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

The Organizers:

Stiofáin Fordham

Luke Oeding (contact: )

Emanuele Ventura

Join the mailing list for announcements, and add the google calendar (iCal).

14:15–15:15, IMPAN 403

**Abstract:**
Berthelot’s conjecture predicts that under a proper and smooth morphism of varieties in characteristic $p$, the higher direct images of an $F$-overconvergent isocrystal are $F$-overconvergent isocrystals. In a joint work with Fabio Tonini and Lei Zhang we prove that this is true for crystals up to isogeny. As an application we prove a Künneth formula for the crystalline fundamental group.

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

14:15–15:15, IMPAN 403

**Abstract:**
Given a projective algebraic variety, the identity component of its automorphism group is a linear algebraic group. This is no longer true for affine varieties where this component can be infinite dimensional. An affine variety $X$ is called flexible if the subgroup $\textrm{SAut} (X)$ generated by the unipotent subgroups of $\textrm{Aut} (X)$ acts transitively on the smooth locus $X_{reg}$. It occurs that for a flexible $X$, the action of $\textrm{SAut}(X)$ on $X_{reg}$ is $m$-transitive for any natural $m$. The mini lecture series consists in two parts. In the first part, flexible affine varieties are defined, criteria of flexibility and some examples and constructions are discussed. The second part is devoted to the flexibility of affine toric varieties.

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

14:15–15:15, IMPAN 403

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

10:30 -12:00, IMPAN 403

**Abstract:**
We're going to get together and talk about the working groups.

14:15–15:15, IMPAN 403

**Abstract:**
In the late 1880's Goursat investigated what we now call rigid local systems, classically described as linear differential equations without accessory parameters. In this talk I will discuss some arithmetic and geometric aspects of certain particular cases of Goursat's in rank four. For example, I will discuss what are likely to be all cases where the monodromy group is finite. This is joint work with Danylo Radchenko.

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

14:15–15:15, IMPAN 403

**Abstract:**

For a Noetherian local ring $(R,{\bf m})$ and an ${\bf m}$-primary ideal $I\subseteq {\bf m}$, the Hilbert-Kunz function $\mbox{HK}(R, I)$ was introduced by Kunz and later studied by Monsky and others. For a Noetherian Standard graded ring $R$ over a perfect field of characteristic $p>0$, and $I\subset R$ a homogeneous ideal of finite colength, the Hilbert-Kunz density function (denoted $\mbox{HK}d(R, I)$) is a new invariant introduced by Trivedi, to study the Hilbert-Kunz multiplicity of $R$ with respect to $I$. In this talk, we give a brief introduction to Hilbert-Kunz theory and discuss the Hilbert-Kunz density function for a projective toric varietiy $X$ with a very ample $T$-Cartier divisor $D$. We discuss the limiting asymptotic growth of the Hilbert-Kunz multiplicity of $(X, D)$ relative to the usual multiplicity and comment on the toric pairs $(X, D)$ for which this limit is minimum. This is joint work with Prof. V. Trivedi.

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

14:15–15:15, IMPAN 403

**Abstract:**

In the 80's of last century, Deligne and Mostow studied the monodromy problem of Lauricella hypergeometric functions and gave a rigorous treatment on the subject, which provides ball quotient structures on $\mathbb{P}^n$ minus a hyperplane configuration of type $A_{n+1}$. Then some 20 years later Couwenberg, Heckman and Looijenga developed it to a more general setting by means of the Dunkl connection, which deals with the geometric structures on projective arrangement complements. In this talk, I will briefly review the Lauricella system first and then explain how to fit it into the picture of Dunkl system.

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

14:15–15:15, IMPAN 403

**Abstract:**

The theory of $(\varphi, \Gamma)$-modules, initiated by Fontaine, provides a simple description of the category of continuous representations of the Galois group of a finite extension of $\mathbb{Q}_p$ on a finite-dimensional $p$-adic vector space. We describe a corresponding theory for representations of the $n$-fold Cartesian product of this Galois group, and indicate how perfectoid spaces play a key role in its analysis. Joint work with Annie Carter and Gergely Zábrádi.

