Jochen Heinloth: 
Stability and existence of good moduli spaces for algebraic stacks
Recently Alper, Hall and Rydh gave general criteria when a moduli problem can locally be described as a quotient and thereby clarified the local structure of algebraic stacks.
We report on a joint project with Jarod Alper and Daniel HalpernLeistner in which we use these results to show general existence results for good coarse moduli spaces. In the talk we will focus on two aspects that illustrate how the geometry of algebraic stacks gives a new point of view on classical methods. Namely we explain a version of HilbertMumford stability and then show how Langton's proof of semistable reduction for coherent sheaves on projective varieties can be reformulated in terms of geometry. This allows to prove a semistable reduction theorem purely in terms of the geometry of the moduli problem. 
Filippo Viviani: 
On the cone of effective cycles on the symmetric products of curves
I will report on a joint work with F. Bastianelli, A. Kouvidakis and A. F. Lopez in which we study the cone of (pseudo)effective cycles on symmetric products of a curve. We first prove that the diagonal cycles span a face of the pseudoeffective cone of cycles in any given dimension. Secondly, we look at the contractibility faces associated to the AbelJacobi morphism towards the Jacobian and in many cases we are able to compute their dimension. 
John Christian Ottem:

The integral Hodge conjecture for threefolds
The Hodge conjecture predicts which rational cohomology classes on a smooth complex projective variety can be represented by linear combinations of complex subvarieties. The integral Hodge conjecture, the analogous conjecture for integral homology classes, is known to be false in general (the first counterexamples were given in dimension 7 by Atiyah and Hirzebruch). I'll survey some known results on this conjecture, and present some new counterexamples in dimension three. This is joint work with Olivier Benoist.

Nicola Pagani:

Double ramification cycles from Jacobians
Let M_{g,n} be the moduli space of smooth npointed curves of genus g. For a given vector of integers (d1,...,dn) one can define a natural cycle of pointed curves (C, p_{i}) such that O_{C}(Σ d_{i} p_{i}) admits a nonzero global section. We discuss how this cycle can be extended to the moduli space of stable curves by interpreting it as the pullback of a cycle on (compactified) universal Jacobians. Because there are multiple ways to compactify the Jacobian, this leads to multiple cycles related by wallcrossing. We explain why this gives an effective approach for the computation of the cohomology class of these cycles (in terms of tautological classes). A joint work with Jesse Kass.

Eloise Hamilton: 
Moduli spaces for Higgs bundles, stable and unstable
The aim of this talk is to illustrate using the example of Higgs bundles how a combination of classical and Non Reductive Geometric Invariant Theory (GIT) can be used to solve classification problems in algebraic geometry. This method can also be applied to the classification of sheaves and of curves.
I will start by defining Higgs bundles and explaining the classification problem for Higgs bundles. After introducing the stack of Higgs bundles, I will explain how it can be described geometrically. As we will see, the stack of Higgs bundles can be decomposed into disjoint strata, each consisting of Higgs bundles of a given 'instability type'. I will explain how classical GIT can be used to obtain a moduli space for the substack of semistable Higgs bundles, and how nonreductive GIT can be applied to obtain moduli spaces for the remaining unstable strata.

Greg Stevenson: 
Homotopy invariants of singularity categories
I'll give a brief, but hopefully gentle, introduction to A1homotopy invariants (e.g. homotopy Ktheory) of DGcategories. I'll then discuss a strategy for computing these invariants for singularity categories of certain noetherian algebras. This strategy leads us to a description of the A1homotopy invariants of the singularity categories of a class of 0dimensional Gorenstein singularities. If time permits I'll speculate on how to use these ideas to produce new, and concrete, examples of phantoms. All of this is based on joint work with Sira Gratz. 
Charles Vial: 
The generalized Franchetta conjecture for hyperKaehler varieties
The original Franchetta conjecture, established by Harer, predicts that the restriction of a linebundle on the universal family of smooth projective curves of given genus g>1 to a fiber is a multiple of the canonical linebundle. Recently, O'Grady proposed an analogue of that conjecture for codimension2 cycles on the universal family of polarized K3 surfaces of given degree. In this talk I will propose a version of the Franchetta conjecture for hyperKaehler varieties (and their powers) and provide some evidence, most notably by focusing on the Fano variety of lines on a smooth cubic fourfold. This is joint work with Lie Fu, Robert Laterveer and Mingmin Shen. 
Giovanni Mongardi: 
Hilbert schemes of points and various equivalences
In this talk, we analyze what happens when we take two k3 surfaces with the same derived category and their Hilbert schemes of points. We will show how the birational geometry of IHS manifolds forces these schemes to be non birational in many cases. This is part of a joint work with C. Meachan and K. Yoshioka. 
Enrica Mazzon: 
Berkovich approach to degenerations of hyperKähler varieties
To a degeneration of varieties, we can associate the dual intersection complex, a topological space that encodes the combinatoric of the central fiber and reflects the geometry of the generic fiber. In this talk I will show how the techniques of Berkovich geometry give an insight in the study of the dual complexes. In this way, we are able to determine the homeomorphism type of the dual complex of some degenerations of hyperKähler varieties. The results are in accordance with the predictions of mirror symmetry, and the recent work about the rational homology of dual complexes of degenerations of hyperKähler varieties, due to Kollár, Laza, Saccà and Voisin. This is joint work with Morgan Brown.

