We study the outer Lipschitz Geometry of a pair of two normally embedded Holder Triangles. We present an invariant and discuss some related questions.

Let X be an affine complex algebraic variety. The "seminormalization of X" is an algebraic variety X^+ obtained by gluing together the points in the fibers of the normalization morphism. One of the interest of the seminormalization comes from the fact that it has nice singularities in codimension 1 while being linked to X by a finite and birational homeomorphism. The main result of this talk is that one can identify the polynomial functions on X^+(C) with the rational functions of X which are continuous for the euclidean topology on all X(C). We will give some characterizations of this type of functions and some explicit constructions of seminormalizations.

Consider map-germs (k^n,o)-> (k^p,o) up to the groups of right/left-right/contact equivalence. The group orbits are complicated and are traditionally studied via their tangent space. This transition is classically done by vector fields integration, thus binding the theory to the real/complex case.

I will present the new approach to this subject. One studies the maps of germs of Noetherian schemes, in any characteristic. The corresponding groups of equivalence admit `good' tangent spaces. The submodules of the tangent spaces lead to submodules of the group orbits. This extends (and sometimes strengthens) classical results on 'determinacy vs infinitesimal determinacy'.

Consider map-germs (k^n,o)-> (k^p,o) up to the groups of right/left-right/contact equivalence. The group orbits are complicated and are traditionally studied via their tangent space. This transition is classically done by vector fields integration, thus binding the theory to the real/complex case.

I will present the new approach to this subject. One studies the maps of germs of Noetherian schemes, in any characteristic. The corresponding groups of equivalence admit `good' tangent spaces. The submodules of the tangent spaces lead to submodules of the group orbits. This extends (and sometimes strengthens) classical results on 'determinacy vs infinitesimal determinacy'.

In this talk, I will introduce the conjecture of Igusa on exponential sums modulo p^m and some recent results on this conjecture. This conjecture relates to Igusa’s idea on a new local-global principle for homogeneous polynomials of higher degree. In particular, the recently joint work with Raf Cluckers and Mircea Mustata on the case of non-rational singularities will be mentioned. On the other hand, I also introduce some further results in case of rational singularities.

Given two morphisms f and g from algebraic varieties X and Y to an algebraic group G, we define their convolution to be the morphism f∗g from X×Y to G by f∗g(x,y):=f(x)g(y). Similarly to the smoothing effect of the convolution operation in analysis, this operation yields morphisms with improved singularity properties.

Given a word w in a free group F_r on a set of r elements, and an algebraic group G, one can associate a word map w:G^r-->G (e.g. the commutator map (x,y)--->[x,y]). We apply the above philosophy and show that word maps on semisimple Lie groups and Lie algebras have very nice singularity properties after sufficiently many self-convolutions, with bounds depending only on the complexity of the word.

The singularity properties we consider are intimately connected to the point count of schemes over finite rings of the form Z/p^kZ (as explained in Yotam Hendel's talk). We utilize this connection to provide applications in group theory, namely to the study of random walks on compact p-adic groups induced by these word maps.

While this talk is connected to Yotam Hendel's talk, it will be self-contained and no prior knowledge is assumed.

Based on a joint work with Yotam Hendel https://arxiv.org/abs/1912.12556.

Let f:X->Y be a morphism between smooth algebraic varieties defined over the integers.

We show its fibers satisfy an extension of the Lang-Weil bounds with respect to finite rings of the form Z/p^kZ uniformly in p, k and in the base point y if and only if f is flat and its fibers have rational singularities, a property abbreviated as (FRS).

This characterization of (FRS) morphisms serves as a joint refinement of two results of Aizenbud and Avni; namely a similar characterization in the case of a single variety, and a characterization of (FRS) morphisms which is non-uniform in the prime p.

Aizenbud and Avni's argument in the case of a variety breaks in the relative case due to bad behaviour of resolution of singularities in families with respect to taking points over Z and Z/p^kZ. To bypass this, we prove a key model theoretic statement on a certain satisfactory class of positive functions (formally non-negative motivic functions), which allows us to efficiently approximate suprema of such functions.

Based on arXiv:2103.00282, joint with Raf Cluckers and Itay Glazer.

I will present a new criterion to test Whitney equisingularity and Thom's (a_f) condition for certain families of (possibly non-isolated) hypersurface singularities that "behave well" with respect to their Newton diagrams. As a corollary, I will show that in such families all members have isomorphic Milnor fibrations. This is a joint work with Mutsuo Oka.

The spectrum of an isolated hypersurface singularity is an important discrete invariant formed by $\mu$ rational numbers, where $\mu$ is the Milnor number of the hypersurface singularity. Despite knowing the interval where these rational numbers live, knowing their distribution throughout that interval is a challenge. In 1981, K. Saito asked if the asymptotic distribution of the spectral numbers of an isolated hypersurface singularity "tends" to the asymptotic distribution of the spectral values of an isolated quasihomogeneous singularity.

