# 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

### Oscar Garcia-Prada (ICMAT, Madrid): Involutions of Higgs bundle

moduli spaces

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

and Oscar Garcia-Prada.

### 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.

### Paulo Lima-Filho (Texas A&M): Bi-graded invariants for real varieties

### 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.

### Pedro F. dos Santos (Lisbon): Bi-graded invariants for real

varieties

### 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 **P**^{3} 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

**P**^{3}.

### 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.