# Facets of Geometry

## General information about the program:

Program formatted for printing

The main scientific program will take place on Monday, Tuesday, Thursday and Friday. Each of these days will open by a talk of one of our plenary speakers.

Each section takes place either in the morning or in the afternoon each day, see the chart below. During the whole week each section will be held twice in the morning and twice in the afternoon. There will be 2 talks in the morning after the plenary talk and 3 talks in the afternoon which amounts to totally 10 section talks for the whole duration of the conference. The idea is to try to minimize the overlapping of the sections to enable mathematicians with different interests to mix and attend talks in all areas.

## Monday, June 3, 2013

#### Conference Registration starts at 09:00

The conference will be held at Stockholm University Kräftriket House 5, right next to the department of Mathematics. The plenary lectures and the "a" sessions will be held at House 5 room 15 and the "b" sessions at House 5 room 14.

### Conference Opening::

#### Anders Karlhede (Stockholm University)

The Dean of School of Science, Anders Karlhede, will open the conference with a short address.

### Monday Plenary Lecture :: Peter Perry

#### Peter Perry (University of Kentucky)

###### Spectral Geometry: From Selberg's Trace Formula to Geometric Scattering Theory

Spectral geometry is the study of how eigenvalues and scattering data on a Riemannian manifold reflect its underlying geometry. In 1956, Selberg proved a beautiful trace formula for finite-volume Riemann surfaces X which gave a precise, quantitative relation between the eigenvalues and scattering resonances of the Laplacian on the one hand, and the lengths of the closed geodesics of X on the other. Selberg's formula motivated the development of scattering theory in non-Euclidean spaces by Faddeev, Lax-Phillips, and others and led to the development of geometric scattering theory. In this lecture, we will trace the history of geometric scattering theory and highlight its contributions to spectral geometry.

### Morning session 1a :: Johannes Sjöstrand

#### Johannes Sjöstrand (Paris 6)

###### Eigenvalue distribution for non-self-adjoint differential operators.

In this talk we will review some earlier results in the analytic case and some more recent ones where the analyticity is destroyed by means of a small random perturbation. In the first case the eigenvalues sometimes obey a complex Bohr-Sommerfeld rule that invalidate the Weyl law and the analysis is based on complex deformations of the real phase space. Non-self-adjoint operators do most of the time exhibit large resolvent norms and corresponding eigenvalue instability. It is then natural to study the effect of small random perturbations. Surprisingly, such perturbations cause the eigenvalues to distribute according to the Weyl law. An heuristic explanation is that the random perturbations destroy analyticity and forbid complex deformations of phase space, leaving the Weyl law as the only natural possibility. We will state some results and develop the underlying ideas of complex phase space deformations and random perturbations, following works of A. Melin, M. Hitrik, S. Vu Ngoc, M. Hager, W. Bordeaux Montrieux, the speaker and other people.

### Morning session 1b :: Oleg Viro

#### Oleg Viro (State University of New York at Stony Brook)

###### Amoeba hyperfield, analysis over it, and amoebas' geometry.

Amoebas of complex algebraic varieties are contained in and sometimes coincide with the corresponding varieties over the hyperfield of non-negative real numbers in which the multiplication is the usual addition while the addition is multivalued and described by the triangle inequality. In the talk these notions will be introduced and discussed.

### Morning session 2a :: Rafe Mazzeo

#### Rafe Mazzeo (Stanford University)

###### Spectral geometry of the Riemann moduli space

I will discuss some aspects of the spectral theory of the Laplacian for the Weil-Peterson metric on the Riemann moduli space (joint with Ji, Mueller and Vasy). This is part of a larger effort to study the analysis of natural elliptic operators on this and other natural singular spaces. This work leads to the necessity of understanding the fine asymptotics of the Weil-Peterson metric, which has recently been obtained with Swoboda. I also describe the consequences of this for the asymptotic structure of the heat kernel.

### Morning session 2b :: Vyacheslav Kharlamov

#### Vyacheslav Kharlamov(Université de Strasbourg)

###### A few examples of Z-valued real Schubert calculus

We intend to show by examples that in various real enumerative problems it is possible to perform some natural Z-valued count of real solutions in such a way that the result becomes invariant under input data, and that, as a consequence of such an invariance, one obtains nontrivial lower bounds on the number of real solutions, and even observes a surprising abundance of real solutions.

