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.[-]

This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories [2, 5, 6]. I will then proceed to explain the concept of wall-crossing, both in theory, and in examples [1, 2, 4, 6].

- Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction [7].

- Applications : Moduli spaces of sheaves on $K3$ surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on $K3$ surfaces [1, 8].[-]

This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories [2, 5, 6]. I will then proceed to explain the concept of wall-crossing, both in theory, and in examples [1, 2, 4, 6].

- Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction [7].

- Applications : Moduli spaces of sheaves on $K3$ surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on $K3$ surfaces [1, 8].[-]

This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland's notion of stability conditions on derived categories [2, 5, 6]. I will then proceed to explain the concept of wall-crossing, both in theory, and in examples [1, 2, 4, 6].

- Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction [7].

- Applications : Moduli spaces of sheaves on $K3$ surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on $K3$ surfaces [1, 8].[-]

The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint work with Mircea Mustata and Chenyang Xu.[-]

The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint work with Mircea Mustata and Chenyang Xu.[-]

The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint work with Mircea Mustata and Chenyang Xu.[-]