spaces".
To appeara
5 ) A. Joyal, "Quasicategories vs simplicial categories".
In preparation.
Adjoint maps. Diagrams. Limits and colimits. Large quasicategories. The quasicategory K of Kan complexes. Complete and cocomplete quasicategories. Grothendieck fibrations. Proper and smooth maps. Kan extensions. Cylinders, distributors and spans. Duality. Yoneda Lemma. The universal left fibration over K. Trace and cotrace. Factorisation systems in quasicategories. Quasialgebra. Locally presentable quasicategories. Quasivarieties. Internal categories. Descent. Exact quasicategories. Quasitopos. Higher quasicategories.
1) A. Joyal, "Quasicategories and Kan complexes" JPAA
vol 175 (2002), 207222.
2) A. Joyal, "The theory of quasicategories II". In preparation.
3) A. Joyal "The theory of quasicategories in perspective".
To appear
4) J. Lurie "Higher topos theory" In preparation.
Schedule:
Monday 15 Jan. 910 room 230 Tuesday 16 Jan. 910 room 230 Wednesday 17 Jan. 910 room 230 Friday 19 Jan. 9:3010:30 room 230 Monday 22 Jan. 9:3010:30 room 230 Tuesday 23 Jan. 9:3010:30 room 230 Wednesday 24 Jan. 9:3010:30 room 230 Friday 26 Jan. 9:3010:30 room 230 Monday 29 Jan. 9:3011:00 room 230 Tuesday 30 Jan. 9:3011:00 room 230 Wednesday 31 Jan. 9:3011:00 room 230 Friday 2 Feb. 9:3011:00 room 230
Simplicial sheaves and presheaves, homology and cohomology, descent,
localization and motivic homotopy theory, sheaves and presheaves
of groupoids, stacks, cocycle categories, torsors, nonabelian cohomology.
lecture
notes
Reading list:
1) P.G. Goerss and J.F. Jardine, "Simplicial Homotopy Theory",
Progress in Mathematics Vol. 174, Birkhäuser BaselBostonBerlin
(1999).
2) J.F. Jardine, "Stacks and the homotopy theory of simplicial
sheaves", Homology, Homotopy and Applications 3(2) (2001),
361384.
3) J.F. Jardine, "Cocycle categories", Preprint (2006),
http://www.math.uwo.ca/~jardine/papers/preprints/index.shtml
4) J.F. Jardine, "Lectures on simplicial presheaves",
http://www.math.uwo.ca/~jardine/papers/sPre/index.shtml
5) Fabien Morel and Vladimir Voevodsky. A^{1}homotopy theory
of schemes. Inst. Hautes Études Sci. Publ. Math., 90:45143
(2001), 1999.
Schedule:
Monday 22 Jan. 1:002:30 room 230 Tuesday 23 Jan. 1:002:30 room 230 Thursday 25 Jan. 1:002:30 room 230 Monday 29 Jan. 1:002:30 room 230
Audio&Slides
Hermitian forms and quadratic forms, positive and negative hermitian
Ktheory, various versions of the periodicity theorem, homotopy
invariance, topological analogs, the case when 2 is not invertible,
a new homology theory on rings : the stabilized Witt group.
Reading list :
M. Karoubi et O. Villamayor : Kthéorie algébrique
et Kthéorie topologique II. Math. Scand. 32, p. 5786 (1973).
A. Bak : Ktheory of forms. Annals of Math. Studies 98. Princeton
University Press (1981).
M. Karoubi : Théorie de Quillen et homologie du groupe orthogonal.
Le théorème fondamental de la Kthéorie hermitienne.
Annals of Math. 112, p. 207282 (1980).
F.JB.J Clauwens : the Ktheory of almost hermitian forms. Topological
structures II, Mathematical Centre Tracts 115, p. 4149 (1979).