In the first part of this talk, I will recall Saito's approach to the distribution of the spectral numbers and its relation with other conjectures in singularity theory, such as Durfee conjecture. In the second part, we will move to the irreducible plane curve case in order to compare the limit distribution with the real distribution of spectral values; as a consequence, we find that the log canonical threshold is strictly bounded below by the doubled inverse of the Milnor number. Finally, we will move to Newton non-degenerate singularities and we will show that in this case we can establish Saito's limit distribution in a natural way.

This is a joint work with M. Schulze.

We will present our new result in collaboration with Lorenzo Fantini, András Némethi and Anne Pichon concerning the program of ``polar explorations`, the quest to determine the generic polar variety of a singular surface germ. This program is related to the general problem of relating resolution of singularities via point blowing-ups and Nash transform, raised by Lê Dung Tráng in 2000.

More precisely, we prove that the topological type of a normal surface singularity $(X,0)$ provides finite bounds for the multiplicity and polar multiplicity of $(X,0)$, as well as for the combinatorics of the families of generic hyperplane sections and of polar curves of the generic plane projections of $(X,0)$.

The talk will focus on our use of the sheaf of Kahler differentials of $(X,0)$ and in a key ingredient in our proof: we show the existence of a topological bound of the growth of the Mather discrepancies of $(X,0)$. This result allows us to bound the number of point blowups necessary to achieve factorization of any resolution of $(X,0)$ through its Nash transform.

In 1979 O. Zariski proposed a general theory of equisingularity for algebraic or algebroid hypersurfaces over an algebraically closed field of characteristic zero. This theory is based on the notion of dimensionality type that is defined recursively by considering the discriminants loci of successive "generic" corank 1 projections. The points of dimensionality type 0 are regular points and the singularities of dimensionality type 1, are generic singular points in codimension 1. Zariski proved that the latter ones are isomorphic to the equisingular families of plane curve singularities.

In this talk we give a similar characterisation for singularities of dimensionality type two, i.e. for generic singularities in codimension two. We show that they are isomorphic to equisingular families of surface singularities, with the equisingularity type determined by the discriminant loci of their “generic” corank 1 projection. Moreover, as we show, in this case the generic linear projections are generic (this is still open for dimensionality type greater than 2). Over the field of complex numbers, in the algebraic or analytic case, we show that such families are bi-Lipschitz trivial, by constructing an explicit Lipschitz stratification.(Based on joint papers with L. Paunescu.)

In this talk we will explain an intrinsic characterization of the geometric monodromy group of an isolated plane curve singularity as the stabilizer in the mapping class group of the Milnor fiber of the relative isotopy class of a canonical vector field. We will also talk about two interesting consequences of this result: an easy and effective criterion to detect if a simple closed curve in the Milnor fiber is a geometric vanishing cycle or not; and the non-injectivity of the natural representation of the versal unfolding of the singularity. This is joint work with Nick Salter.

A stratification of an algebraic set X ⊂ ℂ^n (or ℝ^n) is supposed to capture "how singular" the different points of X are: If x ∈ X lies in a stratum S, then there should exist a neighbourhood of x on which X is roughly translation invariant along S. This intuitive idea can be made precise if one replaces ℂ (or ℝ) by a bigger field K containing infinitesimal elements. Using this approach, we will construct an entirely canonical stratification of X. A motivation for this was to get a better control of the local Poincaré series associated to singularities of X. We will see how the canonical stratification indeed provides some control (though there are also questions that are still open). This is work in progress with David Bradley-Williams.

It is well-known that images of closed sets under continuous mappings are not necessarily closed. Motivated by a question of John N. Mather on properties of generic projection and various problems in analysis and optimization, we consider the question if preserving closedness is a generic property of linear mappings and continuous semi-algebraic mappings. Namely, we study when the image of a closed convex set under a linear mapping or the image of a closed semi-algebraic set under a continuous semi-algebraic mapping is closed. Moreover, the stability of closedness under small linear perturbations will be also investigated.

The talk is devoted to real plane algebraic curves with finitely many real points. We study the following question: what is the maximal possible number of real points of such a curve provided that it has given (even) degree and given geometric genus? This question is related to the first part of Hilbert’s 16-th problem (topology of real algebraic varieties) and to Hilbert’s 17-th problem (more precisely, positivity of real polynomials vs. their representations as sums of squares). We obtain a complete answer to the above question in the case where the degree is sufficiently large with respect to the genus, and prove certain lower and upper bounds for the number in question in the general case. This is a joint work with E. Brugallé, A. Degtyarev and F. Mangolte.

The decomposition of a set into "cylinders" in one of the fundamental tools of semi-algebraic geometry (as well as subanalytic geometry and o-minimal geometry). Defined by means of intervals, these cylinders are an essentially real-geometric construct.

In a recent paper with Novikov we introduce a notion of "complex cells", that form a complexification of real cylinders. It turns out that such complex cells admit a rich hyperbolic geometry, which is not directly visible in their real counterparts. I will sketch some of this theory, and how it can be used to prove some new results in real geometry (for instance a sharpening of the Yomdin-Gromov lemma).

I will discuss an approach to proving the conjecture that a normal rigid surface is smooth. The approach is based on the notion of deficient conormal singularities introduced by the speaker.

