The classical Hodge conjecture for smooth, projective varieties has been open for over 70 years, although it has been proven for some specific varieties. In this talk I will discuss a cohomological version of the Hodge conjecture for singular varieties. I will give a sufficient condition in terms of Mumford-Tate groups for a variety to satisfy the singular Hodge conjecture. If time allows I will give explicit examples of such varieties. This is joint work with Ananyo Dan.[-]

Let $f$ be a homogeneous polynomial, defining a principal Zariski open set $D(f)$ in some complex projective space $\mathbb{P}^n$ and a Milnor fiber $F(f)$ in the affine space $\mathbb{C}^{n+1}$. Let $f_0, . . . , f_n$ denote the partial derivatives of $f$ with respect to $x_0, . . . , x_n$ and consider syzygies $a_0f_0 + a_1f1 + a_nf_n = 0$, where $a_j$ are homogeneous polynomials of the same degree $k$.
Using the mixed Hodge structure on $D(f)$ and $F(f)$, one can obtain information on the possible values of $k$.[-]

In this talk we consider the Milnor fiber F associated to a reduced projective plane curve $C$. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of $F$, also known as the Alexander polynomial of the curve $C$, is presented. This leads to an effective algorithm to detect all the roots of the Alexander polynomial and, in many cases, explicit bases for the monodromy eigenspaces in terms of polynomial differential forms. The case of line arrangements, where there are many open questions, will illustrate the complexity of the problem. These results are based on joint work with Morihiko Saito, and with Gabriel Sticlaru.[-]

I will give a general introduction to the study of the Hodge filtration on local cohomology sheaves associated to closed subschemes of smooth complex varieties, using techniques from both D-module theory and birational geometry. In the case of hypersurfaces, this is essentially the theory of Hodge ideals, which I will recall. This study has applications to various topics, like local vanishing, local cohomological dimension, the Du Bois complex, minimal exponents of singularities, etc. I will discuss a few, and more will appear in M. Mustaja's lecture.[-]