This seminar presents a geometric and analytic derivation of Dirac–Dunkl operators as symmetry reductions of the flat Dirac operator on Euclidean space. Starting from the standard Dirac operator, we restrict to a fundamental Weyl chamber of a finite Coxeter group equipped with the Heckman–Opdam measure, and determine the necessary drift and reflection corrections that ensure formal skew–adjointness under this weighted geometry. This procedure naturally reproduces the Dunkl operators as the unique first–order deformations compatible with reflection symmetry, whose Clifford contraction defines the Dirac–Dunkl operator and whose square yields the Dunkl Laplacian. We then extend the construction to include arbitrary unitary representations of the reflection group, obtaining representation-dependent Dirac–Dunkl operators that act on spinor- or matrix-valued functions. In the scalar and sign representations, these operators recover respectively the bosonic and fermionic Calogero–Moser systems, while higher-dimensional representations give rise to multi-component spin–Calogero models. The resulting framework unifies analytic, geometric, and representation-theoretic aspects of Dirac and Dunkl operators under a single symmetry-reduction principle.

Mirror symmetry can be partially understood by an equivalence of categories of sheaves derived from the work of Mukai and Arinkin which goes by the name of Fourier-Mukai transform. In this talk we will focus on an example of branes lying on the singular locus of the Hitchin system. By studying the Fourier-Mukai transform of these "singular" branes we pose some questions on the current statement of the conjectured duality of Mirror symmetry for branes. 

This is joint work with Johannes Horn (Frankfurt), Robert Hanson (Imperial) and André Oliveira (Porto).

In this talk, we will discuss a generalization of A. Mellit's decomposition for GLn-character varieties to arbitrary reductive groups. As a motivation, we will introduce a motivic version of T. Hausel and M. Thaddeus' conjecture and explain how to use the previous result to approach it.

Slides

The Alday-Gaiotto-Tachikawa (AGT) conjecture in physics predicts a relationship between 2d conformal field theories and certain 4d gauge theories. A precise mathematical version, proved by Maulik-Okounkov, Schiffmann-Vasserot and others, states that the equivariant cohomology of the moduli space of instantons (4d side) is a module of a certain W-algebra (2d side). Moreover, the fundamental class of the moduli space is a Whittaker vector in the W-algebra module, known as the Gaiotto state. I will explain how one can compute this Gaiotto state using the topological recursion formalism of Eynard and Orantin. This means that one can compute the Nekrasov instanton partition function (which is the norm squared of the Gaiotto state) using topological recursion techniques. Time permitting, I will discuss some applications of this relationship, including relations to Seiberg—Witten theory, Hurwitz theory and connections to KP integrability. 

The talk is based on joint papers with Vincent Bouchard, Gaetan Borot, Thomas Creutzig and Giacomo Umer.

In this talk, we will introduce a reinterpretation of Alexander invariants for knots through the study of representation varieties of knot groups into the affine group AGL1(C). We establish connections between these representation varieties and the Alexander module, providing a geometric perspective of Alexander polynomials as singular loci of coherent sheaves associated with knot quandles. As an application, we introduce a Topological Quantum Field Theory (TQFT) approach, recovering the Burau representation of braids, thus offering new methods for understanding and categorifying Alexander-type knot invariants. Results are joint work with Angel Gonzalez-Prieto and Vicente Muñoz.

The relationship between computational models and dynamics has captivated mathematicians and computer scientists since the earliest conceptualisations of computation. Recently, this connection has gained renewed attention, fueled by T. Tao's program aiming to discover blowing-up solutions of the Navier-Stokes equations using an embedded computational model, and by the intriguing links between static solutions and contact structures. In this vibrant context, a series of groundbreaking works have shown that Turing machines can be simulated through the flow lines of certain remarkable dynamical systems.

In this talk, we will introduce a novel perspective on computability in dynamical systems, drawing inspiration from Topological Quantum Field Theory. We will demonstrate that any computable function can be realised through the flow of a vector field on a smooth bordism. This result not only provides a new computational model but also unveils compelling insights into the interplay between the topological characteristics of the flow, the existence of compatible contact-like geometric structures on the bordism, and the computational complexity of the function.

