The normal starting time of this seminar is 16.30 on Monday.
2017, Monday, October 2, 16:30, Battelle, Dmitry Novikov (Weizmann Institute)
Complex cellular parameterization
(joint work with Gal Binyamini)
We introduce the notion of a complex cell, a complex analog of the cell decompositions used in real algebraic and analytic geometry. Complex cells defined using holomorphic data admit a natural notion of analytic continuation called $\delta$-extension, which gives rise to a rich hyperbolic geometric structure absent in the real case. We use this structure to prove that complex cellular decompositions share some interesting features with the classical constructions in the theory of resolution of singularities. Restriction of a complex cellular decomposition to the reals recovers the preparation theorem for subanalytic functions, and can be viewed as an analytic continuation thereof.
A key difference in comparison to the classical resolution of singularities is that the cellular decompositions are intrinsically uniform over (sub)analytic families. We deduce a subanalytic version of the Yomdin-Gromov theorem where $C^k$-smooth maps are replaced by mild maps.
2016, Friday, June 23, 11:00, Battelle, Ernesto Lupercio (CINVESTAV)
Quantum toric varieties
I will describe the theory of quantum toric varieties that generalizes usual toric geometry. Joint with Meersseman, Katzarkov and Verjovsky.
2016, Thursday, June 22, 11:30, Battelle, Conan Leung (CUHK)
Informal introduction to G_2-manifolds III
2016, Wednesday, June 21, 11:30, Battelle, Conan Leung (CUHK)
Informal introduction to G_2-manifolds II
2016, Monday, June 19, 15:00, Battelle, Conan Leung (CUHK)
Informal introduction to G_2-manifolds I
Villa Battelle, May 2, 14:00-15:00; May 3 14:15-15:15; May 5, 14:30-15:30, Aaron Bertram (Utah)
Minicourse: “Moduli Spaces of Complexes in Algebraic Geometry ”
The ideal of the twisted cubic in projective three-space is completely described by a 2×3 matrix of linear forms in four variables. The space of such matrices (modulo the actions of GL(2) and GL(3)) is a smooth, projective variety compactifying the space of twisted cubics. But the objects parametrized by the points at the boundary of this moduli space are not ideals of curves. They are complexes of line bundles that are stable with respect to a “stability condition on the derived category.” What does this mean? Can this be used to systematically find nice models for moduli and relate them to moduli spaces of coherent sheaves?
Day 1) Introduction to Stability Conditions. Ordinary stability of vector bundles on a Riemann Surface relies on two invariants: the rank and degree (first chern class). A stability condition on the derived category of coherent sheaves on a complex manifold relies on a generalized rank and degree, and also on an exotic t-structure on the derived category, with an abelian category of complexes at its heart. On an algebraic surface, there are stability conditions whose underlying heart can be described by a tilting construction. However, finding a single stability condition on a projective Calabi-Yau threefold (e.g. the quintic in P4) remains open.
Day 2) Models of the Hilbert Schemes of Points on a Surface. As the stability condition varies, the moduli spaces of stable objects (with respect to the stability condition) undergo a series of birational transformations. The particular example of the Hilbert scheme of ideal sheaves on an algebraic surface has been studied for various classes of surfaces. We will survey some results.
Day 3) The Euler Stability Condition on Projective Space. An interesting stability condition on P^n has the Euler characteristic playing the role of the rank. We will use this stability condition to study stratifications of the spaces of symmetric tensors, generalizing the secant varieties to the Veronese embeddings of P^n. This is joint work with Brooke Ullery.