We study outer Lipschitz geometry of surface germs definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic). By the finiteness theorems of Mostowski, Parusinski and Valette any definable family has finitely many outer Lipschitz equivalence classes. Our goal is classification of definable surface germs with respect to the outer Lipschitz equivalence.The inner Lipschitz of definable surface germs was described by Birbrair. The outer Lipschitz geometry is much more complicated. Using the $K$-equivalence classification of Lipschitz functions (``pizza decomposition'') of Birbrair \emph{et al.} and the theory of abnormal surface germs (``snakes'') by Gabrielov and Sousa, we obtain a decomposition of a surface germ into normally embedded H\"older triangles, unique up to outer Lipschitz equivalence. This triangulation, with some additional data (``pizza toppings'') is a complete discrete invariant of an outer Lipschitz equivalence class of surface germs.

In the works [1] and [2] we develop a new metric algebraic topology, called the Moderately Discontinuous Homology and Homotopy in the context of subanalytic germs in R^n (with a supplementary metric structure) that satisfies the analogues of the usual theorems in Algebraic Topology: long exact sequences, relative case, Mayer Vietoris, Seifert van Kampen for special coverings... This theory captures Lipschitz information, or in other words, quasi isometric invariants. The typical examples are germs with the inner metric (length metric induced by the euclidean metric) and with the outer metric (the restriction of the euclidean metric). A subanalytic germ is topologically a cone over its link and the moderately discontinuous theory captures the different speeds, with respect to the distance to the origin, in which the topology of the link collapses towards the origin. In this talk, I will present the most important concepts in the theory and some results or applications that we got until the present.

[1] (with J. Fernández de Bobadilla, S. Heinze, E. Sampaio) Moderately discontinuous homology.

To appear in Communications on Pure and Applied Mathematics. Available in arXiv: 1910.12552 or in https://arxiv.org/pdf/1910.12552.pdf

[2] (with J. Fernández de Bobadilla, S. Heinze) Moderately discontinuous homotopy. Submitted. Available in ArXiv:2007.01538 or in https://arxiv.org/pdf/2007.01538.pdf

We present a recent, quite elementary construction of embedded resolutions of Schubert varieties in the Grassmannian. The construction is based on the use of flags with incidence relations running in two directions. Over the relevant Schubert variety the resolution coincides with the well-known Kempf-Laksov construction. Time permitting we will show how bioriented flags can also be used to give resolutions of Schubert varieties in the flag manifold.

I will discuss the alternate universe in which algebraic topology focuses on Lipschitz maps between metric spaces, rather than continuous maps between topological spaces. For "sufficiently nice" spaces (such as finite simplicial complexes with a simplexwise linear metric) the only difference is that one can define quantitative invariants that do not make sense without a metric. I will introduce some of the results and tools, including work of and with Berdnikov, Chambers, Dotterrer, Ferry, Guth, and Weinberger.

In this talk I will explain which three-dimensional Fano manifolds admit a Kahler-Einstein metric.

Symbolic dynamics is a tool that simplifies the study of dynamical systems in various aspects. It is known for almost fifty years that uniformly hyperbolic systems have “good” codings. For non-uniformly hyperbolic systems, Sarig constructed in 2013 “good” codings for surface diffeomorphisms. In this talk we will discuss some recent developments on Sarig’s theory, when the map has discountinuities and/or critical points such as dynamical billiards, and with applications to geodesic flows.

Symbolic dynamics is a tool that simplifies the study of dynamical systems in various aspects. It is known for almost fifty years that uniformly hyperbolic systems have “good” codings. For non-uniformly hyperbolic systems, Sarig constructed in 2013 “good” codings for surface diffeomorphisms. In this talk we will discuss some recent developments on Sarig’s theory, when the map has discountinuities and/or critical points such as dynamical billiards, and with applications to geodesic flows.

I will discuss some constructions of rational quasi-projective real algebraic varieties with infinitely many real forms. I will mainly focus on a construction taken from a joint work with L. Moser-Jauslin and G. Freudenburg, which provides an infinite countable family of pairwise non-isomorphic smooth rational real algebraic affine fourfolds whose complexifications are all isomorphic to the product of a smooth complex affine quadric surface with the affine plane. If time permits, I will explain some basic ideas from a work in progress aiming at the construction of uncountable families of pairwise non-isomorphic real forms of certain singular quasi-projective toric surfaces.

A rather non-standard format. Four to eight mini-talks, as time permits. Dynamics (KAM, kinetic gas models, etc), PDEs, inverse problems, discretization in mm–spaces, geometric group theory, algorithmics, Geometry, after all, control theory, stochastic structures etc. Mixing open problems and results by my collaborators and mine, or, so far, dead-ends, there are younger guys who are obviously smarter than me.