We will discuss the notion of regularity of sheaves on projective varieties. The starting point will the work of Costa and Mirò-Roig using n-blocks. We will go beyond their definition, giving an equivalent definition and discussing when it is possible to relax their hypotheses. We will use also the idea of n-blocks to produce other interesting results, like a generalization of ACM bundles and Horrocks Theorem. This is work in progress with Simone Marchesi.

In this talk I will present my latest work on the deformations of Clarke-Oliveira's instantons. The Bryant-Salamon Manifold - the negative spinor bundle of 4-sphere - is an 8-dimensional (non-compact) asymptotically conical manifold whose holonomy is Spin(7). After reviewing the notions of asymptotically conical manifolds and instantons on them, I will introduce the deformation theory where I will briefly explain how we identify the space of infinitesimal deformations with the kernel of a  Dirac operator on the manifold. Finally I will apply this theory to describe the deformations of Clarke-Oliveira’s Instantons and calculate the (virtual) dimensions of the moduli spaces.

We explore the computation of motivic invariants of the G-character variety of a finitely generated group, where G is an algebraic group over an algebraically closed field of characteristic zero. The G-character variety can be stratified by conjugacy classes of parabolic subgroups of G in terms of root data where each stratum is no longer a GIT quotient but a pseudo-quotient, a relaxed notion capturing the topology and behaving well with respect to motivic calculations. Dealing with these pseudo-quotient strata is further reduced by the notion of a core: it is enough to deal with polystable points with prescribed Levi type. For reductive groups of types ABCD explicit motivic formulae are provided and several mirror symmetry statements and conjectures are derived. 

Results are joint work with Carlos Florentino, Azizeh Nozad and Ángel González Prieto.

SLIDES

We study A_g, the moduli space of principally polarized abelian varieties of dimension g. The subring generated by the Chern classes of the Hodge bundle is called the tautological ring, and it was fully determined by Gerard van der Geer in 1999, but the question of which geometrically defined cycles belong to this subring was open. In 2024, Canning, Oprea and Pandharipande showed that [A_1 \times A_5] is not tautological in A_6, and later I showed that [A_1 \times A_{g-1}] is not tautological for g=12 or g >= 16 even.

The cycle [A_1\times A_{g-1}] is one of the Noether-Lefschetz cycles on the moduli spaces. With Greer and Lian, we conjecture that these cycles are related to modular forms of weight 2g, and have some evidence that points towards this.

A new technique, which was not available in 1999 is the existence of a projection operator by Canning, Oprea, Molcho and Pandharipande onto the tautological ring. This leads to interesting conjectures about Gromov-Witten invariants on a moving elliptic curve, which now have been proven in collaboration with Pandharipande and Tseng, and are also connected to the failure of the lambda_g conjecture on the moduli space of curves.

The talk will be an overview of the intersection theory of A_g and the moduli space of curves, and I will explain briefly the ideas behind some proofs. If time allows, I will go into some work in progress with Feusi and Molcho on what the tautological ring should look like on the toroidal compactifications of A_g.

Given an action of the 1-dimensional complex torus on a normal projective vareity, one gets a set of birationally equivalent GIT quotients. In this talk we will consider the opposite question: given a birational map, we would like to realize it as the induced birational map between two GIT quotients of a certain variety by a C*-action. We will present examples of this two-way correspondence and discuss it's combinatorial part, appearing in the context of toric varieties.

We explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus 1. We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We utilize semistability of trivially twisted very stable bundles to prove that the wobbly locus is always a divisor in the moduli space of semistable bundles on a genus 1 curve. We prove, by extension, a conjecture regarding the closedness and dimension of the wobbly locus in this setting. This conjecture was originally formulated by Drinfeld in higher genus.

In recent decades, exotic phases of matter have emerged as a major focus in condensed matter physics. Among these, topologically ordered phases stand out for their unique properties, such as low-temperature states which are locally indistinguishable and potentially noise-tolerant, geometry-dependent physical properties, or the presence of novel quasiparticles with promising applications in quantum computation. In this talk, I will introduce the framework of quantum spin systems through tensor networks and present the toy models representing these phases, constructed using (C*-weak) Hopf algebras, and outline their expected physical properties. In particular, I will discuss some algebraic challenges that we face in this context.