Alexander Kuznetsov: 
Derived categories of singular toric surfaces
I will talk about semiorthogonal decompositions of singular toric surfaces. If the class group of the surface is torsion free, I will construct a semiortohogonal decomposition with components being derived categories of explicit finite dimensional algebras. Otherwise, I will construct a similar semiorthogonal decomposition for the twisted derived category of the surface. This is a joint work with Joseph Karmazyn and Evgeny Shinder. 
Mattia Talpo: 
A logarithmic version of the derived McKay correspondence
I will talk about a "limit" version of the derived McKay correspondence of Bridgeland, King and Reid in the context of logarithmic geometry. The two sides of the derived equivalence in this case are given by objects called the "infinite root stack" and the "valuativization”, that correspond to different ways of enhancing the central fiber of some simple kind of degenerations. Our result also implies the rather surprising fact that the derived category of parabolic sheaves on a variety with a normal crossings divisor is invariant under certain blowups. This is joint work with Sarah Scherotzke and Nicolò Sibilla. 
Pieter Belmans: 
Moduli of sheaves on (noncommutative) surfaces
The universal object associated to a fine moduli space of sheaves on a surface gives a FourierMukai functor from the derived category of the surface to the derived category of the moduli space. On the other hand we can consider the Hochschild cohomology of both varieties, and their associated HochschildKostantRosenberg decomposition. If the functor is fully faithful we can relate the Hochschild cohomologies. There is an interesting interplay between the (commutative) deformations of the moduli space and deformations of the category of coherent sheaves on the surface. I will explain this interplay, and how studying noncommutative surfaces can lead to a better understanding of deformations of moduli spaces of sheaves. 
Sjoerd Beentjes:

The crepant resolution conjecture for DonaldsonThomas invariants
The crepant resolution conjecture for DonaldsonThomas (DT) invariants is a conjecture in enumerative geometry originating from string theory. It relates the DT generating series of a certain type of threedimensional CalabiYau orbifold to that of a crepant resolution of its coarse moduli space; this setting is a global version of the McKay correspondence.
We discuss joint work with John Calabrese and Jørgen Rennemo in which we interpret the conjecture as an equality of rational functions, and prove it using wallcrossing methods. 
Agnieszka Bodzenta: 
Categorifying noncommutative deformation theory
I will categorify noncommutative deformation theory by viewing underlying spaces of infinitesimal deformations of n objects as abelian categories with n simple objects. If the deformed collection is simple, I will prove the indrepresentability of the deformation functor. For an arbitrary collection I will construct an indhull for the deformation functor and use it to present the deformation functor as a noncommutative Artin stack. 
Sara Angela Filippini: 
Stability data, irregular connections and tropical curves
I will outline the construction of isomonodromic families of irregular meromorphic connections on P^1 with values in the derivations of a class of infinitedimensional Poisson algebras, and describe two of their scaling limits. In the "conformal limit" we recover a version of the connections introduced by Bridgeland and ToledanoLaredo, while in the "large complex structure limit" the connections relate to tropical curves in the plane and, through work of Gross, Pandharipande and Siebert, to tropical/GW invariants. This is joint work with M. GarciaFernandez and J. Stoppa. 
Alexey Bondal: 
Noncommutative moduli
We discuss noncommutative thickenings of families of sheaves. 