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

14:15–15:15, IMPAN 403

**Abstract:**

In knot theory and more generally the theory of 3-manifolds various "quantum invariants" like the Witten-Reshetikhin-Turaev or the Kashaev invariant have been much studied in recent years, in particular because of the famous "volume conjecture" related to the asymptotic growth of the Kashaev invariant. Rather surprisingly, it transpired a few years ago that these invariants also have very non-trivial number-theoretical properties, including a kind of weak invariance under the modular group $SL(2,\mathbb{Z})$ ("quantum modular forms") and the experimental discovery of the appearance of certain units in cyclotomic extensions as factors in the asymptotic expansions. The talk will report on this and specifically on recent joint work with Frank Calegari and Stavros Garoufalidis that constructs such units in a purely algebraic way starting from elements in algebraic $K$-theory or in the more elementary "Bloch group", with the proof of a well-known conjecture of Nahm on the modularity of certain $q$-hypergeometric series as an unexpected application.

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

10:30–12:00, IMPAN 403

**Abstract:** This series of lectures will revolve around a family of global $L$-functions built up by assembling symmetric power moments of Kloosterman sums over finite fields. I will show that they have geometric origin, that they extend meromorphically to the complex plane, and that they satisfy a functional equation which was conjectured by Broadhurst and Roberts. Time permitting, I will also discuss the relation between certain special values of these L-functions and integrals of Bessel functions arising from quantum field theory. This is all joint work with Claude Sabbah and Jeng-Daw Yu. I will assume as little background as possible from the audience and take the subject as an excuse to, on the one hand, recall classical material about $L$-functions and periods and, on the other hand, give a leisure introduction to the new ideas coming from exponential motives, irregular Hodge theory, and automorphic forms. Don't be afraid, just come!

Lecture I

14:15–15:15, IMPAN 321

**Abstract:**

I will start by recalling some basics about the twistor correspondence for quaternionic manifolds which in the case of positive quaternionion-Kahler manifolds yields contact Fano manifolds. Then, I will give the local classification of quaternionic manifolds admitting a circle action with no triholomorphic points provided that its fixed points set component is of maximal dimension. Finally, I will discuss the application of this result for quaternion-Kahler manifolds, hence contact Fano manifolds.

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

12:00–12:45, MIMUW 5050

**Abstract:**

We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over $\mathbb{Z}$. This gives an analogue for K3 surfaces of Deligne's description of the category of ordinary abelian varieties over a finite field, and refines earlier work by N.O. Nygaard and J.-D. Yu. Two important ingredients in the proof are integral $p$-adic Hodge theory, and a description of CM points on Shimura stacks in terms of associated Galois representations.

References: arXiv:1711.09225,
arXiv:1707.01236

The first part of the talk will be introductory and will mostly focus on the motivating precedent of Deligne's theorem for ordinary abelian varieties.

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

13:15–14:00, MIMUW 5050

**Abstract:**

Part II of above.

10:30–12:00, IMPAN 403

Lecture II Abstact

14:15–15:45, IMPAN 403

The vortex equations provide an equivariant generalization of Gromov—Witten theory for Kähler manifolds $X$ equipped with a holomorphic Hamiltonian action of a compact Lie group. Their moduli spaces support Kähler structures that are useful to understand certain gauge theories (for example gauged sigma-models, but not only) at both classical and quantum level. In my talk, I shall describe the geometric quantization of the moduli spaces of vortices in line bundles (i.e when $X=\mathbb{C}$ with usual circle action) on a compact Riemann surface $Y$ with fixed compatible area form $\omega_Y$. As complex manifolds, the moduli spaces identify with symmetric powers of $Y$. A crucial ingredient of our construction is the Deligne pairing of line bundles over a familiy of curves. In a natural complex polarization, the resulting quantum Hilbert spaces are finite-dimensional, and they can be interpreted as spaces of multi-spinors on $Y$ valued in a prequantization of an integral rescaling of $\omega_Y$. I will also address the issue of relating Hilbert spaces corresponding to different quantization data geometrically. From the audience, I shall assume very little background beyond the standard package of algebraic geometry. Joint work with Dennis Eriksson.

10:30–12:00, IMPAN 403

Lecture III Abstact

14:15–15:15, IMPAN 321

**Abstract:** Talk will be an elementary survey of recent work with Masha Vlasenko and Francis Brown.

Keywords: periods associated to connections, Mellin transforms of periods, iterated integrals.