SLIDES

In this talk I will explain how some actions of C^* on smooth projective varieties are related to birational transformations (between the sink and the source of the action), discussing examples and related problems.

These results are scattered in joint works with Jarosław Wiśniewski, Luis E. Solá Conde and Eleonora A. Romano.

SLIDES

The moduli space $\mathcal M_g$, i.e., the space of hyperbolic surfaces of genus $g$ up to isometry, has a natural structure of orbifold given by the action of the mapping class group on Teichmüller space. The branch locus consists on the hyperbolic surfaces which admit some non-trivial isometry, and it is naturally stratified by the topological action of their isometry group. These strata are called equisymmetric loci.

On the other hand, moduli space can be compactified by adding  hyperbolic stable  surfaces, which can be grouped into strata attending to their topological type.  This coincides with the Deligne-Mumford compactification.

In joint work with Víctor  González-Aguilera, from Universidad Técnica Federico Santa María (Chile), we investigate which topological types of stable surfaces appear at the boundary of a given equisymmetric locus. In this talk I will present a theoretical procedure to find  the boundary of any given equisymmetric stratum   and I will particularize to equisymmetric strata determined by abelian actions, which include the hyperelliptic and the $p$-gonal loci.  I will also apply this procedure to give a complete description of the boundary of certain equisymmetric loci given by a dihedral action.

The classical Yang-Baxter equation (CYBE) plays an important role in the modern theory of integrable systems. In a seminal work of Belavin and Drinfeld from the year 1983 it was proven that solutions of CYBE are of the  following three types: elliptic, trigonometric or rational. Moreover, Belavin and Drinfeld gave a complete classification of all elliptic and trigonometric solutions. 

In my talk, I am going to explain a geometric description of solutions of  CYBE. Namely, starting with any geometric datum (E, A), where E is a Weierstraß cubic curve and A a torsion free sheaf of Lie algebras (whose generic fiber  is a given complex simple Lie algebra) with vanishing cohomology, one can  associate to it in a canonical way a solution of CYBE. It turns out that all solutions of CYBE arise in this way. Moreover, the type of the curve in the datum (E, A) (i.e. smooth/nodal/cuspidal)  determines the type of the corresponding solution of CYBE (i.e. elliptic/trigonometric/rational). 

The developed method will be illustrated by explicit examples. This talk is based on my  joint works with Raschid Abedin, Lennart Galinat and Thilo Henrich.

The Picard number $\rho(X)$ of a smooth projective surface $X$ is the rank of its Neron-Severi group which, defined as the  group of divisors of $X$ modulo numerical equivalence, is finitely generated. The Picard number of $X$ is bounded above by the Hodge number $h^{1,1}(X)=\text{dim } H^1(X,\Omega^1_X)$ and  $X$ is said to have maximal Picard number if $\rho(X)=h^{1,1}(X)$.

Although there is no a priori reason to believe the geography of minimal surfaces of general type should not be highly populated by surfaces with maximal Picard number, the examples of such surfaces are scarce in the literature. See \cite{MR3322784} for an overview on the subject.

Let us denote by $K^2$ the self-intersection of the canonical class  and by $\chi$ the holomorphic Euler characteristic of an algebraic surface. The first published non-trivial examples of algebraic surfaces of general type with maximal Picard number are due to Persson \cite{MR661198}, who constructed surfaces with maximal Picard number on the Noether line $K^2=2\chi-6$ for every admissible pair $(K^2,\chi)$ such that $\chi \not\equiv 0 \text{ mod } 6$.

This talk will explore the key ideas from \cite{MR4761778}, where surfaces of general type with maximal Picard number were constructed for every admissible pair $(K^2,\chi)$ such that $2 \chi -6\leq  K^2\leq \frac{5}{2}\chi-11$. These constructions, obtained as bidouble covers of rational surfaces, not only allowed to fill in Persson's gap $\chi \equiv 0 \text{ mod } 6$ on the Noether line, but they provided infinitely many new examples of algebraic surfaces of general type with maximal Picard number above the Noether line.