Let f be a complex polynomial with isolated singularities. In this talk, we will start by recalling classical formulas of the Euler characteristic of a fiber of f in terms of Milnor numbers of the singularities of f and the defect of equisingularity at infinity in a compactification of f. Then, recalling some notions of motivic integration and Denef-Loeser, Guibert-Loeser-Merle theorems, we will consider some motivic zeta functions and define for each value a, a motivic invariant at infinity of the fiber of f at a. This invariant does not depend on the chosen compactification, it is generically equal to zero and, under isolated singularities assumptions, its Euler characteristic is equal to the defect of equisingularity at infinity of f for the value a. In the last part of the talk, we will consider the case of plane curves, where computations of this invariant can be done in terms of Newton polygons at infinity, using an induction process based on Newton transformations and iterated Newton polygons.

This is a joint work with Lorenzo Fantini and Pierrette Cassou-Noguès

I will present new mathematical and computational tools to develop a complete and efficient algorithm for computing the set of non-properness of polynomial maps in the complex (and real) plane. In particular, this is a subset of the plane where a dominant polynomial map as above is not proper.The algorithm takes into account the sparsity of polynomials, and the genericness of the coefficients as it depends on their Newton polytopes. As a byproduct it provides a finer representation of the set of non-properness as a union of algebraic or semi-algebraic sets, that correspond to edges of the Newton polytopes, which is of independent interest. This is a joint work with EliasTsigaridas.

Any regular morphism f: (X,x_0) → (Y,y_0) between normal singularities induces a map f_*:V_X → V_Y at the level of valuation spaces (in the sense of Berkovich).The angular distance ρ_X on V_X plays the role of the Poincaré distance on valuation spaces: the action of f_* is non-expanding for these distances. In a joint work with W. Gignac, we study the geometrical properties of the angular distance in dimension 2. In this case, the angular distance has an interpretation in terms of intersection theory of b-divisors, which allows to study more in detail the contracting properties of the actions f_*. In this talk I will present these constructions, as well as some applications in local dynamics and the geometry of singularities admitting special dynamical data (partially joint work with L.Fantini and C.Favre).

We introduce invariants that control the Whitney equisingularity of families of EIDS. As an application of the results, we give a characterization of generic hyperplanes using the invariants.

We will show a link between the arc space (which is an algebro-geometric object) and the identities of partitions of integer numbers: a partition of a positive integer number is simply a way of writing it as a sum of positive integer numbers. Integer partitions have a long and romantic history in number theory. The link that we will describe is based on an invariant of singularities; it allows a new point of view on known results and gives new partition identities. The talk is accessible to a wide audience. This is joint work with Clemens Bruschek and Jan Schepers (2013), with Pooneh Afsharijoo (2019), with Pooneh Afsharijoo, Jehanne Dousse and Frédéric Jouhet (2021).

Consider a formal arc at a singular point P of an algebraic variety, not entirely contained in the singular locus of the variety. A striking result of Drinfeld-Grinberg-Kazdhan asserts that the infinitesimal deformations of the arc are in some sense entirely described by a finite-dimensional object. Taking such a parameter space of minimal dimension leads to the notion of the minimal formal model of the arc. There seems to be intriguing connections between the minimal formal model and the nature of the singularity at P. We will illustrate them in the case of toric and curve singularities. This is (for a big part of it) joint work with Julien Sebag.

For a finite subgroup G of GL_n(C), we can find a normal abelian subgroup A<G so that its index in G is bounded by a constant c(n), which only depends on n. In this talk, we prove that a similar statement holds for the local fundamental group of n-dimensional klt singularities. Then, we show applications of this statement to the study of iteration of Cox rings of Fano type varieties.

In this talk we will study the real structures on certain complex algebraic varieties endowed with a reductive algebraic group action, namely the almost homogeneous varieties. We will see how to determine when such real structures exist and, if so, how to describe and count them. In particular, we will illustrate our approach with two classical families of almost homogeneous varieties: the horospherical varieties (including toric varieties and flag varieties) and the almost homogeneous SL(2)-threefolds. This is a joint work with Lucy Moser-Jauslin (IMB, Dijon, France).

The classical Tits alternative asserts that any linear group over a field of zero characteristic is either virtually solvable, or contains a nonabelian free sub-group. We survey on recent extensions of the Tits alternative to automorphism groups of algebraic varieties. In the case of affine varieties, these automorphism groups are often infinite-dimensional, hence highly nonlinear. Sometimes, they act highly transitively on the variety, that is, m-transitively for any natural number m. According to Borel, no algebraic group acts 4-transitively on an algebraic variety. We show the relations between the alternatives of Tits type and multiple transitivity.

The classical Tits alternative asserts that any linear group over a field of zero characteristic is either virtually solvable, or contains a nonabelian free sub-group. We survey on recent extensions of the Tits alternative to automorphism groups of algebraic varieties. In the case of affine varieties, these automorphism groups are often infinite-dimensional, hence highly nonlinear. Sometimes, they act highly transitively on the variety, that is, m-transitively for any natural number m. According to Borel, no algebraic group acts 4-transitively on an algebraic variety. We show the relations between the alternatives of Tits type and multiple transitivity.

I will partially report on two works in progress one with N. Dutertre the other one with M. Michalska, dealing with bifurcation values at infinity of tame mappings. This will be the opportunity to review a few definitions and results about this topic, generalize and clarify some of them, as well as pointing out modestly some open questions (old and new).