### Afternoon session 1a :: Mattias Jonsson

#### Mattias Jonsson (Ann Arbor Michigan)

###### Pluripotential theory over complex numbers and non-Archimedean fields

Pluripotential theory is the study of plurisubharmonic (psh) functions and has important applications to complex geometry. There is an emerging parallel theory when replacing the complex numbers by a non-Archimedean field. I will discuss this, as well as some speculative connections to degenerations of complex structures.

### Afternoon session 1b :: Rahul Pandharipande

#### Rahul Pandharipande (ETH Zurich)

###### Cohomology of the moduli space of curves.

The moduil space of curves carries tautological cohomology classes. I will discuss the study of relations amongst these classes starting with ideas of Mumford in 1980s. The subject advanced in the 1990s with conjectures of Faber and Faber-Zagier. I will explain the current state of affairs based on Pixton's conjectures related to cohomological field theories.
The talk represents joint work with A. Pixton and D. Zvonkine.

### Afternoon session 2a :: Elizabeth Wulcan

#### Elizabeth Wulcan (Chalmers)

###### Green functions and Segre numbers

This talk is based on a joint work with Mats Andersson. We give meaning to (higher) Monge-Ampére masses $$(dd^c G)^k$$ of Rashkovskii-Sigurdsson's Green function $$G$$ with poles along an ideal sheaf $$\mathfrak{A}$$ (also for $$k$$ larger than the codimension of $$\mathfrak{A}$$). We show that the Lelong numbers of $$\mathbf 1_Z (dd^c G)^k$$, where $$Z$$ is the variety of $$\mathfrak{A}$$, are the so-called Segre numbers of $$\mathfrak{A}$$. This result generalizes the well-known fact that if $$Z$$ is a point, the top Monge-Ampére mass is just a point mass with mass equal to the Hilbert-Samuel multiplicity of $$\mathfrak{A}$$.

### Afternoon session 2b :: Thomas Willwacher

#### Thomas Willwacher (Harvard University)

###### A family of Drinfeld associators and explicit formulas for graph cohomology classes

I will talk about a construction of a family of Drinfeld associators, interpolating between the Knizhnik-Zamolodchikov, the Alekseev-Torossian and the anti-Knizhnik Zamolodchikiv associator. As a byproduct, we will find explicit integral formulas for the conjectural generators of the Grothendieck-Teichmueller Lie algebra, and their corresponding graph cocycles.
This is a joint project with Carlo Rossi

### Afternoon session 3a :: Timur Sadykov

#### Timur Sadykov (Moscow State University)

###### Analytic complexity of special functions

The Kolmogorov-Arnold theorem [1] yields a representaion of a multivariate continuous function in terms of a composition of functions which depend on at most two variables. In the analytic case, understanding the complexity of such a representation naturally leads to the notion of the analytic complexity of (a germ of) a bivariate (multi-valued) analytic function introduced and studied by V.K.Beloshapka [1]. According to Beloshapka's local definition, the order of complexity of any univariate function is equal to zero while the $$n$$-th complexity class is defined recursively to consist of functions of the form $$a(b(x,y)+c(x,y))$$, where $$a$$ is a univariate analytic function and $$b$$ and $$c$$ belong to the $$(n-1)$$-th complexity class. Such a represenation is meant to be valid for suitable germs of multi-valued holomorphic functions.
With such a hierarchy of complexity classes, one can associate a number of differetial and algebraic invariants. The first complexity class can be alternatively defined as the set of multi-valued analytic solutions to the differential polynomial $$F_x F_y (F_{xxy}F_y - F_{xyy}F_x) + F_{xy}(F_{x}^2 F_{yy} - F_{y}^2 F_{xx})$$ [2].
A randomly chosen bivariate analytic functions will most likely have infinite analytic complexity. However, for a number of important families of special functions of mathematical physics their complexity is finite and can be computed or estimated. Using properties of solutions to the Hopf equation and the Gelfand-Kapranov-Zelevinsky system we obtain estimates for the analytic and polynomial complexity of such functions as well as plane webs [2] and knots on Riemann surfaces.
References

[1] V.I.Arnold,
On the representation of continuous functions of three variables by the superpositions of continuous functions of two variables,
Sbornik Math. 48, no. 1, 1959, 3-74.

[2] V.K.Beloshapka,
Analytic complexity of functions of two variables.
Russian Journal of Mathematical Physics, Vol. 14, no. 3, 2007, pp. 243-249.