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

12:00–12:45, MIMUW 5050

**Abstract:**
Let $X$ be a smooth complete toric variety. I will explicitly describe the obstruction space and the cup product map in combinatorial terms. Using this, I give an example of a smooth projective toric threefolds for which the cup product map does not vanish, showing that in general, smooth complete toric varieties may have obstructed deformations. This is joint work with Charles Turo.

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

13:15–14:00, MIMUW 5050

**Abstract:**
The greatest lower bound on the Ricci curvature, $R(X)$, measures how far a Fano manifold is from being Einstein. In the toric case Li gave a beautiful formula for $R(X)$; it measures how far the barycenter of the associated polytope is from the origin. In this talk I will show how this result can be generalized to complexity one.

10:30–12:00, IMPAN 403

**Abstract: **Full abstract

Let $X$ be an algebraic variety over the base field $k$. The additive
group $\mathbb{G}_a$ is the base field $(k,+)$ seen as algebraic group
with its additive structure. In this mini-course we will give an
overview of the classification of actions of the additive group by
means of certain integrable vector fields on $X$.

The posted plan is subject to changes depending on the interests
and background of the audience. If for some reason you did not attend
the first lecture, you can still enter the mini-course. The first
lecture is not prerequisite for the last two lectures.

Lecture I: We will develop the theory of
$\mathbb{G}_a$-actions on affine varieties. In this case,
$\mathbb{G}_a$-actions are in one to one correspondence with vector
fields satisfying a nilpotency condition. This subject is classical
from affine geometry, see for instance [Fre06].

[Fre06] Gene Freudenburg, *Algebraic theory of locally nilpotent derivations*,
Encyclopaedia of Mathematical Sciences 136. Invariant Theory and Algebraic
Transformation Groups 7. Berlin: Springer, 2006.

14:15–15:45, IMPAN 403

Lecture IV Abstact

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 15:15.

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 15:15.

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 15:15.

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 403

**Abstract:**

TBD

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

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:**

In this talk we apply results on the stable reduction of superelliptic curves to Picard curves defined over a number field. We use this to obtain information on arithmetic invariants like the conductor.

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

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 15:15.

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 and 13:15–14:00, MIMUW 5050

**Abstract:**

We show that the Hilbert scheme of points on a higher dimensional affine space is non-reduced and has components lying entirely in characteristic $p$ for all primes $p$. Our main tool is a generalized version of the Białynicki-Birula decomposition.

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

10:30–12:00, IMPAN 321

**Abstract:**
This talk will address the following questions: How can one characterize tensors with positive dimensional symmetry groups? How large can such groups be assuming natural genericity conditions? What is the relationship between symmetry and complexity (e.g., border rank)? This is joint work with A. Conner, F. Gesmundo, E. Ventura and Y. Wang.

14:15–15:15, IMPAN 321

**Abstract:**

We will explain how to compute the cohomology of $p$-adic analytic curves, using a decomposition into shorts and legs, reminiscent of pants decompositions of Riemann surfaces.

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

9:15–10:15, IMPAN 405

**Abstract:**
Part of the F-isocrystals miniworkshop.

10:45–11:45, IMPAN 405

**Abstract:**
Part of the F-isocrystals miniworkshop.

13:15–14:15, IMPAN 405

**Abstract:**
Part of the F-isocrystals miniworkshop.

14:30–15:30, IMPAN 405

**Abstract:**
Part of the F-isocrystals miniworkshop.

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

9:15–10:15, IMPAN 405

**Abstract:**
Part of the F-isocrystals miniworkshop.

10:30–12:00, IMPAN 403

**Abstract:**
The study of linear subspaces contained in algebraic varieties has a long history, going all the way back to the celebrated result of Cayley-Salmon that a smooth cubic surface contains exactly 27 lines. Most results deal with linear subspaces contained in generic hypersurfaces (or complete intersections) in projective space. However, it can also be useful to study linear subspaces contained in special varieties.

In this series of lectures, I will begin by presenting some of the very basic theory on linear subspaces of varieties, and will briefly sketch some of the results for generic hypersurfaces. In the remaning time, I will focus on linear subspaces of special varieties. Here, the tools used for generic hypersurfaces break down. Examples will include toric varieties and determinantal hypersurfaces. Time permitting, I will also discuss applications to Chow rank of homogeneous forms.

This series of lectures is intended for a wide audience; the only prerequisite is working knowledge of the basics of algebraic geometry.