We introduce invariants that control the Whitney equisingularity of families of EIDS. As an application of the results, we give a characterization of generic hyperplanes using the invariants.

I will review several results about the algebraic closure of the field of power series in several indeterminates. This field can be equipped with a non-archimedean absolute value, and considering its completion allows us to work in a bigger field where we can use Newton's method to find roots of polynomials. Therefore one problem is to determine what are the elements of this completion that are algebraic over the field of power series. I will present analogies with the algebraic closure of the field of rational numbers.

This talk is about Gabrielov's rank Theorem, a fundamental result in local complex and real-analytic geometry, proved in the 1960's. Contrasting with the algebraic case, it is not in general true that the analytic rank of an analytic map (that is, the dimension of the analytic-Zariski closure of its image) is equal to the generic rank of the map (that is, the generic dimension of its image). This phenomenon is involved in several pathological examples in local real-analytic geometry. Gabrielov's rank Theorem provides a formal condition for the equality to hold. Despite its importance, the original proof is considered very difficult. There is no alternative proof in the literature, besides a work from Tougeron, which is itself considered very difficult. I will present a new work in collaboration with André Belotto da Silva and Guillaume Rond, where we provide a complete proof of Gabrielov's rank Theorem, for which we develop formal-geometric techniques, inspired by ideas from Gabrielov and Tougeron, which clarify the proof. I will start with some fundamental examples of the phenomenon at hand, and expose the main ingredients of the strategy of this difficult proof.

Birational transformations, also known as Cremona maps, are capable to transform a local feature into a global one and viceversa. We will revise some of these phenomena concerning plane algebraic curves and planar polynomial differential systems.

In this talk we will address relationships between some analytical invariants of irreducible plane curves and their semi-roots. More specifically, we will explore the Tjurina number of a curve and the set of Kähler differential values of the local ring associated with the curve.

It is well known that two generic quadric surfaces intersect in a nonsingular quartic space curve, but when the intersection is not transverse this intersection curve may degenerate to a finite number of different possible types of singular curves. In the nice paper by Farouki et al. (1989), the authors formulate a way of computing thecondition for a degenerate intersection in this case, which refines in the real case and with an algorithmic point of view a classical treatise by Bromwich (1906). Independently, Schläfli (1953) studied the degenerate intersection of two hypersurfaces described by multilinear equations. In joint work with S. di Rocco and R. Morrison, we present a general framework of iterated sparse discriminants to characterize the singular intersection of hypersurfaces with a given monomial support A, which generalizes both previous situations. We study the connection of iterated discriminants with the notion of mixed discriminant and the singularities of the sparse discriminant associated to A.

H. Whitney introduced several cones, candidates to a possible generalization of tangent space at singular points. Many of these cones have an important use in the local study of a singularity. In particular the cones C_3, C_4 and C_5, are extensively used. The two first ones are quite well understood. In this talk we present a description and a procedure to build the C_5 cone for a complex curve. The tools we use for that purpose happen to be interesting in the Lipschitz behavior of a complex curve.

This talk is about Gabrielov's rank Theorem, a fundamental result in local complex and real-analytic geometry, proved in the 1960's. Contrasting with the algebraic case, it is not in general true that the analytic rank of an analytic map (that is, the dimension of the analytic-Zariski closure of its image) is equal to the generic rank of the map (that is, the generic dimension of its image). This phenomenon is involved in several pathological examples in local real-analytic geometry. Gabrielov's rank Theorem provides a formal condition for the equality to hold. Despite its importance, the original proof is considered very difficult. There is no alternative proof in the literature, besides a work from Tougeron, which is itself considered very difficult. I will present a new work in collaboration with André Belotto da Silva and Guillaume Rond, where we provide a complete proof of Gabrielov's rank Theorem, for which we develop formal-geometric techniques, inspired by ideas from Gabrielov and Tougeron, which clarify the proof. I will start with some fundamental examples of the phenomenon at hand, and expose the main ingredients of the strategy of this difficult proof.

In this talk, we will give a proof for Mostowski's regular projection theorem in o-minimal structures, which is a positive answer to the question of Parusinski about a definable version of the theorem.

I will introduce the central spectrum of a commutative ring and explain what is a rational continuous function and a regulous function on the central locus of a real algebraic affine variety. After that I will show that we can normalize in different ways real algebraic varieties by replacing the field of rational functions by the ring of regular functions on the real closed points, the ring of regulous functions on the central locus and the ring of rational continuous functions on the central locus. I will provide examples of these normalizations. Most of the results of the talk were obtained with G. Fichou and R. Quarez.

Rota's Basis Conjecture is a well known problem, that states that for any collection of n bases in a rank n matroid, it is possible to decompose all the elements into n disjoint rainbow bases. Here an asymptotic version of this is will be discussed - that it is possible to find n − o(n) disjoint rainbow independent sets of size n − o(n).

