The space of formal arcs of an algebraic variety carries part of the information encoded in a resolution of singularities. This series of lectures addresses this fact from two perspectives. In the first two lectures, we focus on the topology of the space of arcs, proving Kolchin's irreducibility theorem and discussing the Nash problem on families of arcs through the singularities of the variety; recent results on this problem are proved in the second lecture. The last two lectures are devoted to some applications of arc spaces toward a conjecture on minimal log discrepancies known as inversion of adjunction. Minimal log discrepancies are invariants of singularities appearing in the minimal model program, a quick overview of which is given in the third lecture.[-]

The space of formal arcs of an algebraic variety carries part of the information encoded in a resolution of singularities. This series of lectures addresses this fact from two perspectives. In the first two lectures, we focus on the topology of the space of arcs, proving Kolchin's irreducibility theorem and discussing the Nash problem on families of arcs through the singularities of the variety; recent results on this problem are proved in the second lecture. The last two lectures are devoted to some applications of arc spaces toward a conjecture on minimal log discrepancies known as inversion of adjunction. Minimal log discrepancies are invariants of singularities appearing in the minimal model program, a quick overview of which is given in the third lecture.[-]

The space of formal arcs of an algebraic variety carries part of the information encoded in a resolution of singularities. This series of lectures addresses this fact from two perspectives. In the first two lectures, we focus on the topology of the space of arcs, proving Kolchin's irreducibility theorem and discussing the Nash problem on families of arcs through the singularities of the variety; recent results on this problem are proved in the second lecture. The last two lectures are devoted to some applications of arc spaces toward a conjecture on minimal log discrepancies known as inversion of adjunction. Minimal log discrepancies are invariants of singularities appearing in the minimal model program, a quick overview of which is given in the third lecture.[-]

The space of formal arcs of an algebraic variety carries part of the information encoded in a resolution of singularities. This series of lectures addresses this fact from two perspectives. In the first two lectures, we focus on the topology of the space of arcs, proving Kolchin's irreducibility theorem and discussing the Nash problem on families of arcs through the singularities of the variety; recent results on this problem are proved in the second lecture. The last two lectures are devoted to some applications of arc spaces toward a conjecture on minimal log discrepancies known as inversion of adjunction. Minimal log discrepancies are invariants of singularities appearing in the minimal model program, a quick overview of which is given in the third lecture.[-]

Let $(V,p)$ be a complex isolated complete intersection singularity germ (an ICIS). It is well-known that its Milnor number $\mu$ can be expressed as the difference:
$$\mu = (-1)^n ({\rm Ind}_{GSV}(v;V) - {\rm Ind}_{rad}(v;V)) \;,$$
where $v$ is a continuous vector field on $V$ with an isolated singularity at $p$, the first of these indices is the GSV index and the latter is the Schwartz (or radial) index. This is independent of the choice of $v$.
In this talk we will review how this formula extends to compact varieties with non-isolated singularities. This depends on two different ways of extending the notion of Chern classes to singular varieties. On elf these are the Fulton-Johnson classes, whose 0-degree term coincides with the total GSV-Index, while the others are the Schwartz-McPherson classes, whose 0-degree term is the total radial index, and it coincides with the Euler characteristic. This yields to the well known notion of Milnor classes, which extend the Milnor number. We will discuss some geometric facts about the Milnor classes.[-]

The Jacobian algebra, obtained from the ring of germs of functions modulo the partial derivatives of a function $f$ with an isolated singularity, has a non-degenerate bilinear form, Grothendieck Residue, for which multiplication by $f$ is a symmetric nilpotent operator. The vanishing cohomology of the Milnor Fibre has a bilinear form induced by cup product for which the nilpotent operator $N$, the logarithm of the unipotent part of the monodromy, is antisymmetric. Using the nilpotent operators we obtain primitive parts of the bilinear form and we compare both bilinear forms. In particular, over $\mathbb{R}$, we obtain signatures of these primitive forms, that we compare.[-]

Pham and Teissier showed in the late 60's that any two plane curve germs with the same outer Lipschitz geometry have equivalent embeddings into $\mathbb{C}^2$. We consider to what extent the same holds in higher dimensions, giving examples of normal surface singularities which have the same topology and outer Lipschitz geometry but whose embeddings into $\mathbb{C}^3$ are topologically inequivalent. Joint work with Anne Pichon.

Keywords: bilipschitz - Lipschitz geometry - normal surface singularity - Zariski equisingularity - Lipschitz equisingularity[-]

In 2007, Fan, Jarvis, and Ruan constructed an analogue of the Gromov-Witten (GW) theory of hypersurfaces in weighted projective spaces. The new theory is attached to quasi-homogeneous polynomial singularities and is usually called Fan-Jarvis-Ruan-Witten theory (FJRW). It is part of the general picture of Witten, where GW and FJRW theories arise as two distinct GIT quotients of the same model. I will first explain this idea under the light of mirror symmetry. Then I will present FJRW theory and the geometric problem it illustrates. In particular, I will highlight a geometric property called concavity. For now, it is a necessary condition for explicit results on GW theory of hypersurfaces. But on the FJRW side, the situation has recently changed and I will describe my method based on Koszul cohomology to overcome this difficulty. As a consequence, I obtain a mirror symmetry theorem without concavity.[-]