Venues: (Tuesdays and Fridays) IMPAN 321/403 and (Thursdays) MIMUW 5050

The Organizers:

Stiofáin Fordham

Luke Oeding (contact: )

Emanuele Ventura

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10:30–12:00, IMPAN 403

Lecture II: We will generalize the classification of $\mathbb{G}_a$-actions by means of vector fields to a wide class of varieties including projective varieties. This is a joint work with Adrien Dubouloz [DL16].

[DL16] Adrien Dubouloz and Alvaro Liendo.
Rationally integrable vector fields and rational additive group
actions. * Internat. J. Math.*, 27 (2016), no. 8,
1650060, 19 pp.

14:15–15:15, IMPAN 321

**Abstract:**

A compact complex submanifold of a complex manifold is said to satisfy the formal principle if its formal neighborhood determines its germ of analytic neighborhoods. In 1981, Hirschowitz conjectured that an unobstructed submanifold satisfies the formal principle if its normal bundle is globally generated. I give an overview of the problem and explain an approach by the equivalence method for geometric structures.

The talk will be followed by tea and cake in 409 IMPAN at 15:45.

12:00–12:45, MIMUW 5050

**Abstract:**

Mathematics always has made progress by violating the rules that were set by earlier generations. In this provocative talk I will report on joint work in progress with V. Golyshev on projective spaces and Grassmanians of fractional dimension.

The talk will be followed by a lunch break at the MIMUW Cafeteria.

13:15–14:00, MIMUW 5050

**Abstract:**

Part II

10:30–12:00, IMPAN 403

Lecture III: We will show how these techniques, in the presence of a torus action, allow for the systematic study of root subgroups. A root subgroup is a closed subgroups of $\operatorname{Aut}(X)$ isomorphic to $\mathbb{G}_a$ that is normalized by the torus. In particular, we will recover the description given in [Dem70] of the connected component of the automorphism group of a complete toric variety, see also [AHHL14].

[AHHL14] Ivan Arzhantsev, Jürgen Hausen, Elaine
Herppich, and Alvaro Liendo. The automorphism group of a
variety with torus action of complexity one.
* Mosc. Math. J.*, 14(3):429--471, 641, 2014.

[Dem70] Michel Demazure. Sous-groupes algébriques de rang maximum du groupe de Cremona. *Ann. Sci. École Norm. Sup. (4)*, 3:507--588, 1970.

14:15–15:15, IMPAN 106

**Abstract:**

Given a variety admitting the action of an algebraic torus we consider the orbit space of the associated compact torus action. One would like to determine this topological space up to homomorphism or homotopy equivalence. In my talk I will discuss how this problem can be approached via Geometric Invariant Theory.

The talk will be followed by tea and cake in 409 IMPAN at 15:45.

10:30–12:00, IMPAN 403

**Abstract:**
The theory of hypergeometric functions in several variables is now described by the Gel'fand-Kapranov-Zelevinski theory of $A$-hypergeometric functions introduced in the end of the 1980's. In these lectures we give an introduction to these functions and approach the problem of the computation of their monodromy.

Lecture I

14:15–15:15, IMPAN 321

**Abstract:**
In this talk I will overview QFT and string theory and will show how the hybrid topology that relates Archimedean and non Archimedean analysis can be used as a tool to understand how string theory converges to QFT in low energies. Joint work with O. Amini, S. Bloch and J. Fresán based on an insight of P. Tourkine.

The talk will be followed by tea and cake in 409 IMPAN at 15:45.

12:00–12:45, MIMUW 5050 and 13:15–14:00, MIMUW 5050

**Abstract:** Projective duality can be used to study singularities. A matrix is singular precisely when its determinant vanishes, or equivalently, when it belongs to the projective dual to rank-one matrices, the Segre variety. A higher order tensor is singular when its hyperdeterminant vanishes, i.e. when it belongs to the dual of a higher order Segre product. Efficient expressions for hyperdeterminants are mostly unknown and they are difficult to compute. We describe a connection to the exceptional Lie algebra $E_8$. This gives an interpretation of certain hyperdeterminants (of formats $2\times 2\times 2\times 2$ and $3\times 3\times 3$) and certain discriminants (of the Grassmannians $Gr(3,9)$ and $Gr(4,8)$) as sparse $E_8$-discriminants. We give expressions of these high degree invariants in terms of lower degree fundamental invariants, which allow evaluation, and may be useful for Quantum Information Theory as measures of entanglement. This is joint work with Frédéric Holweck.

The talk will have a lunch break at the MIMUW Cafeteria.