My talk will be based on my Ph.D thesis, "Real algebraic versions of Cartan's Theorems A and B". It is known that Cartan's Theorem A does not hold in the real algebraic case, however I will show that it holds after a sequence of blowing ups along smooth centers. i.e. for any coherent sheaf on a real algebraic affine variety there exists a multi-blowup such that the pullback sheaf is generated by global sections. Also, an example will be given which shows that this Theorem cannot be generalized to quasi-coherent sheaves. There are similar problems with Cartan's Theorem B, usually Čech cohomology of a coherent sheaf are infinitely dimensional. I will construct blown-up Čech cohomology, and show that for some sheaves these cohomology vanish. Finally, I will use this theory to show universal solvability of a real algebraic first Cousin problem, after blowing up.

In this talk we consider the problem of the approximation of a real or complex analytic set germ by germs of Nash or even algebraic sets which are equisingular with respect to the Hilbert-Samuel function. We show that a Cohen-Macaulay analytic singularity can be arbitrarily closely approximated by germs of Nash sets which are also Cohen-Macaulay and share the same Hilbert-Samuel function. Also, we obtain a result that states that every analytic singularity is topologically equivalent to a Nash singularity with the same Hilbert-Samuel function. A key ingredient in our results is a generalization of Buchberger's criterion to standard bases of power series due to T. Becker in 1990. This talk is based on joint work with Janusz Adamus at the University of Western Ontario, Canada.

I will explain how to transform the collection of Newton polygons generated by a process of toroidal resolution of a plane curve singularity into a lotus. This is a special kind of two-dimensional simplicial complex, which unifies the classical encodings of the combinatorial type of the singularity: its Enriques diagram, its weighted dual graph, its Eggers-Wall tree and its splice diagram. This work, which is about to appear in the first volume of the Handbook of Geometry and Topology of Singularities, was done in collaboration with the spanish mathematicians Evelia Garc\'{\i}a Barroso and Pedro Gonz\'alez P\'erez.

I will introduce a non-archimedean version of the link of a singularity. This object is a space of valuations whose structure can be described in terms of the resolutions of the singularity. I will then discuss two joint works with André Belotto and Anne Pichon where this object appears naturally in the study of a metric surface germ.

We present topological approaches to the Jacobian conjecture in $\C^n$. In particular, we obtain a result on the nonproperness set $S_f$ of a nonsingular polynomial mapping $f:\C^n \to \C^n$.

In this talk we will present a construction taking a singularity of a planar algebraic curve and associating to it a certain configuration space of flags. The obtained variety is of dimension equal to the Milnor number of the singularity and possesses a cluster structure (we will explain what it means in this context), in particular the Poisson structure and explicit coordinates and plenty of other structures. The tropical limit of this variety turns out to be canonically isomorphic to the space of versal deformation of the singularity.

We consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, these level curves may fail to be convex. The aim of this talk is two-fold. Firstly, to study a combinatorial object measuring this non-convexity; it is a planar rooted tree. And secondly, we want to characterise all possible topological types of these objects. To this end, we construct a family of polynomial functions with non-Morse strict local minima realising a large class of such trees.

Let $r\mapsto S(r)$ be a set-valued mapping with nonempty values and a closed graph which is definable in an o-minimal structure. In this talk we are interested in the asymptotic behavior of the orbits of the so-called sweeping process

$$\dot x(r) \in - N_{S(r)}, \quad r>0.$$ (SPO)

We show that an analogous technique to the one used by Kurdyka (Ann. Inst. Fourier, 1998) to generalize the Lojasiewicz inequality and control the asymptotic behavior of the gradient orbits, can be extended to our setting. We obtain a that bounded trajectories of (SPO) have bounded length. Our method recovers the result of Kurdyka for functions if the sweeping process is defined by the sublevel sets of a $C^1$-smooth definable function: indeed, in this case setting $S(r) = [f\leq r]$, we deduce that the orbits of (SPO) are in fact gradient orbitsfor $f$, and the Kurdyka-Lojasiwicz inequality is recovered.

This talk is based on a collaboration with D. Drusvyatskiy (Seattle).

This is the continuation of my talk of July 7, 2020, in which I describe the statement of the problem, the motivation and the classical results of Levi-Civita and Dini in the Riemannian setting. In this continuation we will describe in more detail the recent progress toward the generalization of these classical results to sub-Riemannian metrics. In particular, we will discuss genericity of metrics that do not admit non-constantly proportional affinely/projectively equivalent metrics and two separation of variables results for such metrics: on the level of the nilpotent approximation (i.e. of the tangent cone) of the sub-Riemannian structure and the other one on the level of Jacobi curves, special curves in Lagrangian Grassmannians, which contains all information about Jacobi equatons along a sub-Riemannian geodesic in an intrinsic way. Jacobi equations for metrics that admit non-constantly proportional affinely equivalent metrics. We also describe the sub-Riemannian analog of Weyl theorem that all metrics that are simultaneously projectively equivalent and conformal are constantly proportional. The talk is based on the collaboration with Frederic Jean (ENSTA, Paris) and Sofya Maslovskaya (INRIA, Sophya Antipolis).