M. Karoubi : stabilization of the Witt group. C.R. Math. Acad. Sci.
Paris 342, p. 165168 (2006)
Schedule:
Monday 5 Mar. 11:0012:00 room 230 Wednesday 7 Mar. 11:0012:00 room 230 Friday 9 Mar. 11:0012:00 room 230 Monday 12 Mar. 11:0012:00 room 230 Tuesday 13 Mar. 11:0012:00 room 230 Wednesday 14 Mar. 11:0012:00 room 230
Basic structure of A^1homotopy sheaves as unramified sheaves on the category of smooth $k$schemes. A^1motives and A^1homology of spaces. The Hurewicz theorem. Description of the H_0 of smash powers of G_m's in terms of MilnorWitt Ktheory. Consequences and examples of computations. The first nontrivial A^1homotopy sheaf of algebraic spheres. The Brouwer degree.
A^1homotopy classification of rank n vector bundles. Application to the Euler class for rank n vector bundles over dim n affine smooth ndimensional schemes and to stably free vector bundles.
The theory of A^1coverings, A^1universal coverings and A^1fundamental group. Examples of computations (for surfaces for instance). Some perspectives towards a ``surgery classification" of smooth projective A^1connected varieties and more.
References:
F. Morel. An introduction to A^1homotopy theory <http://www.mathematik.unimuenchen.de/%7Emorel/lectureTrieste.ps>, In Contemporary Developments in Algebraic Ktheory, ICTP Lecture notes, 15 (2003), pp. 357441, M. Karoubi, A.O. Kuku , C. Pedrini (ed.).
F. Morel. A^1algebraic topology over a field <http://www.mathematik.unimuenchen.de/~morel/A1homotopy.pdf>,
also available as preprint *806 *on the Ktheory server, at <http://www.math.uiuc.edu/Ktheory/0806/A1homotopy.pdf>F. Morel. A^1homotopy classification of vector bundles over smooth affine schemes <http://www.mathematik.unimuenchen.de/%7Emorel/bgln.pdf>, Preprint.
F. Morel and V. Voevodsky, A^1homotopy theory of schemes, Publications Mathématiques de l'I.H.E.S, volume 90.
Schedule:
Tuesday 6 Mar. 10:0011:00 room 230 Thursday 8 Mar. 10:0011:00 room 230 Friday 9 Mar. 1:302:30pm room 230 Monday 12 Mar. 1:302:30pm room 230 Tuesday 13 Mar. 1:302:30pm room 230 Wednesday 14 Mar. 1:302:30pm room 230
Presheaves of spectra and symmetric spectra, stable categories, homology and cohomology, motivic stable categories, chain complexes and simplicial abelian presheaves, derived categories, Voevodsky's category of motives (maybe).
Reading list:
1) J.F. Jardine, "Generalized Etale Cohomology Theories",
Progress in Mathematics Vol. 146, Birkhäuser, BaselBostonBerlin
(1997).
2) J.F. Jardine, "Motivic symmetric spectra", Doc. Math.
5 (2000), 445552.
3) J.F. Jardine, "Presheaves of chain complexes", KTheory
30(4) (2003), 365420.
4) J.F. Jardine, "Generalised sheaf cohomology theories",
Axiomatic, Enriched and Motivic Homotopy Theory, NATO Science Series
II 131 (2004), 2968.
5) J.F. Jardine, "Lectures on presheaves of spectra",
http://www.math.uwo.ca/~jardine/papers/Spt/index.shtml.
Schedule:
Tuesday, May 1, 10:0011:30am room 230 Wednesday, May 2, 10:0011:30am room 230 Friday, May 4, 10:0011:30 am room 230 Monday, May 7, 10:0011:30am room 230
Derived algebraic geometry, representability of derived moduli
functors, virtual fundamental classes, derived group schemes and
equivariant cohomology theories, elliptic cohomology, topological
modular forms.
Some references:
M. Hopkins. "Topological modular forms, the Witten genus, and the theorem of the cube." Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 554565, Birkhäuser, Basel, 1995.
B. Toen and G. Vezzosi, "Algebraic Geometry over model categories (a general approach to derived algebraic geometry)". On the archive: math.AG/0110109.
B. Toen. "Higher and derived stacks: a global overview." On the archive: math.AG/0604504.
J. Lurie. "Higher topos theory." Now available on my homepage at http://www.math.harvard.edu/~lurie/.
J. Lurie. "Derived algebraic geometry." (Being rewritten; old version available on my homepage at http://www.math.harvard.edu/~lurie/).