When do two different looking quantum field theories describe the same physics? This is essentially asking when the quantum field theories are isomorphic. In the case of topological quantum field theories, there is sometimes a way to determine them via topological invariants. For a superconformal field theory, what would be the minimal set of “invariants” to determine when they are isomorphic? I will discuss some approaches to this question in the context of a particular infinite class of superconformal field theories that admit Hitchin systems. Isomorphic pairs of such theories must have the same operator contents, such as Schur index and Hall-Littlewood index. Such theories can also be described using curve configurations and this will shine light on finding pairs of isomorphic superconformal field theories, which a priori look like distinct theories. In turn, this result provides a conjecture when two theories will necessarily have the same Schur index and Hall-Littlewood index. If time permitting, I will explain how this result further sheds light on the 3d (symplectic) duality.

SLIDES

It is well-known that the Teichmüller space of a compact surface can be identified with a connected component of the moduli space of representations of the fundamental group of the surface in PSL(2,R). Higher rank Teichmüller spaces are generalizations of this, that exist in the moduli space of representations of the fundamental group into certain real simple non-compact Lie groups of higher rank. As for the usual Teichmüller space, these spaces consist entirely  of discrete and faithful representations.  In this talk, I will give a full classification of all possible higher rank Teichmüller spaces, and a parametrization of them using the theory of Higgs bundles (based on joint work with Bradlow, Collier, Gothen and Oliveira).

SLIDES

Moduli spaces and moduli stacks of bundles depend on several parameters for their construction. The moduli space and the moduli stack of vector bundles with fixed determinant both depend on the choice of an algebraic curve, a rank and a line bundle. Moduli spaces of parabolic vector bundles depend, in addition, on the choice of a set of parabolic weights, which act as stability parameters. Their geometry is known to depend non-trivially on the choice of these parameters.

In this talk I will present some new results on the classification of automorphisms and isomorphisms between moduli built with different parameters for three types of moduli: moduli stacks of vector bundles with fixed determinant, moduli stacks of G-bundles and moduli spaces of parabolic vector bundles, including an ongoing project to combine algebro-geometric and computational tools to obtain an exhaustive classification of isomorphisms classes and automorphism groups of parabolic moduli in low rank.

Joint works with Indranil Biswas, Tomás Gómez, Swarnava Mukhopadhyay, Sergio Herreros, Jaime Pizarroso, José Portela and Javier Rodrigo.

In 2010, Kontsevich and Soibelman defined Cohomological Hall Algebras for quivers and potential as a mathematical construction of the algebra of BPS states. These algebras are modelled on the cohomology of vanishing cycles, which makes these algebras particularly hard to study but often result in interesting algebraic structures. A deformation of a particular case of them gives rise to a positive half of Maulik-Okounkov Yangians. The goal of my talk is to give an introduction to these ideas and if time permits; I will explain how for the case of tripled cyclic quiver with canonical cubic potential, this algebra turns out to be one-half of the universal enveloping algebra of the Lie algebra of matrix differential operators on the torus, while its deformation turn out be one half of an explicit integral form of the Affine Yangian of gl(n).

Scattering amplitudes predict the probability of interaction of elementary particles. Motivated by amplitude computations in string theory, Francis Brown has recently introduced the notion of single-valued periods. These are real numbers originating from an automorphism, induced by complex conjugation, of the de Rham cohomology of an algebraic variety, and they can be seen as analogues of p-adic periods for p=infinity. Examples include odd zeta values, regulators, height pairings and special values of real-analytic Eisenstein series. In this talk I will introduce Brown’s theory and I will explain, without assuming any previous knowledge in physics, how it naturally finds application in the computation of string theory amplitudes. In particular, I will report on my work with Pierre Vanhove on genus-zero amplitudes and on my work with Don Zagier on genus-one amplitudes.