Lecture I

12:15–13:15, IMPAN 405

**Abstract:**
Part of the F-isocrystals miniworkshop.

14:15–15:15, IMPAN 405

**Abstract:**
Part of the F-isocrystals miniworkshop.

15:30–16:30, IMPAN 405

**Abstract:**
Part of the F-isocrystals miniworkshop.

9:15–10:15, IMPAN 321

**Abstract:**
Part of the F-isocrystals miniworkshop.

10:45–11:45, IMPAN 321

**Abstract:**
Part of the F-isocrystals miniworkshop.

13:15–14:15, IMPAN 321

**Abstract:**
Part of the F-isocrystals miniworkshop.

12:00–12:45, MIMUW 5050

**Abstract:**
I will first recall known results about birational contractions of K3 surfaces. Then discuss the difficulties occurring for birational contractionsfor hyper-Kahler manifolds in the next dimension 4. This is a report on a work in progress with B. van Geemen.

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

13:15–14:00, MIMUW 5050

**Abstract:**
We will report on joint work in progress with Chiara Camere, Alice Garbagnati, Grzegorz Kapustka on the classification of IHS fourfolds admitting symplectic involutions. We will start with a review of the classification of symplectic involutions on K3 surfaces, so-called Nikulin involutions. We will then discuss the difference with the case of fourfolds, present our ideas and provide some geometric examples.

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 15:15.

10:30–12:00, IMPAN 403

**Abstract:**

Lecture III: **Abstract**

14:15–15:15, IMPAN 321

**Abstract:**
Abstract: In the talk, I will recall a few geometric and deformation theoretic results concerning varieties with trivial canonical class in characteristic zero contrasting them with characteristic $p>0$ counterexamples. I will then explain how to salvage some part of the characteristic $p>0$ theory using arithmetic assumption of ordinarity. The talk is based on independent works with Piotr Achinger and Zsolt Patakfalvi.

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

10:00–18:00, MIMUW room 2270 (10:00-12:00) and 3170 (12:00-18:00)

More information here: T-varieties Mini-Workshop Homepage

10:00–15:00, IMPAN 321

More information here: T-varieties Mini-Workshop Homepage

15:15–16:15, IMPAN 321

**Abstract:**
To study the complexity of the matrix multiplication tensor, Strassen introduced a class of tensors that vastly generalize it, the tight tensors. Tight tensors are essentially tensors with a ”good” positive dimensional symmetry group. This motivates further investigations of (1-generic) tensors with large symmetry group. In this talk, we discuss some combinatorial and geometric consequences of tightness. This is based on joint works with A. Conner, F. Gesmundo, JM Landsberg, and Y. Wang.

The talk will be preceeded by tea and cake in 409 IMPAN at 14:45.

10:30–12:00, IMPAN 403

**Abstract:** The aim of these lectures is to introduce aspects of projective and birational geometry in the realm of tensor decompositions with particular regards to the identifiability problem.

After a brief review of secant varieties, with the help of Terracini's Lemma and the Infinitesimal Bertini Theorem, I will make a link between identifiability and weakly defective varieties. Then I will study the weakly defectiveness of Veronese varieties and as an application I will state, with a proof sketch, the complete classification of (generically) identifiable homogeneous polynomials.
If time will allow I will discuss similar problems for arbitrary tensors.

Lecture I

14:15–15:15, IMPAN 321

**Abstract:**

Lecture II Abstract

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

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

**Abstract:**

For a fixed projective manifold $X$, we say that a property $P(L)$ (where $L$ is a line bundle on $X$) is satisfied by sufficiently ample line bundles if there exists a line bundle $M$ on $X$ such that $P(L)$ hold for any $L$ with ample $L-M$. I will discuss which properties of line bundles are satisfied by the sufficiently ample line bundles --- for example, can you figure out before the talk, whether a sufficiently ample line bundle must be very ample? A basic ingredient used to study this concept is Fujita's vanishing theorem, which is an analogue of Serre's vanishing for sufficiently ample line bundles. At the end of the talk I will define cactus varieties (an analogue of secant varieties) and sketch a proof that cactus varieties to sufficiently ample embeddings of $X$ are (set-theoretically) defined by minors of matrices with linear entries. This is closely related to conjectures of Eisenbud-Koh-Stillman (for curves) and Sidman-Smith (for any varieties). This is based on a joint work with Weronika Buczyńska and Łucja Farnik.

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