J. Lurie. "A survey of elliptic cohomology." Available on my homepage at http://www.math.harvard.edu/~lurie/).
Schedule:
Monday, June 11, 10:0011:00am  room 230 
Wednesday, June 13, 10:0011:00am  room 230 
Friday, June 15, 10:0011:00 am  room 230 
Monday, June 18, 10:0011:00am  room 230 
Wednesday, June 20, 10:0011:00am  room 230 
Friday, June 22, 10:0011:00 am  room 230 
The interplay between the geometry of formal groups and homotopy theory has been a guiding influence since the work of Morava in '70s, and it has been a thread in homotopy theory ever since. The current language of algebraic geometry allows us to make very concise and natural statements about this relationship.
In these lectures, I will discuss how the geometry of the moduli stack M of smooth onedimensional formal groups dictates the chromatic structure of stable homotopy theory. At a prime, there is an essentially unique filtration of M by the open substacks U(n) of formal groups of height no more than n and the resulting decomposition of coherent sheaves on M gives exactly the chromatic filtration. Topics may include formal groups, the relationship between quasicoherent sheaves and complex cobordism, the role of coordinates, the height filtration, closed points and LubinTate deformation theory, and algebraic chromatic convergence. Since M is in some sense very large and cumbersome, I hope also to give some discussion of small (ie DeligneMumford) stacks over M; these include the moduli stack of elliptic curves and certain Shimura varities, thus bringing us to the current research of Hopkins, Miller, Lurie, Behrens, Lawson, Naumann, Ravenel, Hovey, Strickland, and many others.
Let me remark that I am merely the expositor here, building on the work of many people, especially Jack Morava and Mike Hopkins.
Suggested reading:
Schedule:
Thursday 24 May 10:0011:30  Library 
Friday 25 May 1:002:30  Library 
Monday 28 May 10:0011:30  room 230 
Tuesday 29 May 10:0011:30  room 230 
Wednesday 30 May 10:0011:30  room 230 
Schedule:
Monday 28 May 1:003:00  room 230 
Tuesday 29 May 1:003:00  room 230 
Wednesday 30 May 1:003:00  room 230 
This course will be an introduction to modern deformation theory
using homotopy Lie algebras or Linfinity algebras. These tools
are very important for the local study of moduli spaces. But with
these techniques, one gets more than just the classical moduli space,
one also gets the derived structure. This approach is therefore
also relevant for the local study of derived schemes. Moreover,
these techniques shed light on the obstruction theories involved
in the construction of virtual fundamental classes, which are basic
for many numerical invariants (GromovWitten or DonaldsonThomas,
for example) defined in terms of moduli spaces. We will explain
the general theory and then specialize to the cyclic case, which
corresponds to the case of a symmetric obstruction theory. This
case is of particular importance for the DonaldsonThomas theory
of CalabiYau threefolds.
We hope to make this lecture series accessible to graduate students with a strong background in algebraic geometry.
Literature:
M. Manetti: Lectures on deformations of complex manifolds,
arXiv:math.AG/0507286
M. Manetti: Deformation theory via differential graded Lie algebras,
arXiv:math.AG/0507284
Schedule:
Monday, June 11, 1:303:00pm  room 230 
Wednesday, June 13, 1:303:00pm  room 230 
Monday, June 18, 1:303:00pm  room 230 
Wednesday, June 20, 1:303:00pm  room 230 
As graduate students at any of the Institute's University Partners, you may discuss the possibility of obtaining a credit for one or more courses in this lecture series with your home university graduate officer and the course instructor. Assigned reading and related projects may be arranged for the benefit of students requiring these courses for credit.
As part of the Affiliation agreement with some Canadian Universities,
graduate students are eligible to apply for financial assistance
to attend graduate courses. To apply for funding, apply
here.
Two types of support are available:
Students outside the greater Toronto area may apply for travel
support. Please submit a proposed budget outlining expected costs
if public transit is involved, otherwise a mileage rate is used
to reimburse travel costs. We recommend that groups coming from
one university travel together, or arrange for car pooling (or car
rental if applicable).
Students outside the commuting distance of Toronto may submit an
application for a term fellowship. Support is offered up to $1000
per month.
For more details on the thematic year, see Program Page or contact homotopy@fields.utoronto.ca