Can we construct knot invariants from algebras? In the first half of the talk, I will explain how the Drinfeld double, a purely algebraic construction for (Hopf) algebras, has a very natural knot-theoretical interpretation in terms of the so-called universal invariant. This invariant, which is subject to the choice of an algebra, dominates a family of quantum invariants defined by Reshetikhin and Turaev in the late 1980s using the representation theory of the algebra. The downside of the universal invariant is that, when the algebra is infinite-dimensional, it is in general very hard to compute for a given arbitrary knot. In the second half of the talk, following work by Bar-Natan and van der Veen, I will present an algebra D arising from the Drinfeld double construction and a toolbox called Gaussian calculus that allow to compute the invariant efficiently for any knot. If time allows, I will also explain how this invariant seems to be closely related to what is known as the "rational expansion of the coloured Jones polynomials".

In 1990, Witten conjectured that the generating series of intersection numbers of psi classes is a tau function of the KdV hierarchy. This was first proved by Kontsevich. In 2017, Norbury conjectured that the generating series of intersection numbers of psi classes times a negative square root of the canonical bundle is also a tau function of the KdV hierarchy. In joint work with N. Chidambaram and A. Giacchetto (https://arxiv.org/abs/2205.15621), we prove Norbury’s conjecture and obtain polynomial relations among kappa classes which were recently conjectured by Kazarian-- Norbury. We also introduce a new collection of cohomology classes, which correspond to negative r-th roots (previously r=2) of the canonical bundle and form a cohomological field theory (CohFT), the negative analogue of Witten’s r-spin CohFT, which turns out to be geometrically much simpler. We prove that the corresponding intersection numbers can be computed recursively using topological recursion (which I will briefly introduce) and, equivalently, W-constraints. The strategy draws inspiration from our proof, together with S. Charbonnier (https://arxiv.org/abs/2203.16523), of Witten’s r-spin conjecture from 1993 (Faber—Shadrin—Zvonkine’s theorem from 2010) that claims that (positive) rspin intersection numbers satisfy the r-KdV hierarchy. We also obtain new (tautological) relations on the moduli space of curves in a (negative) analogous way to Pandharipande-- Pixton--Zvonkine. The talk will be an overview of these four topics (r=2/>2; positive/negative) and their connections.

Geometric Invariant Theory (GIT) is a standard tool to construct moduli spaces. The usual procedure involves constructing first an auxiliary space Q where a group G acts, and the moduli space is the quotient M=Q/G.

Alper, Heinloth, and Halpern-Leistner have developed another method. They start from the moduli stack parameterizing all objects. Their method allows us to identify a sub-stack (the sub-stack of "semistable" objects) that admits a "good moduli," i.e., a scheme (more generally, an algebraic space) that is sufficiently similar to the stack.

This talk is an elementary introduction to these new methods.

The notion of Mori dream regions (MDR for short) has been introduced by Y. Hu and S. Keel as a generalization of Mori dream spaces. In this talk we establish a correspondence between MDR associated to small Q-factorial modifications and certain C ∗ -actions on polarized pairs (X, L), with X normal projective and L ample and C ∗ -linearizable, called bordisms. This talk is based on a joint work with E. A. Romano (University of Genova), L. E. Sol´a Conde (University of Trento), and S. Urbinati (University of Udine).

Quantum knot invariants display surprising regularity in the lie group rank N. This was first observed in the extension of the Jones polynomial to the HOMFLY-PT polynomial but has since been extended to other infinite families of invariants. Recently Gukov and Malonescu conjectured the existence of an invariant which serves as an analytic continuation of the coloured Jones polynomials. One immediate question we can ask if this invariant also has a large N limit and, if such a limit existed, how it might be computed.

Jones first introduced the knot invariant Jones polynomial in 1984, which can be defined via a set of relations called “skein relations”. It was later shown by Reshetikhin and Turaev that this Polynomial fits into a larger framework of invariants coming from representations of quantum groups. In this framework, the Jones polynomial is the invariant associated to the fundamental representation of $U_q(\mathfrak{sl}_2)$ and the family of invariants associated to different finite-dimensional representations of $U_q(\mathfrak{sl}_2)$ are known as colored Jones polynomials. In this talk, I will review this construction and its recent analytical continuation to $\mathbb{C}\left(x^{1/2},q^{1/2}\right)$ introduced by Gukov and Malonescu, namely the $F_K(x,q)$ series, whose definition has been formalized to positive braid knots by Park. In the end, I will explain my current work involving the extension of the two-variable knot invariant series to $\mathfrak{sl}_3$ and potentially to the family of $\mathfrak{sl}_N$.

