# THEMATIC PROGRAMS

November 26, 2014

### 2007 Graduate Courses held at the Fields Institute

 Audio& Slides Taking the Course for Credit Thematic Year Homepage

Mailing List : To receive updates on the Program please subscribe to our mailing list at www.fields.utoronto.ca/maillist

Times and dates of courses will be announced when finalized.

 January-February n-categories with duals and TQFT (Jan. 16-19) Instructor: Eugenia Cheng Simplicial presheaves (Jan. 22-29) Instructor: J.F. Jardine March-April Hermitian K-theory (Mar. 5-14) Instructor: Max Karoubi Structure and computations of A^1-homotopy sheaves, with applications (Mar. 6-14) Instructor: Fabien Morel May-June Presheaves of spectra (May1-7) Instructor: J.F. Jardine Derived algebraic geometry and topological modular forms (June 11, 13, 15, 18, 20, and 22) Instructor: Jacob Lurie The Moduli Stack of Formal Groups and Homotopy Theory (May 24-30) Instructor: Paul Goerss Lie n-groupoids (May 28-30) Instructor: Ezra Getzler L-infinity algebras and deformation theory (June 11, 13, 18, 20) Instructor: Kai Behrend

### n-categories with duals and TQFT Instructor: Eugenia Cheng

This course will be aimed at the least experienced portion of the audience so will depend somewhat on the audience. I will hope to cover:

- the three Baez-Dolan Hypotheses, explaining their motivation and the evidence suggesting them
- the periodic table, different ways of moving round it in all sorts of directions
- cobordisms and cobordisms with corners as a motivating and guiding example, the work done in this direction
- the issues involved with defining n-categories with duals, together with what is known and not known in low dimensions

Ideologically, I will emphasise the following viewpoints:
- looking at higher-dimensional structures we know are there, and characterising what those are, rather than just coming up with definitions abstractly

- the importance of working things out very precisely in low dimensions, and indeed at all (rather than just making broad sweeping theories, though these are important too)

- the importance of a theory that is actually useable somehow, rather than something so abstract that we have no idea what it looks like

• J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum
field theory, Jour. Math. Phys., 1995, 36:6073--6105
• J. Baez and J. Dolan, Categorification, in Higher Category Theory, eds.
Ezra Getzler and Mikhail Kapranov, Contemp. Math. 230, American
Mathematical Society, Providence, Rhode Island, 1998, pp. 1-36.
• J. Baez and L. Langford, Higher-dimensional algebra IV: 2-tangles, Adv.
Math., 2003, 180:705--764P.
• Freyd and D. Yetter, Braided compact closed categories with
applications to low dimensional topology, Adv. Math., 1989, 77:156--182
• M. C. Shum, Tortile Tensor Categories, Ph.D. Thesis, Macquarie University 1989

Schedule:

 Tuesday 16 Jan. 3:30-5 p.m. room 230 Wednesday 17 Jan. 3:30-5 p.m. room 230 Thursday 18 Jan. 3:30-5 p.m room 230 Friday 19 Jan. 1:30-3 p.m. room 230

### Project 1: Basic aspects of quasi-categoriesInstructor: Andre Joyal

The fundamental category of a simplicial set. Categorical equivalences. Adjoint
maps. Quasi-categories. Weak categorical equivalences. Inner fibrations, isofibrations. Join and slice. Left and right fibrations. The model structure for quasi-categories. Relation with the classical model structure on simplicial sets. Equivalence with complete Segal spaces, Segal categories and simplicial categories. The fibered model structures. Homotopy factorisation systems. The covariant and contravariant model structures over a base. Initial and terminal objects. Initial and final maps. Fully faithful maps, dominant maps. Minimal quasi-categories. Minimal fibrations. Morita equivalences. Localisation.

1) J.E. Bergner, "A survey of $(\infty,1)$-categories. In preparation.
2) A. Joyal, "Quasi-categories and Kan complexes" JPAA vol 175 (2002), 207-222.
3) A. Joyal, "The theory of quasi-categories I". In preparation.
4) A. Joyal and M. Tierney, "Quasi-categories vs Segal spaces". To appeara
5 ) A. Joyal, "Quasi-categories vs simplicial categories". In preparation.

### Project 2: Extension of category theory to quasi-categories

Adjoint maps. Diagrams. Limits and colimits. Large quasi-categories. The quasi-category K of Kan complexes. Complete and cocomplete quasi-categories. 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 quasi-categories. Quasi-algebra. Locally presentable quasi-categories. Quasi-varieties. Internal categories. Descent. Exact quasi-categories. Quasi-topos. Higher quasi-categories.

