###
25/06/2014, 11:30 — 12:30 — Room P3.10, Mathematics Building

Siye Wu, *University of Hong Kong*

```
```###
Branes and quantization for mathematicians - III

Branes are extended objects that define the boundary conditions of sigma models. These lectures cover the geometry of branes, their roles in duality and mirror symmetry, and the relation to quantization.

###
25/06/2014, 10:00 — 11:00 — Room P3.10, Mathematics Building

Bertrand Toën, *Université de Montpellier 2*

```
```###
Quantization in the context of derived algebraic geometry - III

This last lecture is concerned with the construction of deformation quantization of moduli spaces endowed with shifted symplectic structures. More generally, I will present the notion of shifted Poisson structures as well as a shifted version of Kontsevich's formality theorem. I will explain how this implies the existence of quantizations. The lecture will end with examples, some recovering well known quantum objects (e.g. quantum groups), and some new.

###
24/06/2014, 11:30 — 12:30 — Room P3.10, Mathematics Building

Bertrand Toën, *Université de Montpellier 2*

```
```###
Quantization in the context of derived algebraic geometry - II

In this second lecture I will present more about derived algebraic geometry and will introduce the notion of shifted symplectic structures. I will state several existence theorems and deduce that many moduli spaces, when suitably considered as derived algebraic stacks, are endowed with natural shifted symplectic structures.

###
24/06/2014, 10:00 — 11:00 — Room P3.10, Mathematics Building

Siye Wu, *University of Hong Kong*

```
```###
Branes and quantization for mathematicians - II

Branes are extended objects that define the boundary conditions of sigma models. These lectures cover the geometry of branes, their roles in duality and mirror symmetry, and the relation to quantization.

###
23/06/2014, 11:30 — 12:30 — Room P3.10, Mathematics Building

Siye Wu, *University of Hong Kong*

```
```###
Branes and quantization for mathematicians - I

Branes are extended objects that define the boundary conditions of sigma models. These lectures cover the geometry of branes, their roles in duality and mirror symmetry, and the relation to quantization.

###
23/06/2014, 10:00 — 11:15 — Room P3.10, Mathematics Building

Bertrand Toën, *Université de Montpellier 2*

```
```###
Quantization in the context of derived algebraic geometry - I

I will present the main objective of the series of lectures, namely the construction of deformation quantization of certain moduli spaces. In a second part I will present some of the basic notions of derived algebraic geometry, such as derived schemes and derived algebraic stacks.

###
26/06/2013, 16:00 — 17:00 — Room P3.10, Mathematics Building

Song Sun, *Imperial College*

```
```###
Kahler-Einstein metrics and stability - IV

In these lectures we will explain the recent proof by X-X. Chen, S.
K. Donaldson and the speaker on Yau's conjecture relating existence
of Kahler-Einstein metrics on Fano manifolds to algebro-geometric K
-stability. We will describe the main problem, outline the strategy
and highlight some technical aspects involved in the proof.

###
26/06/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

Song Sun, *Imperial College*

```
```###
Kahler-Einstein metrics and stability - III

In these lectures we will explain the recent proof by X-X. Chen, S.
K. Donaldson and the speaker on Yau's conjecture relating existence
of Kahler-Einstein metrics on Fano manifolds to algebro-geometric K
-stability. We will describe the main problem, outline the strategy
and highlight some technical aspects involved in the proof.

###
25/06/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

Song Sun, *Imperial College*

```
```###
Kahler-Einstein metrics and stability - II

In these lectures we will explain the recent proof by X-X. Chen, S.
K. Donaldson and the speaker on Yau's conjecture relating existence
of Kahler-Einstein metrics on Fano manifolds to algebro-geometric K
-stability. We will describe the main problem, outline the strategy
and highlight some technical aspects involved in the proof.

###
24/06/2013, 16:00 — 17:00 — Room P3.10, Mathematics Building

Song Sun, *Imperial College*

```
```###
Kahler-Einstein metrics and stability - I

In these lectures we will explain the recent proof by X-X. Chen, S.
K. Donaldson and the speaker on Yau's conjecture relating existence
of Kahler-Einstein metrics on Fano manifolds to algebro-geometric
K-stability. We will describe the main problem, outline the
strategy and highlight some technical aspects involved in the
proof.