### Afternoon session 3b :: Alexander Berglund

#### Alexander Berglund (Stockholm University)

###### Rational homotopy theory of automorphisms of highly connected manifolds

By the Harer stability theorem, the cohomology of the diffeomorphism group of an orientable surface stabilizes as the genus increases. A description of the stable cohomology was conjectured by Mumford and later established by Madsen and Weiss.
I will talk about recent results on the cohomology of automorphism groups of high dimensional manifolds, and on the rational homotopy types of their classifying spaces. We prove an analog of Harer's stability theorem for a family of highly connected manifolds. In the calculation of the stable cohomology, certain Lie algebras of symplectic derivations show up that have appeared before in Kontsevich's work on the homology of outer automorphisms of free groups. This leads to an interesting and somewhat surprising correspondence between unstable homology classes of outer automorphism groups of free groups, and stable characteristic classes of fibrations with fiber a highly connected manifold.
This is joint work with Ib Madsen.

## Tuesday, June 4, 2013

### Tuesday Plenary Lecture :: Gerard van der Geer

#### Gerard van der Geer (Universiteit van Amsterdam)

###### Stratifications on Moduli Spaces

In the talk I will report on joint work with Torsten Ekedahl on stratifications on moduli spaces of abelian varieties and K3 surfaces in positive characteristic.

### Morning session 1a :: Nick Shepherd-Barron

#### Nick Shepherd-Barron (University of Cambridge)

###### Stable Schottky forms.

Fay gave formulae for the period matrices of certain carefully constructed degenerating families of algebraic curves. We use these formulae to prove that there are no stable Siegel modular forms that vanish along the moduli space of curves, or even along various special subvarieties of it, such as the trigonal locus.

[Joint work with G. Codogni.]

### Morning session 1b :: Richard Kenyon

#### Richard Kenyon (Brown University)

##### Dimers and simple Harnack curves

We discuss the algebraic curves arising from the determinant of discrete operators associated with the planar dimer model, and in particular focus on the connections between the curve and various probabilistic aspects of the model.

### Morning session 2a :: Bruno Vallette

#### Bruno Vallette (Universite de Nice Sophia Antipolis)

Givental action is homotopy gauge symmetry

I will prove that the Givental action on genus zero cohomological field theories, also known as hypercommutative algebras, is equal to the gauge symmetry action on Maurer-Cartan elements of the homotopy Lie algebra encoding homotopy Batalin-Vilkovisky algebras. This equivalent description allows us to extend the Givental action to homotopy hypercommutative algebras, i.e. from the homology level to the chain level.
Reference: arxiv.org/1304.3343 ]

### Morning session 2b :: Uzy Smiliansky

#### Uzy Smiliansky (Weizman Institure of Science)

###### Tournaments and their spectra

A round robin tournament is a competition between N players, each plays once against all the rest and the outcome of a game is definite, (no draw). The outcome can be summarized in an NxN matrix with the (i,j) entry being 1 if player i wins and 0 otherwise. One can also consider these matrices as the adjacency matrices of directed graph. The spectra of tournament matrices show interesting features, and their statistics can be studies when one studies the entire ensemble of NxN tournaments endowed with uniform probability. No wonder - Random Matrix theory reproduces the observed statistical features, which can be addressed by using a trace formula connecting the spectra to the counting statistics of cycles on the directed graph.

### Afternoon session 1a :: Hannah Markwig

#### Hannah Markwig (Universität des Saarlandes)

###### Tropical moduli spaces of covers and Hurwitz numbers

Tropical double Hurwitz numbers have been introduced intersection-theoretically as the degree of a tropical branch map from the moduli space of covers, in analogy to algebraic Hurwitz numbers. A correspondence theorem holds. The correspondence theorem has been generalized to arbitrary Hurwitz numbers by adding more ends to the target tropical curve and requiring the ramifications to lie above ends. We allow simple ramification to be in the interior. We construct the appropriate tropical moduli space and show that we get the Hurwitz number intersection-theoretically as the degree of the tropical branch map. In the proof of the invariance of the degree of the tropical branch map, we use known facts about the algebraic moduli space of relative stable maps.

Joint work with Arne Buchholz

### Afternoon session 1b :: Jan-Erik Björk

#### Jan-Erik Björk (Stockholm University)

###### Multi-dimensional residues in the complex domain

We explore the construction of Coleff-Herrera currents which, in general, are attached to complex analytic sets. Interplay with analytic D-module theory is high-lighted.

