Informal Geometry Seminar

Informal Geometry Seminar (IGS), as its name says is an informal seminar for graduate students and postdocs at IST to share their ideas with each other and a good place to ask simple questions without any pressure.
Its fixed time and place are every Tuesday at 11:15- 13:00, room 3.10 of IST mathematics department. Contact: Hassan Najafi Alishah <halishah at math.ist.utl.pt>

From Feb. 2014 on, announcements will appear here in the page of Math department.

Spring 2013-14:

GIT and symplectic stability (I)

18 Feb. 2014

Title:: GIT and symplectic stability (I)

Speaker:: Alfonso Zamora

Abstract:: Geometric Invariant Theory (GIT) is a powerful tool to study quotients of algebraic varieties by the action of Lie groups, related to symplectic quotients by the Kempf-Ness theorem. From both points of view a notion of stability for the orbits of the group action plays a prominent role.
In the lectures we will give the basic notions and ideas behind GIT stability, symplectic stability and the Kempf-Ness theorem. We will pay special attention to the unstable orbits for which one can finds a "maximal way to destabilize" them. This idea can be seen from both the algebraic and the symplectic point of view and we will show this coincidence.
All the treatment will be done through three (basic but very illustrative) examples: the obtention of the projective space as a quotient, the moduli space of configurations of n points in the projective line (related to the moduli space of polygons) and the obtention of the grassmanian as a quotient.

Polygon spaces, their Gromov width and Hamiltonian toric actions

3 Feb.(Mon.),4 Feb(Tue.), 12 Feb.(Wed.) 2014

Title:: Polygon spaces, their Gromov width and Hamiltonian toric actions.

Speaker:: Alessia Mandini and Milena Pabiniak

Abstract:: In this talk we want to introduce a very interesting class of symplectic manifolds: moduli spaces of polygons in $\mathbb{R}^3$ with edges of lengths $(r_1,\dots,r_n)$. Under some genericity assumptions on lengths $r_i$, the polygon space is a symplectic manifold. In fact it is a symplectic reduction of Grassmannian manifold of 2-planes in $C^n$. Moreover polygon spaces can be equipped with a toric Hamiltonian action at least on an open dense subset. After introducing this family of manifolds we will concentrate on the spaces of 5-gons and calculate for them a symplectic invariant called Gromov width (definition will be given).

Every symplectic toric orbifold is a centered reduction of …

28 Jan. 2014

Title:: Every symplectic toric orbifold is a centered reduction of a Cartesian product of weighted projective spaces

Speaker:: Aleksandra Marinkovic

Abstract:: We prove that every symplectic toric orbifold is a centered reduction of a Cartesian product of weighted projective spaces. This generalizes the result of Abreu and Macarini that every monotone symplectic toric manifold is a centered reduction of weighted projective space. The idea of the proof is to present the labeled polytope corresponding to symplectic toric orbifold as an intersection of monotone polytopes and then to do reduction in stages. As a corollary we show that every symplectic toric orbifold contains a non-displaceable Lagrangian toric fiber and we identify this fiber.

Computations in (equivariant) cohomology

21 Jan. 2014

Title:: Computations in (equivariant) cohomology

Speaker:: Silvia Sabatini

Abstract:: Equivariant cohomology is an incredibly powerful tool to understand the topology of a manifold endowed with an action of a Lie group. Computing the equivariant cohomology ring may be hard, but in some cases computations can be carried on by using a "magic tool" called the "Localization Theorem". I will (try to) show some of these computations in some simple cases, and convince you that much more can be done in a much greater generality.

Conservative Franks lemma

14 Jan. 2014

Title:: Conservative Franks lemma

Speaker:: Hassan Najafi Alishah

Abstract:: The Franks lemma states that any perturbation of the derivative of a diffeomorphism at a finite set can be realized as the derivative of a nearby diffeomorphism in the $C^1$ topology. I will talk about conservative version of the Franks lemma and a bit about applications.

Fall 2013-14:

G2 and the rolling ball

3 Dec. 2013

Title:: G2 and the rolling ball

Speaker:: John Huerta

Abstract:: The search for simple models of the exceptional Lie groups is a long standing problem in mathematics. In this talk, we use a nonassociative algebra known as the split octonions to explain how the smallest exceptional Lie group, G2, can be thought of as the symmetry group of a 'spinorial ball' rolling on a projective plane precisely 3 times as big.

The symplectomorphism group of 4-manifolds and more

26 Nov. 2013

Speaker:: Sinan Eden

Abstract:: Gromov, among many other things, calculated the symplectomorphism group of $S^2\times S^2$. I will try to motivate why it is interesting, give the main ideas in the proof, and discuss some of the already existing generalizations.

References:: - M. Gromov - Pseudo holomorphic curves in symplectic manifolds [1985]
- M. Abreu - Topology of symplectomorphism groups of $S^2\times S^2$ [1998]
- M. Abreu, D. McDuff - Topology of symplectomorphism groups of rational ruled surfaces [2000]

Keywords:: Symplectic geometry, J-holomorphic curves, symplectomorphism group, almost complex structures.

