Photo by A. Ortega
** Room 1.115, Johann von Neumann Haus, Rudower Chaussee 25, 12489 Berlin
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10:30-11:45, Alessio Sammartano (Politecnico di Milano) **

** On the parity conjecture for Hilbert schemes of points on threefolds **

** The Hilbert scheme of points in affine n-dimensional space, parametrizing zero-dimensional subschemes with a fixed degree, is a fundamental parameter space in algebraic geometry. Quot schemes are a generalization of Hilbert schemes, parametrizing finite length quotients of a locally free sheaf. We will explore some interesting phenomena and problems about these spaces that are special to the three-dimensional case, focusing in particular on the tangent space and on recent progress on the parity conjecture of Okounkov and Pandharipande. Joint work with Ritvik Ramkumar.
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**13:30-14:45, Joachim Jelisiejew (Warsaw) **

** Limits of finite Gorenstein subschemes **

** Finite Gorenstein subschemes in P^n are frequently better behaved than general ones,
for example smoothable for n <= 3. However, being Gorenstein is not a closed property and limits of Gorenstein subschemes in the Hilbert scheme are badly behaved. I will propose another moduli space compactification of the locus of (oriented) Gorenstein subschemes and discuss its geometry and many open questions.
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**15:15-16:30, Martijn Kool (Utrecht) **

** Hilbert schemes on affine 4-space and the geometry of the Magnificent Four **

**Hilbert schemes of affine d-space are naturally cut out by sections of
vector bundles on (smooth) non-commutative Hilbert schemes. In the special
case d = 4, these vector bundles are quadratic and the sections are isotropic. Using the
Oh-Thomas localization formula for Donaldson-Thomas type invariants of Calabi-Yau
4-folds, this leads to a weighted count of the torus fixed points of these Hilbert schemes.
A conjectural formula for this weighted count was given in the physics literature by
Nekrasov-Piazzalunga. The weights associated to the fixed points involve certain
signs, which will be the focus of this lecture. Joint work with Jørgen Rennemo.
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**Organizers:
G. Farkas ,
R. Pandharipande
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**Photos:
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