1) A. Joyal, "Quasi-categories and Kan complexes" JPAA vol 175 (2002), 207-222.
2) A. Joyal, "The theory of quasi-categories II". In preparation.
3) A. Joyal "The theory of quasi-categories in perspective". To appear
4) J. Lurie "Higher topos theory" In preparation.

Schedule:

 Monday 15 Jan. 9-10 room 230 Tuesday 16 Jan. 9-10 room 230 Wednesday 17 Jan. 9-10 room 230 Friday 19 Jan. 9:30-10:30 room 230 Monday 22 Jan. 9:30-10:30 room 230 Tuesday 23 Jan. 9:30-10:30 room 230 Wednesday 24 Jan. 9:30-10:30 room 230 Friday 26 Jan. 9:30-10:30 room 230 Monday 29 Jan. 9:30-11:00 room 230 Tuesday 30 Jan. 9:30-11:00 room 230 Wednesday 31 Jan. 9:30-11:00 room 230 Friday 2 Feb. 9:30-11:00 room 230

### Simplicial presheaves Instructor: J.F. Jardine

Simplicial sheaves and presheaves, homology and cohomology, descent, localization and motivic homotopy theory, sheaves and presheaves of groupoids, stacks, cocycle categories, torsors, non-abelian cohomology.
lecture notes

1) P.G. Goerss and J.F. Jardine, "Simplicial Homotopy Theory", Progress in Mathematics Vol. 174, Birkhäuser Basel-Boston-Berlin (1999).
2) J.F. Jardine, "Stacks and the homotopy theory of simplicial sheaves", Homology, Homotopy and Applications 3(2) (2001), 361-384.
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., 90:45-143 (2001), 1999.

Schedule:

 Monday 22 Jan. 1:00-2:30 room 230 Tuesday 23 Jan. 1:00-2:30 room 230 Thursday 25 Jan. 1:00-2:30 room 230 Monday 29 Jan. 1:00-2:30 room 230

### Hermitian K-theory Instructor: Max Karoubi

Audio&Slides
Hermitian forms and quadratic forms, positive and negative hermitian K-theory, 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.

M. Karoubi et O. Villamayor : K-théorie algébrique et K-théorie topologique II. Math. Scand. 32, p. 57-86 (1973).
A. Bak : K-theory 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 K-théorie hermitienne. Annals of Math. 112, p. 207-282 (1980).
F.J-B.J Clauwens : the K-theory of almost hermitian forms. Topological structures II, Mathematical Centre Tracts 115, p. 41-49 (1979).
M. Karoubi : stabilization of the Witt group. C.R. Math. Acad. Sci. Paris 342, p. 165-168 (2006)

Schedule:

 Monday 5 Mar. 11:00-12:00 room 230 Wednesday 7 Mar. 11:00-12:00 room 230 Friday 9 Mar. 11:00-12:00 room 230 Monday 12 Mar. 11:00-12:00 room 230 Tuesday 13 Mar. 11:00-12:00 room 230 Wednesday 14 Mar. 11:00-12:00 room 230

### Structure and computations of A^1-homotopy sheaves, with applications Instructor: Fabien Morel

Basic structure of A^1-homotopy sheaves as unramified sheaves on the category of smooth $k$-schemes. A^1-motives and A^1-homology of spaces. The Hurewicz theorem. Description of the H_0 of smash powers of G_m's in terms of Milnor-Witt K-theory. Consequences and examples of computations. The first non-trivial A^1-homotopy sheaf of algebraic spheres. The Brouwer degree.

A^1-homotopy classification of rank n vector bundles. Application to the Euler class for rank n vector bundles over dim n affine smooth n-dimensional schemes and to stably free vector bundles.

The theory of A^1-coverings, A^1-universal coverings and A^1-fundamental group. Examples of computations (for surfaces for instance). Some perspectives towards a surgery classification" of smooth projective A^1-connected varieties and more.

References:

F. Morel. An introduction to A^1-homotopy theory <http://www.mathematik.uni-muenchen.de/%7Emorel/lectureTrieste.ps>, In Contemporary Developments in Algebraic K-theory, ICTP Lecture notes, 15 (2003), pp. 357-441, M. Karoubi, A.O. Kuku , C. Pedrini (ed.).

F. Morel. A^1-algebraic topology over a field <http://www.mathematik.uni-muenchen.de/~morel/A1homotopy.pdf>,
also available as preprint *806 *on the K-theory server, at <http://www.math.uiuc.edu/K-theory/0806/A1homotopy.pdf>

F. Morel. A^1-homotopy classification of vector bundles over smooth affine schemes <http://www.mathematik.uni-muenchen.de/%7Emorel/bgln.pdf>, Preprint.