### Afternoon session 2a :: Jean-Jacques Risler

#### Jean-Jacques Risler (Université Pierre et Marie Curie - Paris 6)

###### Curvature of the Real Amoeba Work by Mikael Passare and Jean-Jacques Risler

1. Total Curvature
Let $$X\subset \mathbf{R}^n$$ be a smooth algebraic hypersurface. Then the total curvature of $$X$$ is the "volume" of the Gauss map $$g : X \to \mathbf{RP}^n$$. The total curvature of the real Amoeba is then the volume of the image of the Logarithmic Gauss map.
2. Simple Harnack curves
I will recall the definition of G. Mikhalkin, and the theorem of Mikhalkin- Rullgard which characterize plane Simple Harnack curves by the fact that the Amoeba has maximal area.
3. Total Curvature of the Real Amoeba
I will give a bound for the total curvature of the real Amoeba of a real smooth plane curve X (in term of its Newton Polygon) and prove that this bound is reached if and only if X is a (smooth) simple Harnack curve.
4. Case of surfaces
If time , I will discuss the possiblity of extensions to the case of Surfaces. .

### Afternoon session 2b :: Peter Ebenfelt

##### June 04 :: 15:20-16:05 :: Room 14
###### Partial rigidity of degenerate CR embeddings into spheres

We shall consider degenerate CR embeddings $$f$$ of a strictly pseudoconvex hypersurface $$M\subset {\mathbb C}^{n+1}$$ into a sphere $${\mathbb S}$$ in a higher dimensional complex space $${\mathbb C}^{N+1}$$. The degeneracy of the mapping $f$ will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the speaker, together with X.Huang and D. Zaitsev, established a rigidity result for CR embeddings $$f$$ into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank $$d$$ of the second fundamental form and all of its covariant derivatives is $$< n$$ (here, $$n$$ is the CR dimension of $$M$$), then $$f(M)$$ is contained in a complex plane of dimension $$n+d+1$$. The converse of this statement is also true, as is easy to see. When the total rank $$d$$ exceeds $$n$$, it is no longer true, in general, that $$f(M)$$ is contained in a complex plane of dimension $$n+d+1$$, as can be seen by examples. In this talk, we shall show that (well, explain how) when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension $$n$$, then partial rigidity may still persist, but there is a "defect" $$k$$ that arises from the ranks exceeding $$n$$ such that $$f(M)$$ is only contained in a complex plane of dimension $$n+d+k+1$$. Moreover, this defect occurs in general, as is illustrated by examples.

### Afternoon session 3a :: Mounir Nisse

#### Mounir Nisse (Texas A & M University)

###### Co)amoebas  of linear spaces and higher convexity of (co)amoebas complement of any algebraic variety

The first part of my talk will be a joint work with Passare on (co)amoebas of $$k$$-dimensional affine linear spaces  in $$(\mathbb{C}^*)^n$$. When $n=2k$, we compute the volume of their coamoebas in the generic case. Moreover, if the spaces are real  and generic, then we compute the volume of their amoebas too. In the second part of my talk, I will strengthen Henriques's result about the higher convexity of amoeba complements  when the underlying variety is a complete intersection,  and extend it to coamoebas  complements   in the general case (i.e., without the assumption of complete intersection). Also, if the codimension of our variety is $$r$$, and the complement of its amoeba is $$\mathscr{A}^c$$, then I define a map from the integer homology group $$H_{r-1}(\mathscr{A}^c,\mathbb{Z})$$ to $$H^r((\mathbb{C}^*)^n, \mathbb{Z})$$ generalizing the order map of the hypersurface case. This part is a joint work with Frank Sottile. My talk will be ended by some interesting questions.

### Afternoon session 3b :: Ahmed Zeriahi

#### Ahmed Zeriahi (Universite de Toulouse)

###### Convergence of the normalized Kähler-Ricci flow on Fano varieties

Abstract : Let $$X$$ be a Fano manifold whose Mabuchi functional is proper. A deep result of Perelman-Tian-Zhu asserts that the normalized Kaehler-Ricci flow, starting from an arbitrary Kaehler form in $$c_1 (X)$$ , converges towards the unique Kaehler-Einstein metric on $$X$$. We will give an alternative proof of a weaker convergence result which applies to the broader context of (log)-Fano varieties. This is a joint work with R. Berman, S.Boucksom, P.Eyssidieux and V.Guedj.

