Title: On toric log Fano varieties

**
Abstract. ** I will discuss a combinatorial formula for the alpha-invariant of a polarized
toric log variety. For toric log Fano varieties, I will give a sharp lower bound for the
alpha-invariant, in terms of the global minimal log discrepancy.
.

Title: Syzygies and secant loci

**
Abstract. ** We discuss the interactions between syzygies of curves and the geometry of the secant loci in the symmetric products. We show that a regular behavior of these loci for special line bundles imply the vanishing of linear syzygies. The talk is based on a joint work with Edoardo Sernesi.

Title: Direct images of logarithmic differentials.

**Abstract: **I will talk about some not very recent work about decomposing
derived direct images of sheaves of logarithmic differentials. As as an application,
we give a relatively short proof of Kollar's theorem on direct images of dualizing
sheaves.

Title: Irrationality of projective varieties

**Abstract.**
We will discuss various birational invariants extending the notion of gonality to projective varieties of arbitrary dimension, and measuring the failure of a given variety to satisfy certain rationality properties, such as being uniruled, rationally connected, unirational, stably rational or rational.
We will describe these invariants for different classes of projective varieties, as e.g. hypersurfaces and symmetric products of curves, by means of both ad hoc arguments and techniques relying on positivity properties of their canonical bundles.

Title: Periods of Gushel-Mukai varieties

**Abtract: ** Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic
fourfold and the primitive second cohomology of its variety of lines, a smooth hyperkahler fourfold, are
isomorphic as integral Hodge structures. We prove an analogous statement for smooth Gushel-Mukai
varieties of dimension 4 (resp. 6), i.e., smooth dimensionally transverse intersections of the cone over the
Grassmannian Gr(2, 5), a quadric, and two hyperplanes (resp. of the cone over Gr(2, 5) and a quadric).
The associated hyperkahler fourfold is now a smooth double cover of a sextic fourfold called an EPW
sextic. This is joint work with Alexander Kuznetsov.

Title: Totally invariant subsets of endomorphisms of projective spaces

**Abstract: **An endomorphism of the projective space is a finite morphism f : P^{n} → P^{n} of degree at least two. A totally invariant subset of f is a subvariety D in P^{n} such that we have a set-theoretic equality f^{−1}(D) = D. A well-known conjecture claims that such a totally invariant subvariety is always a union of linear spaces. This elementary question is surprisingly difficult and has been considered by several people over the last ten years. In this talk I will focus on the case where D is a divisor and show how the positivity of the logarithmic cotangent bundle can be used to prove the conjecture for divisors in P^{3}.

Title: Irregular varieties with geometric genus one and fake tori

**Abstract: **We study the Albanese image of irregular varieties with geometric genus
one. We show that if the Albanese map is not surjective,
the Albanese image is closely related to theta divisors. We apply this
result to study varieties with the same Hodge numbers as complex tori.
This is a joint work with Jungkai Chen and Zhiyu Tian.

Title: Construction of singular divisors, Newton-Okounkov bodies, and syzygies on abelian varieties

**Abtract: **Constructing divisors with prescribed singularities is one of the most powerful techniques
in modern projective geometry, leading to proofs of major results in the minimal model program
and the strongest general positivity theorems by Angehrn-Siu and Kollar-Matsusaka.
We present a novel method for constructing singular divisors on surfaces based on infinitesimal
Newton-Okounkov bodies. As an application of our machinery we discuss a Reider-type theorem for
higher syzygies on abelian varieties building on earlier work of Lazarsfeld-Pareschi-Popa.

Title: Rigid representations of the fundamental group

**Abstract: **
I will talk about rigid representations of the topological fundamental group of a smooth complex quasi-projective
variety. Such representations are conjectured to come from some families of smooth projective varieties.
I plan to give a general introduction to this conjecture and show some new results in case of rank 3 representations.
This is a joint work with C. Simpson.

Title: Augmented base loci and restricted volumes on normal varieties

**Abstract: ** We extend to normal projective varieties defined over an arbitrary
algebraically closed field a result of Ein, Lazarsfeld, Mustata,
Nakamaye and Popa characterizing the augmented base locus of a line
bundle on a smooth projective complex variety as the union of
subvarieties on which the restricted volume vanishes. We also give a
proof of the folklore fact that the complement of the augmented base
locus is the largest open subset on which the Kodaira map defined by
large and divisible multiples of the line bundle is an isomorphism. We
also give an interesting extension to real divisors following some
recent work of C. Birkar.

Title:From Convex geometry of certain invaluations to positivity aspect in algebraic geometry

**Abtract: **A few years ago Okounkov associated a convex set (Newton{Okounkov body)
to a divisor, which encodes the asymptotic vanishing behaviour of all sections of all powers
of the divisor along a fixed flag. This, on the other hand, brought to light the following
guiding principle "use convex geometry, through the theory of Newton{Okounkov bodies,
to study the geometrical/algebraic/arithmetic properties of divisors on smooth projective
varieties". The main goal of this talk is to explain some of the philosophical underpinnings
of this principle with a view towards studying local positivity and syzygetic properties of algebraic varieties .

Title: Ample vector bundle generated by sections

**Abtract: ** We derive a vanishing theorem for Dolbault
cohomology of ample vector bundle generated by sections, this is a joint work with F. Laytimi.

Title:Standard canonical support loci

**Abtract: **Inside the cohomological support loci of the canonical bundle of an irregular compact Kahler manifold there are some “easier” irreducible components, called standard. I will discuss some results about their structure.

Title: Cones of positive cycles

**Abtract: **I will discuss some positive and negative results about cones of positive divisors and higher-codimension cycles.

Title:On finiteness of CM jacobians of smooth hyperelliptic curves and superelliptic curves

**Abtract: ** Coleman's conjecture predicts that for g sufficiently large there exists at most finitely
many smooth complex projective curves of genus g (up to isomorphism) whose jacobians are CM abelian varieties. Based on the recent work by Tsimerman on the solution of Andre-Oort conjecture and my recent joint works with Ke Chen and Xin Lu on Oort conjectue for hyperelliptic curves and superelliptic curves
(defined by an equation y^{n}=f(x), where f(x) is a polynomial without multiple roots) we show Coleman's conjecture holds true for those two cases for any g>7.