2024
1 martie 10h30 IMAR Laurenţiu Maxim On the geometry and topology of aspherical compact Kähler manifolds and related questions.
26 March, 13h30 Univ Lille Cezar Joita Tame deformations of highly singular function germs, 1
Abstract. We give analytic and algebraic conditions under which a deformation of real analytic functions with non-isolated singular locus are locally topologically trivial at the boundary.
26 March, 14h30 Univ Lille Ying Chen (Wuhan, China) Fibrations of tamely composable maps
Abstract. We study composed map germs with respect to their local fibrations. Under most general conditions, inspired by the tameness condition introduced recently, we prove the existence of singular tube fibrations, and we determine the topology of the fibres.
26 March, 15h30 Univ Lille Gabriel Monsalve (Sao Paulo, Brasil) Global index of real polynomials
Abstract. We explain two methods for expressing the global index of the gradient of a 2 variable polynomial function f: in terms of the atypical fibres of f, and in terms of the clusters of Milnor arcs at infinity. We derive upper bounds for the global index.
15 April, 14h00 Univ Lille Cezar Joita What is the image of an analytic map germ?
Abstract. The image of a holomorphic map germ is not necessarily locally open, and it is not always well-defined as a set germ. We find the structure of what becomes the image of a map germ in case the target is a surface.
16 April, 10h15 Univ Lille Cezar Joita Tame deformations of highly singular function germs, 2
Abstract. We give analytic and algebraic conditions under which a deformation of real analytic functions with non-isolated singular locus are locally topologically trivial at the boundary.
16 April, 14h00 Univ Lille Laurentiu Maxim Algebraic and topological aspects of optimization
Abstract. I will present formulae for the Chern-Mather classes of (very) affine varieties in terms of their conormal geometry, and discuss some recent applications to the computation of the algebraic complexity degree of various optimization problems. (Based on joint work with Rodriguez, Wang and Wu.)
16 April, 15h30 Univ Lille Mihai Tibar Enumerative Geometry of the Gradient
Abstract. I will discuss some recent results involving the degree of the gradient: the polar degree, the Euclidean distance degree (ED-degree), and the linear Morsification of complex polynomials. Some of these degrees enter in the computations of characteristic classes.
21 May, 11h00 IMAR Mihai Tibar Presentation of the project
Abstract. Aspects of the scientific project, and a presentation of two recent papers published by the team.
21 May, 14h00 IMAR Emil Horobet The Euclidean Distance Discriminant, part I
Abstract. In many real-world applications optimizing the distance from a given data point (for example, coming from a measurement) to a model it should fit plays a crucial role. If the model is an algebraic variety, it was shown that for a generic data point, the number of complex critical points of the distance function is constant. This number is called the Euclidean Distance (ED) Degree of the underlying variety. In concrete applications, one is interested in finding the real solutions
to this optimization problem. However, the number of real solutions is not constant for generic data. Instead, there exists (generically) a hypersurface, called the ED discriminant, which partitions the space of data points into connected regions where the number of the real critical points of the distance function is constant. This number of real critical points varies from region to region, while the ED discriminant contains all data points with a special behavior in the number of critical points. We can have different than (complex) generic numbers of critical points in three ways. One, if two of the complex conjugate critical points collide and gain a multiplicity, for plane curves this locus is called the evolute, or focal locus. Two, if one of the critical points wanders off to an unpermitted region, say the singular locus of the variety; this locus is called the singular data locus. Three, for some varieties there might exist data with more than the expected number of critical points, or equivalently infinitely many. We call the locus of such data points the infinite ED discriminant. In these talks we detail all of these subvarieties.
22 May, 10h00 IMAR Dirk Siersma (Utrecht) Geometric aspects of the square of the Euclidean distance function
Abstract. The concept of focal set of a manifold in Euclideanan space will be discussed in detail, including the connection with tight and taut embeddings. Next an outline will be given to the generalizations to complex manifold and the complex squared distance function.
22 May, 11h30 IMAR Emil Horobet The Euclidean Distance Discriminant, part II
Abstract. See above the complete abstract.
24 May, 10h30 IMAR Laurentiu Paunescu The Lipschitz type of the Geometric Directional Bundle
Abstract. In this work we investigate the behaviour of the geometric directional bundles, associated to arbitrary subsets in $\R^n$, under bi-Lipschitz homeomorphisms, and give conditions under which their bi-Lipschitz type is preserved. The most general sets we consider satisfy the sequence selection property (SSP) and, consequently, we investigate the behaviour of such sets under bi-Lipschitz homeomorphisms as well. In particular, we show that the bi-Lipschitz images of subanalytic sets generically satisfy the (SSP) property. (Joint work with S. Koike)
24 May, 11h40 IMAR Laurentiu Maxim A geometric perspective on generalized weighted Ehrhart theory
Abstract. Classical Ehrhart theory for a lattice polytope encodes the relation between the volume of the polytope and the number of lattice points the polytope contains. In this talk, I will discuss a geometric interpretation, via the (equivariant) Hirzebruch-Riemann-Roch formalism, of a generalized weighted Ehrhart theory depending on a homogeneous function on the polytope and with Laurent polynomial weights attached to each of its faces. In the special case when the weights correspond to Stanley’s g-function of the polar polytope, we recover in geometric terms a recent combinatorial formula of Beck-Gunnells-Materov. (Joint work with J. Schuermann.)