In the 70’s, Hartshorne stated his famous conjecture that a smooth n-dimensional projective variety of P N must be a complete intersection if n > 2/3N. In this talk, we will show how a conjecture of this type could be solved if we were able to understand how to extend vector bundles from a subvariety to an ambient space. In fact, the conjecture can be divided into two different natural problems: 1) When is it possible to extend the normal bundle of a subvariety X of an ambient variety Y ? In other words, when is it possible to express X as the zero locus of a section of a vector bundle over Y whose rank is the codimension of X in Y ? When the codimension is one, the answer is always positive, while in codimension two the so-called Hartshorne-Serre correspondence essentially asserts that the answer is positive if and only if the determinant of the normal bundle extends to Y . We will discuss the kind of results one could expect for this problem. 2) When does a vector bundle of rank r over P N splits as a direct sum of line bundles? We will show that on could expect this to be true when N > 2r, using that, in this range, one could expect the vector bundle to extend to P N+1 . Putting together both speculations, we will conclude that one could expect Hartshorne’s conjecture to be true when n > 3/4(N − 1).

Irreducible holomorphic symplectic (IHS) varieties can be thought as a generalization of hyperkähler manifolds allowing singularities. 

Among them we can find for example moduli spaces of sheaves on K3 and abelian surfaces, which have been recently shown to play a crucial role in non abelian Hodge theory.  After recalling the main features of IHS varieties, I will present several results concerning their intersection cohomology and the perverse filtration associated with a Lagrangian fibration, establishing a compact analogue of the celebrated P=W conjecture by de Cataldo, Hausel and Migliorini for varieties which admit a symplectic resolution.

 It is often convenient to visualize algebraic varieties (and hence systems of polynomial equations) by sampling. The key challenge is to have the right distribution and density in order to recover the shape, i.e the topology of the variety. Bottlenecks are pairs of points on the variety joined by a line which is normal to the variety at both points. These points play a special role in determining the appropriate density of a point-sample. Under suitable genericity assumptions the number of bottlenecks of an affine variety is finite and is called the bottleneck degree.  Estimations of the bottleneck degree and certain generalizations lead to efficient sampling techniques. We will show how classical projective algebraic geometry has proven very useful in this analysis.  The talk is based on joint work with D. Eklund, O. Gäfvert and M. Weinstein.

I will recall the Hitchin system, that Marina Logares already introduced in a previous talk, an abelian fibration in the moduli space of Higgs bundles. This is a global version of the characteristic polynomial map, and is a powerful tool in many branches of mathematics. In this talk, after introducing the objects and playing with some examples, I will speak about a couple applications, notably, to study some geometric properties of the Higgs moduli space, and, related to the latter, to the geometric Langlands programme.

We extend the Topological Quantum Field Theory developed by González-Prieto, Logares, and Muñoz for computing virtual classes of G-representation varieties of closed orientable surfaces in the Grothendieck ring of varieties to the setting of the character stacks. To this aim, we define a suitable Grothendieck ring of representable stacks, over which this Topological Quantum Field Theory is defined. In this way, we compute the virtual class of the character stack over BG, that is, a motivic decomposition of the representation variety with respect to the natural adjoint action.

We apply this framework in two cases providing explicit expressions for the virtual classes of the character stacks of closed orientable surfaces of arbitrary genus. First, in the case of the affine linear group of rank 1, the virtual class of the character stack fully remembers the natural adjoint action, in particular, the virtual class of the character variety can be straightforwardly derived. Second, we consider the non-connected group 𝔾m⋊ℤ/2ℤ, and we show how our theory allows us to compute motivic information of the character stacks where the classical naïve point-counting method fails.