## Wednesday, June 5, 2013

### Wednesday program.

On Wednesday June 5 we plan a memorial session in the morning followed by a boat trip in the archipelago to a nice little town Vaxholm located about 40 km north of Stockholm.

### Morning Program

Memorial Session dedicated to the memory of Torsten Ekedahl and Mikael Passare.

9.00 - 9.20
Anders Björner
9.20 - 9.40
Christer Kiselman
9.40 - 10.00
Jan-Erik Roos
10.30 - 10.50
August Tsikh
10.50 - 11.10
Gerard van der Geer
11.10 - 11.30

### Afternoon Program

Boat trip in the Stockholm archipelago to a nice little town Vaxholm located about 40 km north of Stockholm.

Time of departure::
13:00
Place of departure ::
Nybrokajen Kajplats 14 (see map below)
Boatrip ends at
16:00

Visa Facets Of Geometry Locations på en större karta

Conference buffet at restaurant Kräftan starts at 17:30

## Thursday, June 6, 2013

### Thursday Plenary Lecture :: Jean-Pierre Demailly

#### Jean-Pierre Demailly (Grenoble)

###### On the cohomology of pseudoeffective line bundles

The lecture will present various results concerning the cohomology of pseudoeffective line bundles on compact Kähler manifolds, twisted with corresponding multiplier ideal sheaves. In case the curvature is strictly positive in the sense of currents, the prototype is the well known Nadel vanishing theorem. We are interested here in the case where the curvature is merely semipositive. Various results and applications will be discussed, including a recent vanishing theorem due to Junyan Cao (forthcoming PhD thesis in Grenoble), and a study of simple compact Kähler 3-folds (joint work with F. Campana and M. Verbitsky from April 2013).

### Morning session 1a :: Alain Yger

#### Alain Yger (Université de Bordeaux)

###### Multidimensional analytic residue calculus : averaging versus keeping track of multiplicative aspects

Averaging methods in multivariate analytic residue calculus (such as those that transform the Cauchy kernel into the Bochner-Martinelli one) were suggested by Mikael Passare and settled in our joint paper with Mikael and August Tsikh in 2000. One of our motivations was originally to overcome the crucial observation by Mikael and August that residue integrals do not have unconditional limits. Such averaged residue currents revealed since then to be quite useful tools to rephrase in an analytic context division or intersection problems, among them questions involving normalized blow-up such as integral closures of ideals, their relation with so-called Chow ideals of cycles, and Briançon-Skoda theorem (M. Andersson, J.E. Björk, H. Samuelsson, E. Wulcan, E. Götmark, J. Sznajdman, R. Lärkäng, J. Lundqvist). On the other hand, multiplicative aspects of residue calculus, since they fit with the algebraic Transformation Law, remain essential in order to profit from multidimensional residue technics in an arithmetic context (such a context being unfortunately not preserved when residue currents are averaged). The talk, as a tribute to Mikael's memory, will focus on this dichotomy (averaging on one side, keeping track of multiplicative structure of residue calculus on the other side), illustrated by a selection of examples.

### Morning session 1b :: Anton Alekseev

#### Anton Alekseev (Geneva University)

###### Kontsevich formality and the logarithmic propagator

The original proof of the Kontsevich formality theorem is based on the Stokes formula applied to compactified configuration spaces of points in the upper half-plane. Kontsevich also suggested the second version of the formality morphism where the differential forms of the type d arg(z-w) are replaced by d log(z-w). In this new context, the integrals defining the weights in the formality morphism may contain singularities, and it is not clear whether the Stokes formula applies. We will show convergence of the integrals, and we will explain how one can make the Stokes formula work using some locally defined torus actions.
The talk is based on a joint work with J. Loeffler, C. Rossi, C. Torossian and T. Willwacher.

### Morning session 2a :: Ragnar Sigurdsson

#### Ragnar Sigurdsson (University of Iceland)

###### Pluricomplex Green functions

The lecture is a report on a joint work, still in progress, with Aron Lagerberg and Benedikt Steinar Magnússon. The extremal functions of pluripotential theory are defined as suprema $$\big(\sup {\cal F}\big)$$ or upper regularized suprema $$\big(\sup {\cal F}\big)^*$$ of classes $${\cal F}\subset {\cal PSH }(X)$$ of plurisubharmonic functions on $$X$$, where $$X$$ can be an open domain in $${\mathbb C}^n$$, complex manifold, or even a complex space. It is also natural to look at extremal functions in the quasi-pluri-subharmonic classes $${\cal PSH }(X,\omega)$$.
Among the examples of extremal functions are various pluricomplex Green functions which are usually taken as negative functions with prescribed location and type of singularities.