F. Morel and V. Voevodsky, A^1-homotopy theory of schemes, Publications Mathématiques de l'I.H.E.S, volume 90.

Schedule:

 Tuesday 6 Mar. 10:00-11:00 room 230 Thursday 8 Mar. 10:00-11:00 room 230 Friday 9 Mar. 1:30-2:30pm room 230 Monday 12 Mar. 1:30-2:30pm room 230 Tuesday 13 Mar. 1:30-2:30pm room 230 Wednesday 14 Mar. 1:30-2:30pm room 230

### Presheaves of spectra Instructor: J.F. Jardine

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).

1) J.F. Jardine, "Generalized Etale Cohomology Theories", Progress in Mathematics Vol. 146, Birkhäuser, Basel-Boston-Berlin (1997).
2) J.F. Jardine, "Motivic symmetric spectra", Doc. Math. 5 (2000), 445-552.
3) J.F. Jardine, "Presheaves of chain complexes", K-Theory 30(4) (2003), 365-420.
4) J.F. Jardine, "Generalised sheaf cohomology theories", Axiomatic, Enriched and Motivic Homotopy Theory, NATO Science Series II 131 (2004), 29-68.
5) J.F. Jardine, "Lectures on presheaves of spectra", http://www.math.uwo.ca/~jardine/papers/Spt/index.shtml.

Schedule:

 Tuesday, May 1, 10:00-11:30am room 230 Wednesday, May 2, 10:00-11:30am room 230 Friday, May 4, 10:00-11:30 am room 230 Monday, May 7, 10:00-11:30am room 230

### Derived algebraic geometry and topologicval modular forms Instructor: Jacob Lurie

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), 554--565, 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, 10:00-11:00am room 230 Wednesday, June 13, 10:00-11:00am room 230 Friday, June 15, 10:00-11:00 am room 230 Monday, June 18, 10:00-11:00am room 230 Wednesday, June 20, 10:00-11:00am room 230 Friday, June 22, 10:00-11:00 am room 230

### The Moduli Stack of Formal Groups and Homotopy Theory Instructor: Paul Goerss

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 one-dimensional 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 quasi-coherent sheaves and complex cobordism, the role of coordinates, the height filtration, closed points and Lubin-Tate 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 Deligne-Mumford) 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.

• Behrens, Mark, "A modular description of the {$K(2)$}-local sphere at
the prime 3", Topology 45 No. 2 (2006), 343--402.
• Goerss, Paul, "(Pre-)sheaves of ring spectra over the moduli stack of
formal group laws" , Axiomatic, enriched and motivic homotopy
theory,NATO Sci. Ser. II Math. Phys. Chem., 131, 101-131, Kluwer Acad.
Publ., Dordecht 2004.
• Hopkins, M. J., "Algebraic topology and modular forms", Proceedings of
the International Congress of Mathematicians, Vol. I (Beijing, 2002),
291--317, Higher Ed. Press, Beijing, 2002.
• Hopkins, M. J. and Gross, B. H., The rigid analytic period mapping,
Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc.
(N.S.), 30 No. 1 (1994), 76-86.
• Hovey, Mark and Strickland, Neil, "Comodules and Landweber exact
homology theories", Adv. Math., 192 No. 2 (2005), 427-456.
• Naumann, Niko, "Comodule categories and the geometry of the stack of
formal groups", available at http://front.math.ucdavis.edu/math.AT/0503308.
• Smithling, Brian, "On the moduli stack of commutative, 1-parameter
formal Lie groups" available at http://www.math.uchicago.edu/~bds/fg.pdf

Schedule:

 Thursday 24 May 10:00-11:30 Library Friday 25 May 1:00-2:30 Library Monday 28 May 10:00-11:30 room 230 Tuesday 29 May 10:00-11:30 room 230 Wednesday 30 May 10:00-11:30 room 230

### Lie n-groupoids Ezra Getzler

Schedule:

 Monday 28 May 1:00-3:00 room 230 Tuesday 29 May 1:00-3:00 room 230 Wednesday 30 May 1:00-3:00 room 230

### L-infinity algebras and deformation theory Instructor: Kai Behrend

This course will be an introduction to modern deformation theory using homotopy Lie algebras or L-infinity 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 (Gromov-Witten or Donaldson-Thomas, 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 Donaldson-Thomas theory of Calabi-Yau 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, 1:30-3:00pm room 230 Wednesday, June 13, 1:30-3:00pm room 230 Monday, June 18, 1:30-3:00pm room 230 Wednesday, June 20, 1:30-3:00pm room 230

### Taking the Institute's Courses for Credit

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.

### Financial Assistance

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