27-30 May, 9h00-12h00 IMAR Mihai Tibar Eight lectures on "Topology of complex plane curves at IMAR, București.
27 May : 9h00-12h00 Lectures 1-2
28 May : 9h00-12h00 Lectures 3-4
29 May : 9h00-12h00 Lectures 5-6
30 May : 9h00-12h00 Lectures 7-8
17 June, 14h00 IMAR Nicolas Dutertre (Angers, France) Topology of functions with non-isolated stratified critical points
Abstract. Let f : (Rn,0)-->(R,0) be a definable function germ of class C2 and let (X,0) in (Rn,0) be a germ of a closed definable set. We investigate topological invariants associated with f| X.
In particular, we give several topological formulae for the Euler characteristics of related sets.
We also relate the topology of f| X to the topology of a definable function with isolated critical point in the stratified case. Joint work with Juan Antonio Moya Pérez (Universitat de Valencia, Spain).
17 June, 15h15 IMAR Mihai Tibar Enumerative geometry of the gradient
Abstract. We will present, under a unifying viewpoint, although using different techniques, some recent results involving the degree of the gradient: on the polar degree, on the Euclidean distance degree (ED-degree), and on the linear Morsification of complex polynomials.
27 June, 11h00 IMAR Tamas Laslo (UBB-Cluj) Seifert 3-manifolds and numerical semigroups
6 September, 11h00 IMAR Francisco Braun (Univ. Federal de Sao Carlos, Brasil) Polynomial foliations of the plane
Abstract. I will address the study of foliations of R2 whose leaves are the solutions of a system of differential equations ˙x = P (x, y), ˙y = Q(x, y), where P, Q : R2 → R are polynomial functions without common (real) zeros of degree at most n. It is known that the topological classification of these foliations depend on the number of inseparable leaves and their configuration in the plane, together with a single leaf in each canonical region. I will recall this in the first part of the talk. Then I will recall that s(n), the maximal number of separatrices such a foliation can have is at most 2n. But despite it is known that s(0) = s(1) = 0 and s(2) = s(3) = 3, it is not known up to now the exact value of s(n) when n ≥ 4. In the special class of Hamiltonian systems, that is, when P (x, y) = −h_y (x, y) and Q(x, y) = h_x(x, y) where h : R^2 → R is a polynomial of degree at most n + 1, we denote the maximal number of separatrices such a foliation can have by s_H (n). (Here the leaves of the foliation are the connected components of the fibers of h.) Clearly s_H (n) ≤ s(n). Recently the lower bound of s(n), when n ≥ 4, was improved to 2n− 1. Actually, it was proved that s_H (n) ≥ 2n − 1 for any n ≥ 4. (s_H (n) = s(n) for n = 0, 1, 2, 3). That is, we now know that 2n − 1 ≤ s_H (n) ≤ s(n) ≤ 2n, but there are up to now no known examples of P and Q with degree n ≥ 4 such that the related foliation has 2n inseparable leaves. I will also show the complete clasification of foliations given by Hamiltonian systems of differential equations of degree less than or equal to 4. In particular s_H (4) = 7. So far I will quickly recall the concept of Reeb component of a foliation. Here a sharp result will be given.
9-13 September, Univ. Ovidius Constanta Research School “Topology and Algebra of Singularies and their Applications”.
25 October, 11h00 IMAR Piotr Migus (Warsaw) Topological and bi-Lipschitz invariants of two variable function germs
Abstract. We present the idea of grouping the polar curves of two-variable function germs in polar clusters. We first get a bijective correspondence between certain partitions of the set of polar quotients of two topologically equivalent function germs. Then, we will explain how this bijective correspondence may be refined in the Lipschitz category in terms of the associated gradient canyons.
11 November, 11h00 IMAR Alex Dimca (Nice, France) Free curves and Briançon-type polynomials
Abstract: We explore several Briançon-type polynomials constructed by E. Artal Bartolo, Pi. Cassou-Noguès and I. Luengo Velasco and show that the exceptional fibers of them give rise to free curves, or to nearby classes of projective plane curves. Using one of this curves, we disprove a conjecture about supersolvable curves being free.
15 November, 11h00 IMAR Alex Dimca (Nice) From Pascal's Theorem to the geometry of Ziegler's line arrangements
Abstract: Ziegler (1989) and Yuzvinsky (1993) have notices that certain arrangements of 9 lines in the projective plane behave differently if their six triple points are, or respectively are not, on a conic. We give a geometric understanding of this fact in full generality, which was previously just a result by direct computer computations on numerical examples.
18 November, 11h00 Univ. Ovidius, Constanta Alex Dimca (Nice) -- Outreach event
Title: “Mathematics: a type of language created by necessity, or a revelation of eternal harmonies?”
Abstract: I will discuss the evolution of ideas and concepts which lead to the statement and proof of the theorem by Bézout on the number of intersection points of two plane curves.