The Hilbert scheme whose general points parametrize smooth, irreducible, curves of degree d, genus g, which are non–degenerate in the projective space $P^R$ has a distinguished component, dominating the moduli space $M_g$ of smooth genus–g curves, but in general it also has many other irreducible components. There are few explicits examples of these, for instance  the one parametrizing curves which are multiple covers of  $P^1$, and the one parametrizing double covers of irrational curves. Under some numerical assumptions on d, g and R, I present new irreducible components. They are generically smooth and turn out to be of dimension higher than the expected one. The general point of any such a component corresponds to a curve $X ⊂ P^R$ which is a suitable ramified m–cover of an irrational curve  $Y⊂ P^{R−1}$, m > 2, lying in a surface cone over $Y$.

Ranging from the discrete behaviour of finite groups to the involved topology of surface groups, moduli spaces of representations may exhibit very complex geometries. In this talk, we shall review the fundamentals of this rich theory with branches reaching very distant areas of mathematics. In particular, we will discuss how these moduli spaces that parametrizes representations of finitely generated groups are constructed through Geometric Invariant Theory, in the incarnation of the so-called character varieties.

We will pay special attention to analyzing a beautiful isomorphism, the non-abelian Hodge correspondence, that relates these character varieties with moduli spaces of flat connections and of Higgs bundles, as well as their role in Chern-Simons theory. Time permitting, we will also talk about character varieties of complements of knots and how to use them to create new algebraic invariants of knots.

Physicists have recently been interested in studying Hitchin systems on nodal curves. Interestingly, the required mathematical technology had not been fully developed and in joint work with R. Donagi and J. Distler, we developed the required framework to study type A Hitchin systems as a family over the Deligne-Mumford space. In particular, we study the Hitchin system over a nodal base curve. In this talk, I will explain this work and also discuss a surprising (partly conjectural) application of our work to the Deligne-Simpson problem. In the end, I will discuss a few open questions about Higgs bundles and character varieties and speculate how the perspective from physics might help answering these.

Since their origin in the late 80’s, Higgs bundles manifest as fundamental objects which are ubiquitous in contemporary mathematics and theoretical physics. Some prominent examples of this ubiquity are their role as integrable systems, in Langlands duality and Mirror Symmetry, and in representation theory as character varieties. In this talk we shall give an introduction to Higgs bundles, together with a glimpse of these objects play all the roles mentioned above.

We will present models of equivariant cohomology for differentiable stacks with Lie group actions extending classical results for smooth manifolds due to Borel, Cartan and Getzler. We also derive various spectral sequences for the equivariant cohomology of a differentiable stack generalising among others Bott's spectral sequence which converges to the cohomology of the classifying space of a Lie group. (joint work with Luis Alejandro Barbosa-Torres (University of São Paulo)).

Dada una variedad algebraica proyectiva \(X\) y un morfismo \(\varphi : X \to \mathbb{P}^n\), finito y de grado \(n\) (\(n > 1\)) sobre su imagen, queremos saber en qué casos el grado \(n\) cae cuando deformamos \(\varphi\), y en particular, en qué casos \(\varphi\) se puede deformar a un morfismo birracional o, incluso, a una inmersión. Este estudio de las deformaciones de los morfismos finitos tiene aplicaciones como:

1. La construcción sistemática de subvariedades en el espacio proyectivo con invariantes de interés, en particular, superficies canónicas y canónicas simples y subvariedades de codimensión baja.

2. El hallazgo de lugares de salto del grado de la aplicación canónica en el moduli de superficies de tipo general.

3. El estudio del comportamiento de las deformaciones de las variedades polarizadas hiperelípticas.

4. El hallazgo de ciertos subesquemas no reducidos llamados "ropes" en la clausura de componentes irreducibles del esquema de Hilbert.

En esta charla daré una visión rápida del problema, de las ideas y la teoría desarrollada para abordarlo y de los resultados obtenidos relativos a las aplicaciones mencionadas. Hablaré en más detalle de la construcción de subvariedades de codimensión baja, incluyendo subvariedades en el rango de la conjetura de Harstshone, y de la construcción de subvariedades que no son intersección completa.