Villa Battelle, Monday, Apr 3, 16:30-17:30, Lionel Lang (Uppsala University)
The vanishing cycles of curves in toric surfaces : the spin case
If the interior polygon of a lattice polygon $\Delta$ is divisible by 2, any generic curve $C$ of the linear system associated to $\Delta$ admits a spin structure $q$. If a loop in $C$ is a vanishing cycle, then the Dehn twist along the loop has to preserve $q$. As a consequence, the image of the monodromy of the linear system is a subgroup of the mapping class group $MCG(C,q)$ that preserves $q$. The main goal of this talk is to compare the image of the monodromy with $MCG(C,q)$. To this aim, we will show on one side that $MCG(C,q)$ admits a very explicit set of generators. On the other, we will construct elements of the monodromy by tropical means. The conclusion will be that the image of the monodromy is the full group $MCG(C,q)$ if and only if the interior polygon admits no other divisors than 2. (joint with R. Crétois)
Villa Battelle, Wednesday, Mar 8, 12:00, Maksim Karev (PDMI)
Monotone Hurwitz Numbers
Usual Hurwitz numbers count the number of covers over CP^1 with a fixed ramification profile over point \infty and simply ramified over a specified set of points. They also can be treated as a weighted count of factorizations in the symmetric group. It is known, that Hurwitz numbers can be calculated via intersection indices on the moduli spaces of complex curves by so-called ELSV-formula.
In my talk, I will discuss monotone Hurwitz numbers, which also arise as factorizations count with restrictions. It turns out, that they also can be related to the intersection indices on the moduli spaces of complex curves. I will give a definition of monotone Hurwitz numbers, and try to explain the origin of the monotone ELSV. If time permits, I will speak about the further development of the subject.
The talk is based on the joint work with Norman Do (Monash University).
Villa Battelle, Tuesday, Feb 21, 15:30, Yang-Hui He (London, Nankai and Oxford)
Calabi-Yau Varieties: From Quiver Representations to Dessins d'Enfants
We discuss how bipartite graphs on Riemann surfaces encapture a wealth of information about the physics and the mathematics of gauge theories. The correspondence between the gauge theory, the underlying algebraic geometry of its space of vacua as a quiver variety, the combinatorics of dimers and toric varieties, as well as the number theory of dessin d'enfants becomes particularly intricate under this light.
Joint session of “Fables géométriques” and “Groupes de Lie et espaces des modules” seminars.
Villa Battelle, Monday, Feb 20, 16:30, Yang-Hui He (London, Nankai and Oxford)
Sporadic and Exceptional
There tends to be exceptional structures in classifications: in geometry, there are the Platonic solids; in algebra, there are the exceptional Lie algebras; in group theory, there are the sporadic groups, to name but a few. Could these exceptional structures be related in some way? A champion for such Correspondences is Prof. John McKay. We take a casual promenade in this land of exceptionology, reviewing some classic results and presenting some new ones based on joint work with Prof. McKay.
Special lecture for “Geometry, Topology and Physics” masterclass students.
Villa Battelle, Friday, December 9, 14:30-15:30, Ozgur Ceyhan (Luxembourg)
Backpropagation, its geometry and tropicalisation
The algorithms that make current artificial neural networks successes possible are decades old. They became applicable only recently as these algorithms demand huge computational power. Any technique which reduces the needs for computation have a potential to make great impact. In this talk, I am going to discuss the basics of backpropagation techniques and tropicalisation of the problem that promises to reduce the time complexity and accelerate computations.
2016,Monday, November 7, 16:30, Battelle, Vladimir Fock.
Separation of variables in cluster integrable systems
Cluster integrable systems can be viewed from five rather different points of view. 1. As a double Bruhat cell of an affine Lie -Poisson group; 2. As a space of pairs (planar algebraic curve, line bundle on it); 3. Space of Abelian connection on a bipartite graph on a torus; 4. Hilbert scheme of points on algebraic torus. 5. Collection of flags in an infinite space invariant under the action of two commuting operators. We will see the relation between all these descriptions and discuss its quantization and possible generalizations.
2016,Friday, Nov 4, 14:30-15:15 part I, 15:30-16:15 part II, Johannes Walcher (Heidelberg).
Ideas of D-branes
Abstract: I will give an introduction to D-branes from the point of view of their origin in the physics of string theory. I will discuss both world-sheet and space-time aspects.
2016, Monday, 23 mai, 16.30, Battelle, Frédéric Bihan.
