**Geometry Webinars**

**3 ^{rd} December 2021 | 15:00 (Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871

**João Pedro dos Santos **(Universidade de Paris-Sorbonne e Universidade de Montpellier)

**Algebraic theory of ODEs depending on parameters**

**Abstract:**

Differential Galois theory has the objective to study linear ODEs with the help of algebraic groups. Roughly and explicitly, to a matrix $A\in \mathrm{Mat}_n( \mathbf C(x) )$ and a differential system $y'=Ay$, we associate a subgroup of ${\rm GL_n}(\mathbf C)$, the differential Galois group, whose function is to measure the complexity of the solutions. There are three paths to this theory: Picard-Vessiot extensions (similar to standard Galois theory), monodromy representations (analytic approach) and Tannakian categories (the categorical way).

If instead of working with complex coefficients we deal with a discrete valuation ring $R$, the construction of the differential Galois groups are less obvious and the theory of groups gives place to that of group schemes. This puts forward the Tannakian approach and relevant concepts from algebraic geometry. In this talk, I shall explain how to associate to these differential equations certain flat $R$-group schemes, what properties these may have and, most importantly, how to compute them with the monodromy. In doing so, I shall explain our method to establish a certain "Deligne-Riemann-Hilbert" theorem with "parameters" (this is in connection to recent work of Fiorot-Monteiro-Sabbah), which is a way of finding extensions of differential equations having mild singularities. I shall emphasise the case of formal differential equations for the sake of clarity.

(The talk is a horizontal report on several works done in collaboration with P. H. Hai and his students N. D. Duong and P. T. Tam over the past years.)

**29 October 2021 | 15:00 (Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871** **

Também em formato presencial na sala **6.2.33, da FCUL**

**Giosuè Muratore** (CMAFcIO)

**Enumeration of rational contact curves via torus actions**

**Abstract:**

Complex projective spaces of odd dimension have a unique contact structure. So, in these spaces we have contact (Legendrian) rational curves. We are interested in enumeration of such curves. We prove that some Gromov-Witten numbers associated with rational contact curves in projective space with arbitrary incidence conditions are enumerative. Also, we use the Bott formula on the Kontsevich space to find the exact value of those numbers.

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**23 ^{rd} July | 15:00 **

**(Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871

videoconf-colibri.zoom.us Zoom is the leader in modern enterprise video communications, with an easy, reliable cloud platform for video and audio conferencing, chat, and webinars across mobile, desktop, and room systems. Zoom Rooms is the original software-based conference room solution used around the world in board, conference, huddle, and training rooms, as well as executive offices and classrooms. Founded in 2011, Zoom helps businesses and organizations bring their teams together in a frictionless environment to get more done. Zoom is a publicly traded company headquartered in San Jose, CA. |

**Gonçalo Tabuada **(Univ. Nova de Lisboa e Univ. de Warwick)

**Noncommutative Riemann hypothesis**

**Abstract:**

*I will explain how the classical generalized Riemann hypothesis can be extended from the realm of algebraic geometry to the broad setting of geometric noncommutative schemes in the sense of Orlov. *

*Moreover, I will provide some applications of this result to commutative geometry and to noncommutative geometry.*

**25 ^{th} June | 15:00 **

**(Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871

videoconf-colibri.zoom.us Zoom is the leader in modern enterprise video communications, with an easy, reliable cloud platform for video and audio conferencing, chat, and webinars across mobile, desktop, and room systems. Zoom Rooms is the original software-based conference room solution used around the world in board, conference, huddle, and training rooms, as well as executive offices and classrooms. Founded in 2011, Zoom helps businesses and organizations bring their teams together in a frictionless environment to get more done. Zoom is a publicly traded company headquartered in San Jose, CA. |

**François Petit **(Univ. Paris)

**Thickening of the diagonal, interleaving distance and Fourier-Sato transform**

**Abstract:**

We will explain how to construct a distance on the bounded derived category of abelian sheaves on a large class of metric spaces (including compact Riemannian manifolds) and present the notion of Lipschitz kernels. As an application, we get a new interpretation of the Fourier-Sato transform and prove that it is an isometry. This is a joint work with P. Schapira.