Quantization, algebroids, groupoids and all that (II)

19 Nov. 2013

Speaker:: Giorgio Trentinaglia

Abstract:: Since the talk is supposed to be informal, my abstract will be informal too. I will talk about what's in the title. How much I will be able to say, well, this will depend on the audience. I will almost surely need more than one meeting to go reasonably deep into the topic.

Quantization, algebroids, groupoids and all that (I)

12 Nov. 2013

Speaker:: Giorgio Trentinaglia

Abstract:: Since the talk is supposed to be informal, my abstract will be informal too. I will talk about what's in the title. How much I will be able to say, well, this will depend on the audience. I will almost surely need more than one meeting to go reasonably deep into the topic.

Spring 2011-12:

Spring 2011-12
8 May 2011

Title:: Mathematics as a Natural Science

Speaker:: Sinan Eden

Abstract:: What is mathematics? Why do we think that mathematics is more rigorous that other sciences? Why is mathematics considered as a "formal" science, as opposed to a "natural" science? This talk deals with the question of axiomatization of mathematics, starting with Euclid and ending with Gödel.

References:: Davis, Philip; Hersh, Reuben; The Mathematical Experience, Boston: Birkhäuser, 1981.
Hersh, Reuben; What Is Mathematics, Really? , Oxford Univ. Press, 1997.
Kline, Morris; Why Johnny Can't Add: The Failure of the New Mathematics, St. Martin's Press, 1973.
Kline, Morris; Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1972.

24 April 2012

Title:: Topological recursion and enumeration of surfaces

Speaker:: Nicolas Orantin

Abstract:: How many different surfaces can we build by gluing together 3 squares and two triangles by their edges? What topology do they have? This is the kind of questions which one can answer by studying random matrix integrals. In this talk, I will explain how, by combinatorial arguments, one can solve such a problem of enumeration of discrete surfaces by very basic algebraic geometry. If times allows it, I will explain how the answer generalizes amazingly to the enumeration of Riemann surfaces embedded in a given toric manifold, i.e. to the computation of Gromov-Witten invariants of some toric manifolds or the computation of simple Hurwitz numbers through mirror symmetry.

References:: A review of the subject can be found in arXiv:0811.3531v1

IGS will be resumed after working seminar is finished

31 Jan. 2012

Title:: Polynomial automorphisms of $\mathbb C^n$ and the Jacobian conjecture

Speaker:: Stavros Papadakis

Abstract:: The aim of the talk is to give a gentle introduction to the subject of the title.

References:: Shestakov, Ivan P. ; Umirbaev, Ualbai U. The tame and the wild automorphisms of polynomial rings in three variables. J. Amer. Math. Soc. 17 (2004), no. 1, 197-227
van den Essen, Arno Polynomial automorphisms and the Jacobian conjecture. Progress in Mathematics, 190. Birkhäuser Verlag, Basel, 2000. xviii+329 pp.
van den Essen, Arno Polynomial automorphisms and the Jacobian conjecture. Algébre non commutative, groupes quantiques et invariants (Reims, 1995), 55-81, Sémin. Congr., 2, Soc. Math. France, Paris, 1997. Available here

24 Jan. 2012

Title:: What is an algebraic surface of general type ?

Speaker:: Xavier Roulleau

Abstract:: We will explain the words in the title and give examples. If there is enough time, we will explain the classifications of algebraic surfaces, and how to construct them.

Fall 2011-12:

Stanley's proof of the g-conjecture for simplicial convex polytopes, parts I and II

29 Nov. & 6 Dec. 2011

Title:: Stanley's proof of the g-conjecture for simplicial convex polytopes, parts I and II

Speaker:: Stavros Papadakis

Abstract:: For a simplicial complex D, the f-vector of D is the finite sequence with entries the number of vertices, 1-faces, 2 faces, ... of D. For fixed n, the g-conjecture characterizes the possible number of f-vectors for simplicial complexes D triangulating the sphere $S^n$. We will sketch Stanley's celebrated proof, using toric geometry and the hard Lefschetz theorem, of the g-conjecture for the case when D is the boundary of a simplicial convex polytope.

References:: Billera, Louis J.; Lee, Carl W., A proof of the sufficiency of McMullen's conditions for $f$-vectors of simplicial convex polytopes. J. Combin. Theory Ser. A 31 (1981), no. 3, 237-255.
Stanley, Richard P., The number of faces of a simplicial convex polytope. Adv. in Math. 35 (1980), no. 3, 236-238.
Fulton, William, Introduction to toric varieties. Annals of Mathematics Studies, 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton, NJ, 1993. xii+157 pp. ISBN: 0-691-00049-2

Existence of KAM tori for degenerate Hamiltonian Systems

22 Nov. 2011

Title:: Existence of KAM tori for degenerate Hamiltonian Systems

Speaker:: Hassan Najafi Alishah

Abstract:: I will start with a brief reminder of the talk I gave last semester, then try to explain the beautiful technic that been used to extend the results of non-degenerate systems to degenerate ones. I will state some facts from Riemannian geometry used in this technic and talk a bit about Whitney differentiability.