We will discuss very recent progress on face rings and face numbers of triangulations of manifolds with boundary. The main idea is as follows: given a simplicial complex $\Delta$ whose geometric realization is a connected, orientable, homology manifold with boundary, we define the completion of $\Delta$ --- a complex $\hat{\Delta}$ obtained from $\Delta$ by coning the boundary of $\Delta$ with a single new vertex. We then show that, despite the fact that $\hat{\Delta}$ has a singular vertex, the face ring (also known as the Stanley-Reisner ring) of $\hat{\Delta}$ enjoys several properties that face rings of triangulations of spheres have. This allows us to show that several results on face numbers established in the last decade for triangulations of closed manifolds have very natural extensions to triangulations of manifolds with boundary. The talk will assume no background on face rings and face numbers. This is joint work with Ed Swartz.

\L ojasiewicz inequality asserts that, given a ${\cal C}^1$ globally subanalytic function-germ $f:(X,a) \to \mathbb{R}$, with $X$ globally subanalytic ${\cal C}^1$ submanifold of $\mathbb{R}^n$ and $a\in X$, there is a neighborhood $U$ of $a$ in $X$, a constant $C$, and a rational number $\theta \in [0,1)$ such that for all $x\in U$: $$|f(x)-f(a)|^\theta\le C |\nabla_x f|.$$ In this talk we will explore the case where $a$ does not belong to $X$ but lies in its closure. In particular, the point $a$ might be a singular point of the closure of $X$. We will discuss a \L ojasiewicz type inequality which is valid even if $X$ is not locally closed near $a$. As $f$ does not necessarily extend continuously at $a$, our approach relies on the study of the asymptotic critical values of the considered function.

Sub-Riemannian metrics are defined by a distribution (a subbundle of the tangent bundle) together with an Euclidean structure on each fiber. The Riemannian metrics correspond to the case when the distribution is the whole tangent bundle. Two sub-Riemannian metrics are called projectively equivalent if they have the same geodesics up to a reparameterization and affinely equivalent if they have the same geodesics up to affine reparameterization. In the Riemannian case both equivalence problems are classical: local classifications of projectively and affinely equivalent Riemannian metrics were established by Levi-Civita in 1898 and Eisenhart in 1923, respectively. In particular, a Riemannian metric admitting a nontrivial (i.e. non-constant proportional) affinely equivalent metric must be a product of two Riemannian metrics i.e. a separation of variables (the de Rham decomposition) occur, while for the analogous property in the projectively equivalent case a more involved ("twisted") product structure is necessary. The latter is also related to the existence of sufficiently many commuting nontrivial integrals quadratic with respect to velocities for the corresponding geodesic flow. We will describe the recent progress toward the generalization of these classical results to sub-Riemannian metrics. In particular, we will discuss genericity of metrics that do not admit non-constantly proportional affinely/projectively equivalent metrics and the separation of variables on the level of linearization of geodesic flows (i.e. on the level of the Jacobi equations) for metrics that admit non-constantly proportional affinely equivalent metrics. We also describe the sub-Riemannian analog of Weyl theorem that all metrics that are simultaneously projectively equivalent and conformal are constantly proportional. The talk is based on the collaboration with Frederic Jean (ENSTA, Paris) and Sofya Maslovskaya (INRIA, Sophya Antipolis).

A germ of a real or complex analytic space (X,0) embedded in (R^n ,0) or in (C^n ,0) is equipped with two natural metrics: its outer metric d_o, induced by the standard metric of the ambient space, and its inner metric d_i, which is the associated arc-length metric on the germ. The germ (X,0) is said to be Lipschitz normally embedded (LNE for short) if the identity map of (X,0) is a bilipschitz homeomorphism between the inner and the outer metric, that is if there exist a neighborhood U of 0 in X and a constant K ≥ 1 such that d_i (x,y) ≤ Kd_o(x,y) for all x and y in U. This property only depends on the analytic type of (X,0), and not on the choice of an embedding in some smooth ambient space (R^n ,0) or (C^n ,0). The study of Lipschitz Normal Embedded singularities is a very active research area with many recent results, for example by Birbrair, Bobadilla, Fernandes, Heinze, Kerner, Mendes, Misev, Neumann, Pedersen, Pereira, Pichon, Ruas, and Sampaio, but despite the current progress it is still in its infancy. I will give a panorama on recent results on LNE of germs of normal complex surfaces. In particular, I will present the results obtained recently in collaboration with André Belotto and Lorenzo Fantini for LNE surface germs, which generalize results of Spivakovsky and Bondil that were known for minimal surface singularities.

Let β > 1 be a rational number, say β = r/s, with r > s > 0, gcd(r,s) = 1. A β-quasihomogeneous polynomial is a quasihomogeneous polynomial of type (s,r). Consider the following problem: Given any two real β-quasihomogeneous polynomials F(X,Y), G(X,Y) of degree d≥1, determine whether there exists a germ of semialgebraic bi-Lipschitz homeomorphism Φ: (ℝ2,0) ⟶ (ℝ2,0) such that G○Φ = F. By extending the methods devised by Lev Birbrair, Alexandre Fernandes, and Daniel Panazzolo on their joint work “Lipschitz Classification of Functions on a Hölder Triangle, St. Petersburg Math. J., Vol. 20 (2009)”, we can obtain some fairly satisfactory partial solutions to this problem. That is the subject of my talk. We will see how the problem above, under appropriate hypotheses, can be reduced to the Lipschitz classification of real polynomial functions of a single variable, and we will also see how the classification in the single variable case can be carried out by analyzing the critical values of the given polynomials.