###
25/07/2012, 15:45 — 16:45 — Room P3.10, Mathematics Building

Bruno Oliveira, *University of Miami and New York University*

```
```###
Symmetric differentials and fundamental group

The relationship between the algebra of symmetric differentials
(sections of the symmetric powers of the holomorphic cotangent
bundle) and the topology of a projective manifold is still
considered quite mysterious. In general this relationship will be
quite loose, since for example it is known that there are manifolds
with the same topology but diametrically distinct spaces of
symmetric differentials (e.g. no symmetric differentials and
asymptotically as many as possible). On the other hand, it is
expected that properties of the fundamental group to be reflected
on the space of symmetric differentials.
The goal of these lectures is to show that there is a class of
symmetric differentials that is quite topological in nature. This
class constitutes an extension to all degrees of the class of
closed symmetric differentials of degree 1 (i.e. closed holomorphic
1-forms) which are well known to reflect topological properties. We
will: discuss primarily the case of degree 2; describe examples;
connect to the theory of foliations and fibrations; and show that
the presence of closed symmetric differentials imply that the
fundamental group has to be infinite.

###
25/07/2012, 14:00 — 15:00 — Room P3.10, Mathematics Building

Yongnam Lee, *Sogang University*

```
```###
$Q$-Gorenstein deformation theory and its applications

The 2nd lecture will explain several methods for constructions of
surfaces of general type via $Q$-Gorenstein smoothings. After the
paper by Lee and Park, which constructs a simply connected
Campedelli surface, several interesting examples of surfaces of
general type with ${p}_{g}=0$ were constructed via $Q$-Gorenstein
smoothings. Now, these $Q$-Gorenstein smoothing methods are
extended to some other type of surfaces and to surfaces in positive
characteristics.

#### References

- J. Kollár and N. I. Shepherd-Barron, Threefolds and
deformations of surface singularities, Invent. Math. 91 (1988),
299-338.
- Y. Lee and J. Park, A simply connected surface of general type
with ${p}_{g}=0$ and ${K}^{2}=2$, Invent. Math. 170 (2007), 483-505.
- Y. Lee and N. Nakayama, Simply connected surfaces of general
type in positive characteristic via deformation theory, preprint
2011 (arXiv:1103.5185, to
appear in PLMS).

###
24/07/2012, 15:45 — 16:45 — Room P3.10, Mathematics Building

JongHae Keum, *Korean Institute for Advanced Study*

```
```###
Fake projective planes

It is known that a compact complex manifold of dimension $2$ with
the same Betti numbers as the complex projective plane is
projective. Such a manifold is called "a fake projective plane" if
it is not isomorphic to the complex projective plane. So a fake
projective plane is exactly a smooth surface $X$ of general type
with ${p}_{g}(X)=0$ and ${c}_{1}(X{)}^{2}=3{c}_{2}(X)=9$. By a result of Aubin and
Yau, its universal cover is the unit $2$-ball, hence its
fundamental group ${\pi}_{1}(X)$ is a discrete torsion-free cocompact
subgroup of $\mathrm{PU}(2,1)$ satisfying certain conditions. The
classification of such subgroups has been done by G. Parasad and
S.-K. Yeung, with the computer based computation by D. Cartwright
and J. Steger. This has settled the arithmetic side of the
classification problem. I will go over this, and then report recent
progress on the other side of the problem-geometric construction.

###
24/07/2012, 14:00 — 15:00 — Room P3.10, Mathematics Building

Bruno Oliveira, *University of Miami and New York University*

```
```###
Symmetric differentials and fundamental group

The relationship between the algebra of symmetric differentials
(sections of the symmetric powers of the holomorphic cotangent
bundle) and the topology of a projective manifold is still
considered quite mysterious. In general this relationship will be
quite loose, since for example it is known that there are manifolds
with the same topology but diametrically distinct spaces of
symmetric differentials (e.g. no symmetric differentials and
asymptotically as many as possible). On the other hand, it is
expected that properties of the fundamental group to be reflected
on the space of symmetric differentials.
The goal of these lectures is to show that there is a class of
symmetric differentials that is quite topological in nature. This
class constitutes an extension to all degrees of the class of
closed symmetric differentials of degree 1 (i.e. closed holomorphic
1-forms) which are well known to reflect topological properties. We
will: discuss primarily the case of degree 2; describe examples;
connect to the theory of foliations and fibrations; and show that
the presence of closed symmetric differentials imply that the
fundamental group has to be infinite.