References:: Chong-Qing Cheng, Yi-Sui Sun, Existence of KAM tori in Degenerate Hamiltonian Systems. J. Differential Equations, 114-1 (1994), pp. 288-335
H. Whitney, Analytical extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36(1934), 63-89

***I can provide hard copies of the above papers upon request***

Natural (and less natural) deduction systems for classical (and less classical) logics

15 Nov. 2011

Title:: Natural (and less natural) deduction systems for classical (and less classical) logics

Speaker:: Marco Volpe

Abstract:: In this introductory talk, we will describe a calculus for logical reasoning called "natural deduction". We will present it in the case of classical logic and have a look at possible variants in order to capture some non-classical logics.

Stability of the Steiner symmetrization of convex sets

25 OCt. & 8 Nov. 2011

Title:: Stability of the Steiner symmetrization of convex sets

Speaker:: Filippo Cagnetti

Abstract:: The goal of the talk is the study of the isoperimetric inequality for the Steiner symmetrization in any codimension. First, we give a characterization of the cases of equality. Then, we will prove a quantitative version of the inequality, in the case of convex sets. This is a joint work with Marco Barchiesi and Nicola Fusco.

Open books and pseudo-Anosov maps II

18 Oct. 2011
Again, due to the informality of the seminar we will have the rest of Sinan's unfinished talk.

Open books and pseudo-Anosov maps I

11 Oct. 2011

Title:: Open books and pseudo-Anosov maps

Speaker:: Sinan Eden

Abstract:: One of the natural ways to study the topology of 3-manifolds is to consider its open book decomposition (which is given by (S,h) where S is any compact oriented surface with boundary and h is a diffeomorphism of S that fixes the boundary point-wise). So, a topological 3-dimensional question is turned into a geometrical 2-dimensional question. The aim of my talk will be to investigate one particular relation between the two. Namely, I will try to convince you that after a sequence of positive stabilizations, we can get a pseudo-Anosov diffeomorphism.

References:: Vincent Colin, Ko Honda - Stabilizing the monodromy of an open book decomposition
Ko Honda, William H. Kazez, Gordana Matic -Right-veering diffeomorphisms of compact surfaces with boundary I
Albert Fathi, François Laudenbach, Valentin Poénaru - Thurston's Work On Surfaces
John B. Etnyre - Lectures on open book decompositions and contact structures

Introduction to the McKay Correspondence II

4 Oct. 2011 Due to the informality of talks:-), we will continue with Stavros's unfinished talk.
This is a video which gives you an intuition about blowup that Stavros was talking about last week.

Introduction to the Mckey correspondenc I

27 Sept. 2011

Title:: Introduction to the McKay Correspondence

Speaker:: Stavros Papadakis

Abstract:: The talk will be a gentle introduction to the McKay correspondence. Let $\mathbb C$ be the field of complex numbers. In its simplest form, McKay correspondence relates for a finite subgroup $G\subset SL(2,\mathbb C)$ the geometry of the quotient space $\mathbb C^2 / G$ with the representation theory of $G$.

References:: Dolgachev, I., McKay correspondence, Lecture note is available here.
Reid, M., La correspondance de McKay. Sêminaire Bourbaki, Vol. 1999/2000. Astêrisque No. 276 (2002), 53-72. A copy is available here

Closed forms on fiber bundles

20 Sept. 2011

Title:: Closed forms on fiber bundles

Speaker:: Olivier Brahic

Abstract:: Given a 3-form defined on the total space of a fibration, I will explain how to interpret geometrically the equations for it to be closed in geometric terms. Namely we have Courant algebroid structures defined fiberwise, and preserved by a higher analogue of a connection.

Hopf Algebras and Hopf-Galois Extensions

13 Sept. 2011

Title:: Hopf Algebras and Hopf-Galois Extensions

Speaker:: Marcin Szamotulski

Abstract:: I will deliver an introduction to Hopf algebras, Hopf-Galois extensions, and try to justify the notions. I will present some basic examples, and a way to present classical Galois theory in an 'extravagant' way. If time permits I'll try to present some of my results. Quite soon I will give a more advanced Version of this talk at UL.

Spring 2010-11:

3 May 2011

Title:: Gradification of Everything, Supermetry and Superbock

Speaker:: Sebastian Guttenberg

12 April 2011

Title:: KAM theory (Kolmogorov-Arnold-Moser)

Speaker:: Hassan Najafi Alishah

5 April 2011

Title:: On the moduli spaces of polygons and hyperpolygon

Speaker:: Alessia Mandini

29 March 2011

Title:: Introduction to the Hamiltonian diffeomorphism group (with particular focus on Hofer's distance)

Speaker:: Remi Leclercq

References:: Polterovich. The geometry of the group of symplectic diffeomorphisms (Birkhäuser 2001).

15 March 2011

Title:: Automatic Theorem Proving for Euclidean Geometry

Speaker:: Stavros Papadakis

References:: Cox, Little and O'Shea. Ideals, varieties, and algorithms (Springer 2007).