# Schedule

### Monday

09h00 Welcome and coffee
09h30 Andrei Teleman: Intrinsic signs and lower bounds in real algebraic geometry
10h30 Coffee break
11h00 Jörg Schürmann: Introduction to characteristic classes of singular spaces
12h30 Lunch
14h30 Melissa Liu: Yang-Mills equations over Klein surfaces
15h30 Coffee break
16h00 Jacques Hurtubise: Pseudo-real principal bundles on a compact Kähler manifold

### Tuesday

09h30 Melissa Liu: Yang-Mills equations over Klein surfaces
10h30 Coffee break
11h00 Pedro F. dos Santos: Bi-graded invariants for real varieties
12h30 Lunch
14h30 Jörg Schürmann: Introduction to characteristic classes of singular spaces
15h30 Coffee break
16h00 Oscar Garcia-Prada: Involutions of Higgs bundle moduli spaces

### Wednesday

09h30 Andrei Teleman: A wall crossing formula for degrees of real central projections
10h30 Coffee break
11h00 Melissa Liu: Yang-Mills equations over Klein surfaces
12h30 Lunch
14h30 Excursion

### Thursday

09h30 Jörg Schürmann: Introduction to characteristic classes of singular spaces
10h30 Coffee break
11h00 Paulo Lima-Filho: Bi-graded invariants for real varieties
12h30 Lunch
14h30 Florent Schaffhauser: Modular compactifications of moduli spaces of geometrically stable real and quaternionic vector bundles
15h30 Coffee break
16h00 Wojciech Kucharz: Stratified-algebraic vector bundles
20h00 Dinner

### Friday

09h30 Michael Atiyah: Characteristic classes for KR bundles
10h30 Coffee break
11h00 Paulo Lima-Filho: Bi-graded invariants for real varieties
12h30 Lunch
14h30 Andrei Teleman: Abelian Yang-Mills theory on Real tori
and Theta divisors of Klein surfaces
15h30 Coffee

All lectures will be given in Lecture room E of the Faculty of
Sciences of the University of Brest, 6 avenue Le Gorgeu,
Brest
.

# Abstracts

### Michael Atiyah (Edingburgh): Characteristic classes for KR bundles

Abstract: Since i introduced KR bundles many years ago there has been
much work on developing Chern class theory for these bundles. I will
use old ideas of Grothendieck to give yet another treatment. The
advantage may be that information mod powers of 2 may be useful in
real algebraic geometry.

### Jean Fasel (Duisburg-Essen): A1-homotopic classification of vector bundles

Abstract: We consider the moduli space of G-Higgs bundles over a
compact Riemann surface X, where G is a complex semisimple Lie group,
and study various involutions of the moduli space involving
conjugations in G and X. We describe the fixed points and study their
relations with representations of the fundamental group of X in G. We
will also comment on the relation of the fixed points with Langlands
duality and mirror symmetry for Higgs bundles.

### Jacques Hurtubise (McGill, Montreal): Pseudo-real principal bundles on a compact Kähler manifold

Abstract: Let X be a compact connected Kähler manifold equipped with
an anti-holomorphic involution which is compatible with the Kähler
structure. Let G be a connected complex reductive affine algebraic
group equipped with a real form. We define pseudo-real principal
G–bundles on X; these are generalizations of real algebraic principal
G–bundles over a real algebraic variety. We prove that a
Donaldson–Uhlenbeck–Yau type correspondence holds: a pseudo-real
principal G–bundle admits a compatible Einstein-Hermitian connection
if and only if it is polystable. The link with representations of the
fundamental group is also considered. Joint work with Indranil Biswas

### Wojciech Kucharz (Jagiellonian, Krakow): Stratified-algebraic vector bundles (joint work with Krzysztof Kurdyka)

Abstract: We investigate stratified-algebraic vector bundles on a real
algebraic variety X. A stratification of X is a finite partition of X
into Zariski locally closed subvarieties. A topological vector bundle
on X is called a stratified-algebraic vector bundle if, roughly
speaking, its restriction to each stratum of some stratification of X
is an algebraic vector bundle on that stratum. It turns out that
stratified-algebraic vector bundles have many desirable features of
algebraic vector bundles but are more flexible. Recently first
significant steps have been made toward real algebraic geometry based
on continuous rational functions – called regulous
geometry. Stratified-algebraic vector bundles can be also regarded as
the appropriate class of vector bundles in regulous geometry.