###
23/07/2012, 15:45 — 16:45 — Room P3.10, Mathematics Building

Yongnam Lee, *Sogang University*

```
```###
$Q$-Gorenstein deformation theory and its applications

The 1st lecture will review the singularity of class $T$ and
$Q$-Gorenstein deformation theory. The notion of singularity of
class $T$, which is defined as a quotient surface singularity
admitting a $Q$-Gorenstein smoothing, was introduced by Kollár and
Shepherd-Barron. They also gave an explicit description of the
singularity of class $T$. The notion of $Q$-Gorenstein deformation
is popular in the study of degenerations of normal algebraic
varieties in characteristic zero related to the minimal model
theory and the moduli theory since the paper by Kollár and
Shepherd-Barron. A typical example of $Q$-Gorenstein deformation
appears as a deformation of the weighted projective plane $P(1,1,4)$: Its versal deformation space has two irreducible
components, in which the one-dimensional component corresponds to
the $Q$-Gorenstein deformation and its general fibers are
projective planes. By developing the theory of $Q$-Gorenstein
deformation functor, we can generalize their results to surfaces
in positive characteristics.

#### References

- J. Kollár and N. I. Shepherd-Barron, Threefolds and
deformations of surface singularities, Invent. Math. 91 (1988),
299-338.
- Y. Lee and J. Park, A simply connected surface of general type
with ${p}_{g}=0$ and ${K}^{2}=2$, Invent. Math. 170 (2007), 483-505.
- Y. Lee and N. Nakayama, Simply connected surfaces of general
type in positive characteristic via deformation theory, preprint
2011 (arXiv:1103.5185, to
appear in PLMS).

###
23/07/2012, 14:00 — 15:00 — Room P3.10, Mathematics Building

JongHae Keum, *Korean Institute for Advanced Study*

```
```###
Fake projective planes

It is known that a compact complex manifold of dimension $2$ with
the same Betti numbers as the complex projective plane is
projective. Such a manifold is called "a fake projective plane" if
it is not isomorphic to the complex projective plane. So a fake
projective plane is exactly a smooth surface $X$ of general type
with ${p}_{g}(X)=0$ and ${c}_{1}(X{)}^{2}=3{c}_{2}(X)=9$. By a result of Aubin and
Yau, its universal cover is the unit $2$-ball, hence its
fundamental group ${\pi}_{1}(X)$ is a discrete torsion-free cocompact
subgroup of $\mathrm{PU}(2,1)$ satisfying certain conditions. The
classification of such subgroups has been done by G. Parasad and
S.-K. Yeung, with the computer based computation by D. Cartwright
and J. Steger. This has settled the arithmetic side of the
classification problem. I will go over this, and then report recent
progress on the other side of the problem-geometric construction.

###
30/06/2011, 14:00 — 15:00 — Room P3.10, Mathematics Building

Nitu Kitchloo, *Johns Hopkins and UCSD*

```
```###
Geometry, Topology and Representation Theory of Loop Groups

In this sequence of three talks, I will aim to introduce the
algebraic and geometric structure of Loop groups and their
representations. We will begin with the basic structure of Affine
Lie algebras. This will lead us to the algebraic theory of positive
energy representations indexed by the level. On the geometric side,
we will introduce the Affine Loop group and relate it to the
central extension of the smooth loop group. We will also study the
example of the special unitary group in some detail. In the
remaining time, I will go into some of the deeper structure of Loop
groups. This includes fusion in the representations of a given
level (via the geometric notion of conformal blocks). Time
permitting, I will also describe the homotopy type of the
classifying space of Loop groups. No special background is
required. It would be helpful to know the basic theory of root
systems for semisimple Lie algebras, though this is not a strict
requirement.

#### References

- Arnaud Beauville, Conformal
blocks, fusion rules and the Verlinde formula, Proc. of the
Hirzebruch 65 Conf. on Algebraic Geometry, Israel Math. Conf. Proc.
9, 75-96 (1996).
- Victor Kac, Infinite dimensional Lie algebras, Cambridge
University Press (1990).
- Nitu Kitchloo, On the
topology of Kac-Moody groups (2008).
- Andrew Pressley and Graeme Segal, Loop Groups, Oxford
University Press (1986).