We let $$\nu_v(a)$$ denote the Lelong number of a plurisubharmonic function $$v$$ at a point $$a\in X$$ and look at the class ${\cal F}_{\omega,\varphi,\alpha}=\{u\in {\cal PSH}(X,\omega) \,; u\leq \varphi, \nu_{u+\psi}\geq \alpha, \text{ for local potentials } \psi \text{ of } \omega\},$ for given functions $$\varphi:X\to \overline {\mathbb R}$$ and $$\alpha:X\to {\mathbb R}_+$$, and define the Green function corresponding to $$\omega$$, $$\varphi$$, and $$\alpha$$ as $G_{\omega,\varphi,\alpha}=\sup {\cal F}_{\omega,\varphi,\alpha}$ Our main result is a disc envelope formula for $$G_{\omega,\varphi,\alpha}$$ which generalizes and unifies many known disc envelope formulas for extremal functions in pluri-potential theory.

### Morning session 2b :: Sergei Shadrin

#### Sergei Shadrin (Universiteit van Amsterdam)

###### An ELSV formula for the space of $r$-spin structures.

A famous formula of Ekedahl-Lando-Shapiro-Vainshtein relates Hurwitz numbers to the intersection theory of the moduli spaces of curves. In 2005-2006 Zvonkine conjectured a generalization of this formula that relates Hurwitz numbers with completed cycles to the intersection theory of the moduli spaces of $$r$$-spin structures. In a recent work with Spitz and Zvonkine we found a mathematical physics proof of this "r-ELSV formula". The proof goes through a construction of a matrix model for Hurwitz numbers with completed cycles and its analysis via the spectral curve topological recursion and its link to the Givental theory. I am going to explain the main steps of this argument.

### Afternoon session 1a :: Carolyn Gordon

#### Carolyn Gordon (UC Berkeley)

###### Isospectrality: a Partial Survey

We will discuss conditions for isospectrality and for spectral rigidity of the Laplace operators or other natural operators on Riemannian manifolds. We may briefly also address analogous questions for quantum graphs.

### Afternoon session 1b :: Diane Maclagan

#### Diane Maclagan (University of Warwick)

###### Computational algebra aspects of tropical geometry

One of the many different definitions of a tropical variety uses a variant of Groebner bases that takes the valuations of the coefficients into account. I will discuss joint work with Andrew Chan on algorithms to compute these Groebner bases, and give an overview of other computational algebra challenges in this area.

### Afternoon session 2a :: Toshikazu Sunada

#### Toshikazu Sunada (Meiji University, Tokyo)

###### Random walks, Diophantine problems, and rational points in complex quadrics -- A rambling talk on mathematical crystallography--

A certain Diophantine problem and 2D crystallography are linked through the notion of standard realizations which was introduced originally in the study of random walks. In the discussion, a complex projective quadric is associated with a finite graph. Rational points" on this quadric turns out to be related to standard realizations of 2D crystal structures.

### Afternoon session 2b :: August Tsikh

#### August Tsikh (Siberian Federal University)

###### On amoebas of higher codimension and singular strata of the classical discriminant.

An amoeba of a complex algebraic set is its image under the projection onto the real subspace in the logarithmic scale. We study homological properties of the complements to amoebas for sets of codimension greater than 1. In particular, we use the multidimensional residue theory to refine A.Henriques’ result saying that the complement of an amoeba of a codimension k set is (k-1)-convex. We also describe the relationship between critical points of the logarithmic projction and the logarithmic Gauss map of algebraic sets. Using these tools we give a parameterization for some singular strata of the classical discriminant (a generalization of our result with M.Passare). We conjecture that these strata have the so-called maximum likelihood degree one and, by the recent result of J. Huh, they are A-discriminantal surfaces of codimension greater than 1.

### Afternoon session 3a :: Anders Karlsson

#### Anders Karlsson (Univ. of Geneva)

###### Extensions of Thurston's spectral theorem for surface homeomorphism

In 1976, as a consequence of his groundbreaking work on surfaces, in particular the Nielsen-Thurston classification of homeomorphisms, Thurston deduced a sort of spectral theorem in terms of exponential growth rates of the length of simple closed curves under iteration. I will discuss some partial extensions of this, including a version for random products of homeomorphisms.