10:30–12:00, IMPAN 403

Lecture II: **Abstract**

14:15–15:15, IMPAN 321

**Abstract:**

TBD

The talk will be followed by tea and cake in 409 IMPAN at 15:45.

10:30–12:00, IMPAN 403

**Abstract:** Hodge Theory has an analog for algebraic varieties over local fields: $p$-adic Hodge Theory. In the first part of the course I will survey the main results of this theory and explain how they can be extended to analytic varieties, proper and Stein. In the second part I will discuss the example of Drinfeld upper half-spaces and, if time permits, their coverings. Their cohomologies carry interesting representations of $GL_n$.

Lecture I

14:15–15:15, IMPAN 321

**Abstract:**

TBD

The talk will be followed by tea and cake in 409 IMPAN at 15:45.

10:30–12:00, IMPAN 403

Lecture II Abstract

11:00-12:00, IMPAN 321

**Abstract** The idea of tame topology was introduced by Grothendieck in ”Esquisse d’un programme” and developed by model theorists under the name ”o-minimal structures”. In these lectures I will define o-minimal structures, study their basic properties and their relevance to complex geometry, in particular Hodge theory. As an illustration I will explain that period maps are tame. As an immediate corollary one recovers the algebraicity of Hodge loci (a classical result of Cattani-Deligne-Kaplan).

Lecture I

14:15-15:15, IMPAN 321

Lecture II: **Abstract**

10:00-11:00, IMPAN 321

Lecture III: **Abstract**

12:00–12:45, MIMUW 5050

**Abstract:**

TBD

The talk will be followed by a lunch break at the MIMUW Cafeteria.

13:15–14:00, MIMUW 5050

**Abstract:**

TBD

14:15–15:15, IMPAN 321

**Abstract:**

TBD

The talk will be followed by tea and cake in 409 IMPAN at 15:45.

10:30–12:00, IMPAN 403

**Abstract:**

Lecture I

14:15–15:15, IMPAN 321

**Abstract:**

TBD

The talk will be followed by tea and cake in 409 IMPAN at 15:45.

12:00–12:45, MIMUW 5050

**Abstract:**

TBD

The talk will be followed by a lunch break at the MIMUW Cafeteria.

13:15–14:00, MIMUW 5050

**Abstract:**

TBD

10:30–12:00, IMPAN 403

Lecture II: **Abstract**

14:15–15:15, IMPAN 321

**Abstract:**

In the early nineties, the Buonos Aires Cyclic Homology group calculated the Hochschild and cyclic homology of hypersurfaces, in general, and of the coordinate rings of planar cuspical curves, in particular. With Cortiñas' birelative theorem, proved in 2005, this gives a calculation of the relative K-theory of planar cuspical curves over a field of characteristic zero. By a $p$-adic version of Cortiñas' theorem, proved by Geisser and myself in 2006, the relative K-groups of planar cuspical curves over a perfect field of characteristic $p > 0$ can similarly be expressed in terms of topological cyclic homology, but the relevant topological cyclic homology groups have resisted calculation. In this talk, I will show that the new setup for topological cyclic homology by Nikolaus and Scholze has made this calculation possible. This is joint work with Nikolaus and similar results have been obtained by Angeltveit.

The talk will be followed by tea and cake in 409 IMPAN at 15:45.

10:30–12:00, IMPAN 403

**Abstract:**

Lecture III: **Abstract**

14:15–15:15, IMPAN 321

**Abstract:**

TBD

The talk will be followed by tea and cake in 409 IMPAN at 15:45.

12:00–12:45, MIMUW 5050

**Abstract:**

TBD

The talk will be followed by a lunch break at the MIMUW Cafeteria.

13:15–14:00, MIMUW 5050

**Abstract:**

TBD

10:30–12:00, IMPAN 403

**Abstract:**

Lecture IV: **Abstract**

14:15–15:15, IMPAN 321

**Abstract:**

TBD

The talk will be followed by tea and cake in 409 IMPAN at 15:45.

10:30–12:00, IMPAN 403

**Abstract:**

Lecture I

14:15–15:15, IMPAN 321

**Abstract:**

TBD

The talk will be followed by tea and cake in 409 IMPAN at 15:45.

12:00–12:45, MIMUW 5050

**Abstract:**

TBD

The talk will be followed by a lunch break at the MIMUW Cafeteria.

13:15–14:00, MIMUW 5050

**Abstract:**

TBD

10:30–12:00, IMPAN 403

**Abstract:**

Lecture II Abstract

14:15–15:15, IMPAN 321

**Abstract:**

TBD

The talk will be followed by tea and cake in 409 IMPAN at 15:45.