Une généralisation de la règle de Descartes pour les systèmes polynomiaux dont le support est un circuit
Résumé : La règle de Descartes borne le nombre de racines positives d'un polynôme réel en une variable par le nombre de changements de signe consécutifs de ses coordonnées dans la base monomiale (ordonnée suivant les puissances croissantes). La borne obtenue est optimale et généraliser la règle de Descartes aux systèmes polynomiaux en plusieurs variables est un problème très difficile. Dans un travail avec Alicia Dickenstein (Université de Buenos Aires), nous avons obtenu une généralisation partielle de la règle de Descartes en plusieurs variables. Notre règle s'applique aux systèmes polynomiaux en un nombre arbitraire n de variables dont le support consiste en n+2 monômes quelconques. Comme pour la règle de Descartes usuelle, notre borne est optimale et s'exprime comme un nombre de changement de signes d'une suite de nombres obtenus en considérant les mineurs maximaux de la matrice des coefficients ainsi que de celle des exposants du système.
(in English)Descartes' Rule of Signs for Polynomial Systems Supported on Circuits
Descartes’ rule of signs bounds the number of positive roots of an univariate polynomial by the number of sign changes between consecutive coefficients. In particular, this produces a sharp bound depending on the number of monomials. Generalizing Descartes’ rule of signs or the corresponding sharp bound to the multivariable case is a challenging problem. In this talk, I will present a generalization of Descartes’ rule of signs for the number of positive solutions of any system of n real polynomial equations in n variables with at most n+2 monomials. This is a joint work with Alicia Dickenstein (Buenos Aires University).
2016, Monday, 9 mai, Battelle, Eugenii Shustin, 18.30-19.15
On refined tropical invariants of toric surfaces.
We discuss two examples of refined count of plane tropical curves. One of them is the refined broccoli invariant. It was introduce by Goettsche and Schroeter for genus zero case, and it turns into some descendant invariant or the broccoli invariant according as the parameter takes value 1 or -1. A possible extension of broccoli invariant to positive genera appeared to be rather problematic. However, the refined version turns to be easier to treat. Jointly with F. Schroeter, we have defined a refined broccoli invariant, counting elliptic tropical curves. This can be done for higher genera as well (work in progress). Another example (joint work with L. Blechman) is the refined descendant tropical invariant (involving arbitrary powers of psi-classes). We discuss also the most interesting related question: What is the complex and real enumerative meaning of these invariants?
2016, Monday, 4 april, 16.30, Battelle.
Lionel Lang (Uppsala)
The vanishing cycles of curves in toric surfaces (joint work with Rémi Crétois)
In [Do], Donaldson addressed the following : Do all Lagrangian spheres in a complex projective manifold arise from the vanishing cycles of a deformation to singular varieties? The answer might depend on the choice of the moduli space in which we are allowed to deform our manifold. Already for curves, it leads to interesting questions. In the Deligne-Mumford moduli space M_g, any loop inside a smooth curve can be contracted along a deformation towards a nodal (stable) curve, provided that the genus g>1. What happens if one restricts to a chosen linear system on a toric surface? Degree d curves in the projective plane, for instance. In the latter, two obstructions occur: the loop should not be separating for d>2 (Bezout), the Dehn twist along the loop should preserve a certain spin structure on the curve for d odd (see [Beau]). In the latter, Beauville proves (in particular) that any non-obstructed loop is homologous to a vanishing cycle. In this talk, we suggest a tropical proof of Beauville's result as well as an extension to any (big enough) linear systems on any smooth toric surface. This problem is directly related to the monodromy group given by the complement of the discriminant in the considered linear system. The proof will involve simple Harnack curves, introduced by Mikhalkin, and monodromy given by partial tropical compactifications of the linear system. If time permits, we will also discuss this problem at the isotopic level, problem that is still open.
[Beau] : Le groupe de monodromie des familles universelles d'hypersurfaces et d'intersections complètes. A. Beauville, 1986. [Do] : Polynomials, vanishing cycles and Floer homology. S.K. Donaldson, 2000.
2016, Monday, 21 mars, 16.30, Battelle.