## Friday, June 7, 2013

### Friday Plenary Lecture :: Grigory Mikhalkin

#### Grigory Mikhalkin (Université de Geneve)

###### Amoebas and related geometric concepts

The talk will survey amoebas of complex algebraic varieties in several frameworks as well as other concepts intrinsically related to amoebas.

### Morning session 1a :: Erwan Brugallé

#### Erwan Brugallé (Université Pierre et Marie Curie - Paris 6)

###### Welschinger invariants of minimal conic bundles

At least two methods are available to compute genus 0 real enumerative invariants: a real version of WDVV equations due to Solomon, and a real version of stretching the neck technic in symplectic field theory. I will explain how to combine these two methods to compute Welschinger invariants of minimal conic bundles.
This is a joint work with Jake Solomon

### Morning session 1b :: Håkan Samuelsson

#### Håkan Samuelsson (Chalmers)

###### Regularizing residue currents

In the complete intersection case it is possible to define products of residue currents so that natural and suggestive computation rules hold. This was achieved in the '70s by Coleff and Herrera by taking certain restricted limits of the so called residue integral. The behaviour of the residue integral was further studied by e.g. M. Passare, A. Tsikh, and J.-E. Björk in the '80s and '90s and they found, somewhat surprisingly in view of intersection theory, that the residue integral does not have an unconditional limit. In the talk I will explain how one can regain an unconditional limit by taking a certain mild average of the residue integral; this corresponds to certain regularizations of residue currents.

The talk is based on joint work with J.-E. Björk.

### Morning session 2a :: Eugenii Shustin

#### Eugenii Shustin (Tel Aviv University)

###### Relative enumerative invariants of real rational surfaces

In 2006 Welschinger introduced enumerative invariants of real rational surfaces involving real rational curves tangent to a given divisor at one point as well as cuspidal rational curves and reducible curves. We suggest a series of new relative invariants, which count real rational curves on real rational surfaces that are tangent to a given divisor at each intersection point. We discuss properties of these invariants including their relations to Welschinger invariants. (Joint work with I. Itenberg and V. Kharlamov.)

### Afternoon session 1a :: David Rydh

#### David Rydh (KTH)

###### Algebraicity and coherence of functors.

When Grothendieck laid the modern foundations of algebraic geometry in the late 50s, a central question became when a set-valued functor is representable by a scheme. This turned out to be very difficult to answer.
In the 70s, M. Artin completed Grothendieck's vision by introducing a mild generalization of schemes – algebraic spaces – and giving a precise criterion for when a set-valued functor is algebraic.

In this talk, I will describe a parallel story for additive functors of modules over a commutative ring. The notion of coherence for additive functors was introduced by Auslander in 1965. I will argue that the coherence of a functor is analogous to the algebraicity of a space. In particular, I will present a criterion for the coherence of a half-exact additive functor. This vastly generalizes previous coherence criteria which require the base ring to be a discrete valuation ring.
Although the new criterion is strikingly similar to Artin's criterion and there are interesting connections via deformation theory, we have not been able to directly relate them.

This is joint work with Jack Hall.

### Afternoon session 1b :: Olga Rossi

#### Olga Rossi (Ostrava & La Trobe U)

###### Calculus of Variations 250 years after Lagrange: a symbiosis of analysis, geometry and physics

The idea of using differential geometry in the calculus of variations goes back to Élie Cartan who in 1921 introduced a differential form, now called the Cartan form, to study the variational integral. The work of Cartan and his successors opened new horizons for modern developments in the calculus of variations as a mathematical discipline, with close relationships to differential geometry, the topology of smooth manifolds, global analysis, the theory of exterior differential systems and PDEs, algebraic topology and algebraic geometry. It also extended the scope of classical variational calculus: its techniques now are used beyond analytical mechanics to investigate deep theoretical issues in physics, for instance in relativity, gauge theory, string theory and geometric quantization, as well as in engineering and control theory. In this talk I will discuss the current state of the field with its challenging applications.