### Melissa Liu (Columbia, New York): Yang-Mills equations over Klein surfaces

Abstract: In "The Yang-Mills equations over Riemann surfaces," Atiyah
and Bott studied Yang-Mills functional over a Riemann surface from the
point of view of Morse theory, and derived results on topology of the
moduli space of algebraic bundles over a complex algebraic curve. In
these lectures, I will discuss Yang-Mills functional over a Klein
surface from the point of view of Morse theory, and derive recursive
and closed formulae of mod 2 Poincare polynomials of the moduli space
of real or quaternionic vector bundles over a real algebraic
curve. These lectures are based on "The Yang-Mills equations over
Klein surfaces" by Florent Schaffhauser and myself.

### Florent Schaffhauser (Los Andes, Bogota): Modular compactifications of moduli spaces of geometrically stable real and quaternionic vector bundles

Abstract: In a paper published in 1967, Seshadri introduced the notion
now known as S-equivalence on categories of semi-stable holomorphic
vector bundles of fixed slope, which enabled him to construct a
modular compactification of Mumford's moduli spaces of stable
holomorphic vector bundles of fixed rank and degree. The goal of the
present talk is to explain the analogous notion for semi-stable real
and quaternionic vector bundles. We then study the relationship
between this construction and symplectic geometry.

### Jörg Schürmann (Münster): Introduction to characteristic classes of singular spaces

Abstract: First we explain the Lagrangian approach to Stiefel-Whitney
resp. Chern classes of singular real resp. complex algebraic or
analytic varieties, motivated by Poincare–Hopf index theorems for such
singular spaces. Based on stratified Morse theory for constructible
functions, we discuss the corresponding calculus of characteristic
cycles and its relation to these characteristic classes,
e.g. functoriality under proper pushdown as well as specialization.
We also explain the relation between these characteristic classes for
algebraic or analytic varieties defined over the reals.

Then we move to "motivic constructible" functions, i.e. to relative
Grothendieck groups of algebraic varieties, and introduce the motivic
Chern and Hirzebruch class transformation, where the later unifies the
MacPherson Chern class, the Baum-Fulton-MacPherson Todd class, as well
as the Cappell-Shaneson L-class transformation for singular spaces.

Finally we compute all these characteristic classes for singular toric
varieties. If time permits, we also explain the relation to weighted
lattice points counting in lattice polytopes.

### Andrei Teleman (Marseille): Intrinsic signs and lower bounds in real algebraic geometry (joint with Ch. Okonek)

Abstract: A classical result due to Segre states that on a real cubic
surface in P3 there exists two kinds of
real lines: elliptic and hyperbolic lines. These two kinds of real
lines are defined in an intrinsic way, i.e., their definition does not
depend on any choices of orientation data. Segre's classification of
smooth real cubic surfaces also shows that any such surface contains
at least 3 real lines. Starting from these remarks and inspired by the
classical problem mentioned above - I will explain a general principle
which leads to lower bounds in real algebraic geometry, - I will
explain the reason for the appearance of intrinsic signs in the
classical problem treated by Segre, showing that the same phenomenon
occurs in a large class of enumerative problems in real algebraic
geometry. - I will illustrate these principles in the enumerative
problem for real lines in real hypersurfaces of degree 2m−3 in
P3.

### Andrei Teleman (Marseille): A wall crossing formula for degrees of real central projections (joint with Ch. Okonek)

Abstract: I will discuss a recent result in real algebraic geometry: a
wall crossing formula for central projections defined on submanifolds
of a real projective space. This formula gives the jump of the degree
of such a projection when the center of the projection varies. The
fact that the degree depends on the projection is a new phenomenon,
specific to real algebraic geometry. We illustrate this phenomenon in
many interesting situations. The crucial assumption on the class of
maps we consider is relative orientability, a condition which allows
us to define a Z-valued degree map in a coherent way. I will
discuss several consequences of this result.

### Andrei Teleman (Marseille): Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces (joint with Ch. Okonek)

Abstract: I will explain a method to compute determinant index bundles
of certain families of Real Dirac type operators on Klein surfaces as
elements in the corresponding Grothendieck group of Real line bundles
in the sense of Atiyah. On a Klein surface these determinant index
bundles have a natural holomorphic description as theta line
bundles. In particular we compute the first Stiefel-Whitney classes of
the corresponding fixed point bundles on the real part of the Picard
torus. The computation of these classes is important, because they
control the orientability of certain moduli spaces in Real gauge
theory and Real algebraic geometry.