Boris Shapiro (Stockholm)
On the Waring problem for polynomial rings
We discuss a natural analog of the classical Waring problem for $C[x_1,…,x_n]$. Namely, we show that a general form p from $C[x_1,…,x_n]$ of degree kd where k>1 can be represented as a sum of at most $k^n$ k-th powers of forms of degree d. Noticeably, $k^n$ coincides with the number obtained by naive dimension count if d is sufficiently large.
2016, Friday, March 18, 14.15, villa Battelle.
Sergey Galkin (Moscow)
Gamma conjectures and mirror symmetry
I will speak about an exotic integral structure in cohomology of Fano manifolds that conjecturally can be expressed in terms of Euler's gamma-function, how one can observe it by computing asymptotics of a quantum differential equation, and how one can prove the conjectures using mirror symmetry. This is a joint work with Vasily Golyshev and Hiroshi Iritani (1404.6407, 1508.00719).
2016, Thursday, March 17 Colloquium, villa Battelle
Vassily Golyshev (Moscow), 16:15
Around the gamma conjectures.
Abstract: We will state the gamma conjectures for Fano manifolds and explain how quantum cohomology makes it possible to enhance the classical Riemann-Roch-Hirzebruch theorem by relating the curve count on a variety to its characteristic classes. We will indicate how the gamma conjectures are proved in the known cases.
2016, Monday, 14 mars, 16.30, Battelle.
E. Abakoumov (Paris-Est)
Growth of proper holomorphic maps and tropical power series
How fast a proper holomorphic map, say, from C to C^n can grow? It turns out that the tropical power series appear naturally in answering this question, as well as in some related approximation problems on the complex plane. The talk is based on joint work with E. Dubtsov.
2015, Tuesday, 8 December, 14.30, Battelle. (joint with Séminaire "Groupes de Lie et espaces des modules”)
Bernd Sturmfels (UC Berkeley)
Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Another special class, including Gaussian graphical models, are inverses of symmetric matrices satisfying linear constraints. We present a general theory of exponential varieties, with focus on those defined by hyperbolic polynomials. This is joint work with Mateusz Michalek, Caroline Uhler, and Piotr Zwiernik.
2015, Tuesday, December 8, 11:15 -- 12:15, Battelle.
Renzo Cavalieri (Colorado State)
Tropical geometry: a graphical interface for the GW/Hurwitz correspondence.
In their study of the Gromov-Witten theory of curves [OP], Okounkov and Pandharipande used the degeneration formula to express stationary descendant invariants of curves in terms of Hurwitz numbers and one point descendant relative invariants. Then they use operator formalism to organize the combinatorics of the degeneration formula, and the one point invariants into completed cycles. In joint work with Paul Johnson, Hannah Markwig and Dhruv Ranganathan, we revisit their formalism and show that the Feynmann diagrams that are secretly behind the scenes in [OP] are in fact tropical curves. This yields some mild refinements of the Gromov-Witten/Hurwitz correspondence of [OP]. Time permitting we will describe how a generalization of these tecniques should lead to unveiling a similar structure in the stationary/descendant GW theory of sliceable surfaces.
2015, Monday, 7 December, 16.15, Villa Battelle
Israel Vainsencher (Universidade Federal de Minas Gerais, Brasil)
A twisted cubic curve in 3-space is known to define a (non-integrable) distribution of planes. The planes of the distribution osculate the original twc. We show how to define virtual numbers N_d which enumerate the rational curves of degree d which are tangent to that distribution and further meet 2d+1 general lines. (Based on Eden Amorim thesis)
The next lecture of the course
“Imaginary time in Kaehler geometry, quantization and tropical amoebas” by José Mourão
will be on Monday 9 November, 17.00 Battelle.
2015, October 27, 15.15 and October 29, 16.15, and November 2, 17.00, Villa Battelle
(Minicourse) Imaginary time in Kaehler geometry, quantization and tropical amoebas.
José Mourão, Mathematics Department, Instituto Superior Tecnico, Portugal.
For a compact Kaehler manifold $M$ and a function $H$ on $M$ we give a simple definition of the continuation of the flow defined by $H$ to complex time, $\tau$, using the Groebner theory of Lie series. The resulting complexified (or complex time) symplectomorphisms are diffeomorphisms for some $|\tau|< R_H$. For larger values of $|\tau|$ they may correspond e.g. to the collapse of $M$ to a totally real submanifold. Simple examples will be discussed.