**14 ^{th} May | 15:00 **

**(Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871

videoconf-colibri.zoom.us Zoom is the leader in modern enterprise video communications, with an easy, reliable cloud platform for video and audio conferencing, chat, and webinars across mobile, desktop, and room systems. Zoom Rooms is the original software-based conference room solution used around the world in board, conference, huddle, and training rooms, as well as executive offices and classrooms. Founded in 2011, Zoom helps businesses and organizations bring their teams together in a frictionless environment to get more done. Zoom is a publicly traded company headquartered in San Jose, CA. |

**Stéphane Guillermou **(CNRS, Univ. Grenoble)

**Stable Gauss map of nearby Lagrangians**

**Abstract:**

The stable Gauss map of a Lagrangian $L$ in a cotangent $T^*M$ is a map $g\colon L \to U/O$ obtained by stabilization of the usual Gauss map from $L$ to the Lagrangian Grassmannian of $T^*M$. Arnold's conjecture on nearby Lagrangians implies in particular that $g$ is homotopic to a constant map. We will see the weaker result that the map induced by $g$ on the homotopy groups is trivial.

{This is a joint work with Mohammed Abouzaid, Sylvain Courte and Thomas}

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*Financiado por Fundos Nacionais através da FCT – Fundação para a Ciência e a Tecnologia no âmbito do projeto UIDB/04561/2020*

**16 ^{th} April | 15:00 (Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871

**Pierre Schapira **(Professor emeritus Sorbonne University)

**Euler calculus of constructible functions and applications**

**Abstract:**

In this elementary talk, we will recall the classical notions of subanalytic sets, constructible sheaves and constructible functions on a real analytic manifold and explain how to treat such objects ``up to infinity’’.

Next, we will describe the Euler calculus of constructible functions, in which integration is purely topological, with applications to tomography. Finally we will show how the gamma-topology on a vector space allows one to embed the space of constructible functions in that of distributions.

For more details, see: arXiv:math/2012.09652

**19**^{th}** March | 15:00 (Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871**Ricardo Campos **(CNRS/University of Montpellier)

**Configuration spaces of points and their homotopy type**

**Abstract:**

Given a topological space X, one can study the configuration space of n points on it: the subspace of X^n in which two points cannot share the same position. Despite their apparent simplicity such configuration spaces are remarkably complicated; the homology of these spaces is reasonably unknown, let alone their homotopy type. This classical problem in algebraic topology has much impact in more modern mathematics, namely in understanding how manifolds can embed in other manifolds, such as in knot theory. In this talk I will give a gentle introduction to this topic and explain how using ideas going back to Kontsevich we can obtain algebraic models for the rational homotopy type of configuration spaces of points.

**19 ^{th} February | 14:00 (Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871**Thom****a****s Kr****ä****mer** (Humboldt Universität zu Berlin)

videoconf-colibri.zoom.us |

**Semicontinuity of Gauss maps and the Schottky problem**

**Abstract:**

*We show that the degree of the Gauss map for subvarieties of abelian varieties is semicontinuous in families, and we discuss its jump loci. In the case of theta divisors this gives a finite stratification of the moduli space of ppav's whose strata include the Torelli locus and the Prym locus. More generally we obtain semicontinuity results for the intersection cohomology of algebraic varieties with a finite morphism to an abelian variety, leading to a topological interpretation for various jump loci in algebraic geometry. *

*This is joint work with Giulio Codogni.*

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*Financiado por Fundos Nacionais através da FCT – Fundação para a Ciência e a Tecnologia no âmbito do projeto UIDB/04561/2020*

**22 ^{nd} January | 14:00 (Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871**André Oliveira **(CMUP)

**Lie algebras and higher Teichmüller components**

**Abstract:**

*Consider the moduli space M(G) of G-Higgs bundles on a compact Riemann surface X, for a real semisimple Lie group G. Hitchin components in the split real form case and maximal components in the Hermitian case were, for several years, the only known source of examples of higher Teichmüller components of M(G). These components (which are not fully distinguished by topological invariants) are important because the corresponding representations of the fundamental group of X have special properties, generalizing Teichmüller space, such as being discrete and faithful. Recently, the existence of new such higher Teichmüller components was proved for G = SO(p,q) which, in general, is not neither split nor Hermitian.*

*In this talk I will explain the new Lie theoretic notion of magical nilpotent, which gives rise to the classification of groups for which such components exist. It turns out that this classification agrees with the one of Guichard and Wienhard for groups admitting a positive structure. We provide a parametrization of higher Teichmüller components, generalizing the Hitchin section for split real forms and the Cayley correspondence for maximal components in the Hermitian (tube type) case. *