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for generically arc-analytic maps. Finally, as an application, we characterize in terms of the motivic measure, germs of arc-analytic homeomorphism between real algebraic varieties which are bi-Lipschitz for the inner metric.

In this talk we recall The Poincaré-Hopf Theorem in the smooth case then, using the notion of index of vector fields on singular varieties and the Milnor number for determinantal surfaces, we state a Poincaré-Hopf Type Theorem in this situation.

Complete minimal surfaces of finite total curvature in R4 can be seen as generalization of affine complex curves in C2. I will give two alternate definitions, describe their infinite {\it ends} and explain how these define a link at infinity which can be seen as a braid in the case of a single end. I will discuss how the knot and braid at infinity can help us (or not) address basic questions like: what are the minimal embeddings of R2 into R4 of total finite curvature?

A holomorphic symplectic manifold X is a compact Kaehler manifold equipped with a nowhere degenerate holomorphic 2-form. If D is a smooth hypersurface in X then D is equipped with a regular rank-one foliation (the kernel of the restriction of this form). I shall survey some results concerning the dimension of the minimal invariant submanifold through a general point (that is, closure of a general leaf) of this foliation.

Based on joint works with Marina Sobolevsky, Pavel Sobolevskii, Sergio Alvarez, Daniel Berend and Darlan Girao. It is a development of ideas of Mauricio Peixoto for Geometric Theory of Differential Equations of the Second Order and some related topics in Number Theory.

J. Milnor intersects a plane branched algebraic disk with a small sphere centered at the branch point; this defines a knot and such knots are all (possibly iterated) torus knots. Minimal surfaces in R4 generalize algebraic curves; when they have a branch point which is an isolated singularity, we can reproduce Milnor's construction and derive a class of knots called minimal knots. How large is this class is still a mystery but I will describe a sub-class of these knots called Lissajous toric knots and discuss leads to derive more complicated knots. Finally I will talk of complete minimal surfaces in R4 of finite total curvature where we get knots at infinity (cf. W. Neumann's construction for affine algebraic curves).

The Mather-Gaffney criterion states that a holomorphic map germ $f\colon(\mathbb C^n,0)\to(\mathbb C^p,0)$ is finitely determined for $\mathcal A$-equivalence (that is, right-left equivalence) if and only if it has isolated instability. In the real case, if $f\colon(\mathbb R^n,0)\to(\mathbb R^p,0)$ is finitely determined then we can assume it is analytic (in fact polynomial). Taking the complexification one proves that $f$ has isolated instability. But the converse is false in general and it is not difficult to find an example of a real analytic map germ with isolated instability which is not finitely determined. For analytic functions $f\colon(\mathbb R^n,0)\to(\mathbb R,0)$, Kuo showed that isolated singularity implies finite $C^0$-determinacy. Also Wall proved that if $f\colon(\mathbb R^n,0)\to(\mathbb R^p,0)$ has isolated instability for $\mathcal K$-equivalence (that is, contact equivalence), then it is finitely $C^0$-$\mathcal K$-determined. A natural question is if this is also true in general for $\mathcal A$-equivalence. In this talk we will answer this question for $n=2$, based on recent joint works with R. Mendes and J.A. Moya-Perez.

Nesta palestra vou explicar em que contexto aparecem os laços rápidos, conjuntos de separação e suas generalizações no estudo da Geometria Lipschitz de Singularidades Complexas. Público Alvo: alunos e professores.

The germ of an algebraic variety (X,0) is naturally equipped with two different metrics: the outer metrics which is the restriction of the Euclidean metrics and the inner metrics which is defined using lengths of paths in X. The bilipschitz equivalence class of these metrics is an intrinsic property of the variety and does not depend on the choice of embeddings. If the inner metric is bilipschitz equivalent to the outer metric, we say that (X,0) is Lipschitz normally embedded. In this talk we prove Lipschitz normally embeddedness of some algebraic subsets of the space of matrices. Let X be a subset of the space of matrices, then let X(r) be the set of matrices in X of rank r, and X(<r) be the set of matrices in X of rank less than r. We prove that if X is either the space of all matrices, the space of symmetric matrices or the space of anti-symmetric matrices, then X(r) and X(<r) are Lipschitz normally embedded for any r. We also prove that the set of upper triangular matrices of determinant 0, is Lipschitz normally embedded and that transversal intersections of linear spaces and X(<r) where X is the space of all matrices are Lipschitz normally embedded for any r. This last result has generalizations in the language of determinantal singularities which we will briefly discuss. This is joint work with Dmitry Kerner and Maria A. S. Ruas.