#### See also

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/Loop.pdf

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/LoopGps.pdf

###
29/06/2011, 14:00 — 15:00 — Room P3.10, Mathematics Building

Mark Behrens, *MIT*

```
```###
Topological Automorphic Forms

#### Topological Automorphic Forms II: examples, problems, and
applications

I will survey some known computations of Topological Automorphic
Forms. K-theory and TMF will be shown to be special cases to TAF.
Certain TAF spectra have been identified with $\mathrm{BP}\u27e82\u27e9$
by Hill and Lawson, showing these spectra admit ${E}_{\mathrm{oo}}$ ring
structures. $K(n)$-local TAF gives instances of the higher real
K-theories ${\mathrm{EO}}_{n}$, one of which shows up in the solution of the
Kervaire invariant one problem. Associated to the TAF spectra are
certain approximations of the $K(n)$-local sphere, which are
expected to see "Greek letter elements" in the same manner that TMF
sees the divided beta family. Finally, I will discuss some partial
results and questions concerning an automorphic forms valued genus
which is supposed to generalize the Witten genus.

#### References

- Mark Behrens, Notes
on the construction of TMF (2007).
- Mark Behrens and Tyler Lawson,
Topological Automorphic Forms, Memoirs of the AMS 958
(2010).
- Paul Goerss,
Topological modular forms (after Hopkins, Miller and Lurie),
Séminaire Bourbaki, 2009.
- Mike Hopkins,
Topological modular forms, the Witten genus and the Theorem of the
cube, Proceedings of the 1994 ICM.
- Mike Hopkins,
Algebraic Topology and Modular Forms, Proceedings of the 2002
ICM.
- Tyler Lawson, An overview of
abelian varieties in homotopy theory (2008).

Doug Ravenel's web page for a seminar on topological
automorphic forms contains a comprehensive list of references.

#### See also

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/Behrenstalk3.pdf

###
29/06/2011, 11:00 — 12:00 — Room P3.10, Mathematics Building

Nitu Kitchloo, *Johns Hopkins and UCSD*

```
```###
Geometry, Topology and Representation Theory of Loop Groups

In this sequence of three talks, I will aim to introduce the
algebraic and geometric structure of Loop groups and their
representations. We will begin with the basic structure of Affine
Lie algebras. This will lead us to the algebraic theory of positive
energy representations indexed by the level. On the geometric side,
we will introduce the Affine Loop group and relate it to the
central extension of the smooth loop group. We will also study the
example of the special unitary group in some detail. In the
remaining time, I will go into some of the deeper structure of Loop
groups. This includes fusion in the representations of a given
level (via the geometric notion of conformal blocks). Time
permitting, I will also describe the homotopy type of the
classifying space of Loop groups. No special background is
required. It would be helpful to know the basic theory of root
systems for semisimple Lie algebras, though this is not a strict
requirement.

#### References

- Arnaud Beauville, Conformal
blocks, fusion rules and the Verlinde formula, Proc. of the
Hirzebruch 65 Conf. on Algebraic Geometry, Israel Math. Conf. Proc.
9, 75-96 (1996).
- Victor Kac, Infinite dimensional Lie algebras, Cambridge
University Press (1990).
- Nitu Kitchloo, On the
topology of Kac-Moody groups (2008).
- Andrew Pressley and Graeme Segal, Loop Groups, Oxford
University Press (1986).

#### See also

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/Loop.pdf

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/LoopGps.pdf

###
28/06/2011, 16:00 — 17:00 — Room P3.10, Mathematics Building

Nitu Kitchloo, *Johns Hopkins and UCSD*

```
```###
Geometry, Topology and Representation Theory of Loop Groups

In this sequence of three talks, I will aim to introduce the
algebraic and geometric structure of Loop groups and their
representations. We will begin with the basic structure of Affine
Lie algebras. This will lead us to the algebraic theory of positive
energy representations indexed by the level. On the geometric side,
we will introduce the Affine Loop group and relate it to the
central extension of the smooth loop group. We will also study the
example of the special unitary group in some detail. In the
remaining time, I will go into some of the deeper structure of Loop
groups. This includes fusion in the representations of a given
level (via the geometric notion of conformal blocks). Time
permitting, I will also describe the homotopy type of the
classifying space of Loop groups. No special background is
required. It would be helpful to know the basic theory of root
systems for semisimple Lie algebras, though this is not a strict
requirement.

#### References

- Arnaud Beauville, Conformal
blocks, fusion rules and the Verlinde formula, Proc. of the
Hirzebruch 65 Conf. on Algebraic Geometry, Israel Math. Conf. Proc.
9, 75-96 (1996).
- Victor Kac, Infinite dimensional Lie algebras, Cambridge
University Press (1990).
- Nitu Kitchloo, On the
topology of Kac-Moody groups (2008).
- Andrew Pressley and Graeme Segal, Loop Groups, Oxford
University Press (1986).

#### See also

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/Loop.pdf

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/LoopGps.pdf