*This is joint work with S. Bradlow, B. Collier, O. García-Prada and P. Gothen.*

**16 ^{th} December | 14:00 (Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871**Giordano Cotti **(GFM)

**Quantum differential equations, isomonodromic deformations, and derived categories**

**Abstract:**

*The quantum differential equation (qDE) is a rich object attached to a smooth projective variety X. It is an ordinary differential equation in the complex domain which encodes information of the enumerative geometry of X, more precisely its Gromov-Witten theory. Furthermore, the asymptotic and monodromy of its solutions conjecturally rules also the topology and complex geometry of X. These differential equations were introduced in the middle of the creative impetus for mathematically rigorous foundations of Topological Field Theories, Supersymmetric Quantum Field Theories and related Mirror Symmetry phenomena. Special mention has to be given to the relation between qDE's and Dubrovin-Frobenius manifolds, the latter being identifiable with the space of isomonodromic deformation parameters of the former. The study of qDE’s represents a challenging active area in both contemporary geometry and mathematical physics: it is continuously inspiring the introduction of new mathematical tools, ranging from algebraic geometry, the realm of integrable systems, the analysis of ODE’s, to the theory of integral transforms and special functions. This talk will be a gentle introduction to the analytical study of qDE’s, their relationship with derived categories of coherent sheaves (in both non-equivariant and equivariant settings), and a theory of integral representations for its solutions. The talk will be a survey of the results of the speaker in this research area.*

**20**^{th}** November | 13:45 (Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871**César Rodrigo **(EST Setúbal, CMAFcIO)

**Discretization of gauge theories: Application to elastic rod dynamics**

**Abstract:**

Classical numerical schemes for field theories are usually described in terms of a linear finite element space, assuming the existence of a linear structure that is fundamental for the numerical algorithm. In a non-discrete (smooth) setting, gauge field theories are defined on principal bundles, and have a rich geometric structure that leads to Noether conserved currents but is, in general, incompatible with any linear structure. This is a reason for the lack of energy-conservation properties in several numerical integration schemes for field theories.

Key elements in the smooth geometrical formulation of such physical models are the reduction by some Lie group, or a trivialization choice that allows to identify each bundle fiber with a Lie group. Invariance of the theory from a "gauge" choice shows that the natural space to formulate the corresponding physical laws is a bundle of principal connections, possibly coupled with an associated bundle.

In this talk we shall present geometrical tools that allow to discretize, linearize, trivialize, and reduce gauge field theories, relating the continuous and discrete models by forward difference operators that are covariant for the action of the structure group.

The movement of an elastic rod can be reduced to a principal connection on a trivial principal bundle modeled on the group SE(3). The dynamics of elastic rods is characterized by a choice of two metrics on the Lie algebra, to represent the inertia and elastic components of each rod element, an auxiliary principal connection that represents the minimal energy configuration of the rod, and an additional function to represent forces acting on the rod. We use our discretization tools to generate a family of discrete models for the rod, and present a numerical scheme for the integration of the elastic rod dynamics. The geometrical nature of the tools ensures then corresponding energy conservation properties.

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*Financiado por Fundos Nacionais através da FCT – Fundação para a Ciência e a Tecnologia no âmbito do projeto UIDB/04561/2020*

**23 ^{rd} October | **

**14:15 (Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871**Davide Masoero **(Grupo de Física Matemática, FCUL)

**The Painlevé I equation and the A2 quiver**

**Abstract:**

We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlevé equation. We use the generalised monodromy map for this equation to give solutions to the Bridgeland's Riemann-Hilbert problem arising from the Donaldson-Thomas theory of the A2quiver.

The talk is partially based on a work in collaboration with Tom Bridgeland (https://arxiv.org/abs/2006.10648)

**23 ^{rd} October | 14:00 (Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871**Davide Masoero **(Grupo de Física Matemática, FCUL)

**The Painlevé I equation and the A2 quiver**

**Abstract:**

We study a second-order linear differential equation known as the deformed cubic oscillator, whose isomonodromic deformations are controlled by the first Painlevé equation. We use the generalised monodromy map for this equation to give solutions to the Bridgeland's Riemann-Hilbert problem arising from the Donaldson-Thomas theory of the A2quiver.

The talk is partially based on a work in collaboration with Tom Bridgeland (https://arxiv.org/abs/2006.10648)

**24 ^{th} July | 13:30 (Lisbon time)**

Location: Zoom Meeting: https://videoconf-colibri.zoom.us/j/7992972871

Meeting ID: 799 297 2871**Gonçalo Tabuada** (FCTUNL)

**Noncommutative Weil conjectures**

**Abstract:**

The Weil conjectures (proved by Deligne in the 70's) played a key role in the development of modern algebraic geometry. In this talk, making use of some recent topological "technology", I will extended the Weil conjectures from the realm of algebraic geometry to the broad noncommutative setting of differential graded categories. Moreover, I will prove the noncommutative Weil conjectures in some interesting cases.