### Afternoon session 2a :: Dan Petersen

#### Dan Petersen (KTH)

###### The tautological ring in genus two is not Gorenstein in general

Consider the moduli space of stable n-pointed curves of genus g. Inside its rational Chow ring is a natural subring called the tautological ring. This subring has been conjectured to always be Gorenstein by Faber and Pandharipande, by analogy with the Faber conjectures on M_g.
First I will explain why in genus one, all even cohomology classes are tautological. This is implied by results announced without proof by Getzler in the mid-90s, which were proven only recently by myself. Then I will discuss how similar ideas can be modified to study the case g=2. One can prove that at the "first time" that there exists a non-tautological even cohomology class on the moduli space (i.e. we consider the smallest value of n for which such a class exists), then this class lives below the middle degree. As a direct consequence, the tautological ring cannot possibly be Gorenstein for this value of n.
This is joint work with Orsola Tommasi.

### Afternoon session 2b :: Michael Loss

#### Michael Loss (Georgia Tech)

##### Proof of sharp inequalities via flows

In this review talk a number of sharp functional inequalities will be revisited. Examples are Brascamp-Lieb inequalities and certain types of Sobolev inequalities. Common to these examples is the fact that they can be proved in their sharp form using well adapted non-linear flows. This is part of a general idea of using transportation theory for proving sharp inequalities.

### Afternoon session 3a :: Michael Shapiro

#### Michael Shapiro (Michigan State University)

###### Finite mutation type and growth rate of cluster algebras.

In 2003, Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). Ia a series of papers with A.Felikson, P.Tumarkin and H.Thomas we classified all mutationally finite cluster algebras. Except finitely many cases, almost all mutationally finite cluster algebras are associated with triangulations of 2-dimensional surfaces (generally speaking, surfaces with orbifold points). All mutationally finite non skew-symmetric cases are obtained from skew-symmetric cases by construction of folding (notion due to A.Zelevinsky). Based on the mutational finite classification we described growth rate of cluster algebras.

### Afternoon session 3b ::

#### Tropical Geometry and Amoebae

##### (I. Itenberg and B. Shapiro)
###### Plenary speaker: Grigory Mikhalkin (Université de Geneve)

Section speakers:
1. Erwan Brugallé (Université Pierre et Marie Curie - Paris 6)
2. Mikhail Kapranov (Yale University)
3. Viatcheslav Kharlamov (Université de Strasbourg)
4. Diane Maclagan (University of Warwick)
5. Hannah Markwig (Universität des Saarlandes)
6. Mounir Nisse (Texas A&M university)
7. Jean-Jacques Risler (Université Pierre et Marie Curie - Paris 6)
8. Eugenii Shustin (Tel Aviv University)
9. August Tsikh (Siberian Federal University)
10. Oleg Viro (State University of New York at Stony Brook)

##### C. Faber and S. Merkulov
###### Plenary speaker: Gerard van der Geer (Universiteit van Amsterdam)
Section speakers:
1. Anton Alekseev (Geneva University)
2. Alexander Berglund (Stockholm University)
3. Rahul Pandharipande (ETH Zurich)
4. Dan Petersen (KTH)
5. David Rydh (KTH)
6. Sergei Shadrin (Universiteit van Amsterdam)
7. Michael Shapiro (Michigan State University)
8. Nick Shepherd-Barron (University of Cambridge)
9. Bruno Vallette (Universite de Nice Sophia Antipolis)
10. Thomas Willwacher (Harvard University)

#### Geometry in Several Complex Variables

##### M. Andersson and Bo Berndtsson
###### Plenary speaker: Jean-Pierre Demailly, (Grenoble)
Section speakers:
1. Peter Ebenfelt (UC San Diego)
2. Mattias Jonsson (Ann Arbor Michigan)
3. Jan-Erik Björk (Stockholm University)
4. Timur Sadykov (Moscow State University)
5. Håkan Samuelsson (Chalmers)
6. Ragnar Sigurdsson (University of Iceland)
7. Elizabeth Wulcan (Chalmers)
8. Xing Yang (Lund University)
9. Alain Yger (Université de Bordeaux)
10. Ahmed Zeriahi (Université de Toulouse)

#### Spectral Geometry

##### A. Laptev and P. Kurasov
###### Plenary speaker: Peter Perry (University of Kentucky)
Section speakers:
1. Carolyn Gordon (UC Berkeley)
2. Anders Karlsson (Univ. of Geneva)
3. Richard Kenyon (Brown Univ)
4. Michael Loss (Georgia Tech)
5. Rafe Mazzeo (Stanford University)
6. Olga Rossi (Ostrava & La Trobe U)
7. Johannes Sjöstrand (Paris 6)
8. Uzy Smilansky (Weizmann Institute of Science)
9. Toshikazu Sunada (Meiji University, Tokyo)