Kahler geometry applications: Imaginary time symplectomorphisms correspond to Mabuchi geodesics in the infinite dimensional space of Kaehler metrics with fixed cohomology class. We get thus a explicit way of constructing Mabuchi geodesics from Hamiltonian flows.
Quantum theory applications: By lifting the imaginary time symplectomorphisms to the quantum bundle we get generalized coerent state transforms and are able to study the unitary equivalence of quantizations corresponding to nonequivalent polarizations.
Tropical geometry applications: For toric varieties the toric geodesics of the Mabuchi metric are straight lines in the space of Guillemin-Abreu symplectic potentials. Taking a strictly convex function $H$ (as a function on the moment polytope) one has that, for large geodesic times s, there is a simple relation between the moment map $\mu_s$ and the $Log_t$ map of amoeba theory ($t=e^s$) . This relation further simplifies if one takes as $H$ the full symplectic potential, which is continuous but not smooth on $M$ and corresponds to a geodesic of Kaehler metrics with cone angle singularities. The tropical limit corresponds thus, in this setting, to the infinite geodesic time limit corresponding to convex hamiltonians.
Lecture 1 (introduction and different definitions of complex time evolution): http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L1.pdf
Lecture 2: Kahler tropicalization of C^*: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L2.pdf
Lecture 3: Kahler tropicalization of C and (strange) actions of G_C on Kahler structures: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L3.pdf
Lecture 4: C^infty Kahler tropicalization of toric varieties and of hypersurfaces in toric varieties: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L4.pdf
Lecture 5: C^0 Kahler tropicalization of toric varieties and of hypersurfaces in toric varieties: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L5.pdf
2015, October 27, Tuesday, 15.15, Villa Battelle ( together with Séminaire “Groupes de Lie et espaces des modules”)
Imaginary time in Kaehler geometry, quantization and tropical amoebas.
José Manuel Cidade Mourão, Mathematics Department, Instituto Superior Tecnico, Portugal.
For a compact Kaehler manifold $M$ and a function $H$ on $M$ we define a continuation of the Hamiltonian flow of $H$ to complex time $\tau$. The resulting complexified (or complex time) symplectomorphisms are diffeomorphisms for some $|\tau|< R_H$. For larger values of $|\tau|$ they may correspond e.g. to the collapse of $M$ to a totally real submanifold. We'll discuss some simple examples and applications to Kaehler geometry, quantization and tropical geometry. This talk is the first lecture of a mini-course to be given during October-November 2015.
2015, October 5, Monday, 16.20, Villa Battelle
Tropicalization of Poisson-Lie groups
Anton Alexeev (UniGe)
In the first part of the talk, we recall the notion of Poisson-Lie groups and cluster coordinates for some simple examples.
In the second part, we use the notion of tropicalization to construct completely integrable systems, and for the Poisson-Lie group SU(n)^* match it with the Gelfand-Zeiltin integrable system.
The talk is based on joint works with I. Davydenkova, M. Podkopaeva and A. Szenes.
2015, September 28, Monday, 16.15, Villa Battelle
What is moonshine?
Sergey Galkin (HSE, Moscow)
I will describe a few instances of geometric moonshines: surprising appearance of modular forms and sporadic groups as the answers to seemingly unrelated geometric and topological questions.
2015, 21 September, Monday, 16.15, Villa Battelle.
Cohomology of superforms on polyhedral complexes and Poincare duality for tropical manifolds
Superforms introduced by Lagerberg are bigraded differential forms on $\mathbb R^n$ which can be restricted to polyhedral complexes. We extend these forms to $\mathbb T^n = [-\infty, \infty)^n$ and show that their de Rham cohomology is equivalent to tropical $(p, q)$ cohomology Furthermore, we establish Poincaré duality for cohomology of tropical manifolds. As in the classical theory, the Poincaré pairing can be formulated in terms of integration of superforms.
old page of the seminar http://www.unige.ch/math/folks/langl/fables/