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**Schedule **

Speaker: Ruadhaí Dervan (University of Glasgow)

Title: The universal structure of moment maps in complex geometry

Abstract: Much of complex geometry is motivated by linking the existence of solutions to geometric PDEs (producing "canonical metrics") to stability conditions in algebraic geometry. I will discuss a more basic question: what is the recipe to actually produce interesting geometric PDEs in complex geometry? The construction will be geometric, using a combination of universal families and tools from equivariant differential geometry. This is joint work with Michael Hallam.

Past Talks

Speaker: Xin Wang (KIAS)

Title: 5D Wilson Loops and Topological Strings on Fano-threefolds

Abstract: Geometric engineering provides a rich class of 5D supersymmetric gauge theories with eight supercharges, arising from M-theory compactification on non-compact Calabi-Yau threefolds. The counting of BPS states in the low-energy gauge theory is determined by the degeneracies of M2-branes wrapping holomorphic two-cycles in the Calabi-Yau threefold X. These degeneracies can also be calculated from the (refined) topological strings on the same manifold X. In this talk, I will explore the BPS spectrum of the 5D gauge theory with the insertion of a half-BPS Wilson loop operator. Utilizing M-theory realization, we derive the BPS expansion of the expectation value of the Wilson loop operator in terms of BPS sectors. It is found that the BPS sectors can be realized and computed from topological string theories. In the unrefined limit, the BPS sectors act as generating functions for Gromov-Witten invariants on compact (semi)-Fano threefolds constructed from X. We further conjecture a new refined holomorphic anomaly equations for the generating functions of the Wilson loops. These equations can completely solve the refined BPS invariants of Wilson loops for local \mathbb{P}^2 and local \mathbb{P}^1\times\mathbb{P}^1.

Speaker: Abdellah Lahdili (University of Aarhus) (Last talk of this semester)

Title: The Einstein-Hilbert functional and K-stability.

Abstract: In this talk, I will discuss a correspondence between the problem of finding a constant scalar curvature K\"ahler metric (cscK for short) on a polarized complex manifold and the CR-Yamabe problem on the associated circle bundle. I will then use a CR version of the Einstein-Hilbert functional to show K-semistability of polarised complex manifolds admitting cscK metrics. This is a joint work with Eveline Legendre and Carlo Scarpa.

Speaker: Denis Auroux (Harvard)

Title: Holomorphic discs and extended deformations in mirror symmetry

Abstract: Given a Lagrangian torus fibration on the complement of an anticanonical divisor in a Kahler manifold, one usually constructs a mirror space by gluing local charts (moduli spaces of local systems on generic torus fibers) via wall-crossing transformations determined by counts of Maslov index 0 holomorphic discs; this mirror also comes equipped with a regular function (the superpotential) which enumerates Maslov index 2 holomorphic discs. However, holomorphic discs of negative Maslov index deform this picture by introducing inconsistencies in the wall-crossing transformations; the geometric features of the resulting mirror can be understood in the language of extended deformations of Landau-Ginzburg models. We illustrate this phenomenon (and show that it actually occurs) on an explicit example (a 4-fold obtained by blowing up a Calabi-Yau toric variety), and discuss a family Floer approach to the geometry of the corrected mirror in this setting.

Speaker: Yuta Kusakabe (Kyoto University )

Title: Oka tubes in holomorphic line bundles

Abstract: A complex manifold is called an Oka manifold if continuous maps from Stein manifolds can be deformed into holomorphic maps with approximation and interpolation. Oka manifolds are characterized by a certain kind of ellipticity which is the opposite of Kobayashi-Eisenman-Brody hyperbolicity. Based on the fact that the zero section of a negative line bundle on a compact complex manifold admits a basis of pseudo-Kobayashi hyperbolic neighborhoods, we discuss the question of when the zero section of a line bundle admits a basis of Oka neighborhoods. This talk is based on joint work with Franc Forstnerič.

Speaker: Timothée Bénard (Warwick Mathematic Institute)

Title: Limit theorems on nilpotent Lie groups.

Abstract: I will present limit theorems for random walks on nilpotent Lie groups, obtained in a recent work with Emmanuel Breuillard. Most works on the topic assumed the step distribution of the walk to be centered in the abelianization of the group. Our main contribution is to authorize non-centered step distributions. In this case, new phenomena emerge: the large-scale geometry of the walk depends on the increment average, and the limit measure in the central limit theorem may not have full support in the ambient group.

Speaker: Martin de Borbon (Loughborough University)

Title: Bubbling of Kahler-Einstein metrics: examples and conjectures

Abstract: I will report on joint work with Cristiano Spotti. Chi Li's theory of normalized volumes permits the algebraic determination of tangent cones of singular Kahler-Einstein metrics. The general principle is to extend this to the dynamic setting of families and determine all possible rescaled limits using algebraic geometry. I will discuss examples in low dimensions, including polyhedral metrics on the sphere and Kahler-Einstein metrics on complex surfaces with cone singularities along curves.

Speaker: Marc Troyanov (Institut de Mathématiques EPFL)

Title: Double Forms, Curvature Integrals and the Gauss-Bonnet Formula

Abstract: In this talk I will briefly present the historical development behind Chern's proof in 1944 of the higher dimensional Gauss-Bonnet formula, starting with the work of H. Hopf back in 1925. I will then introduce the notion of double forms (which basically are sums of tensor products of differential forms) and explain how these tensors help us reformulate the Gauss-Bonnet-Chern-Formula and in particular clarify the meaning of Chern's boundary term. I will give some application and if time permits, some perspectives on non compact and singular Riemannian manifolds.

Speaker: Fernando Galaz-García (Durham University)

Title: Decompositions of 3-dimensional Alexandrov spaces

Abstract: Alexandrov spaces (with curvature bounded below) are metric generalizations of complete Riemannian manifolds with a uniform lower sectional curvature bound. Instances of Alexandrov spaces include compact Riemannian orbifolds and orbit spaces of isometric compact Lie group actions on compact Riemannian manifolds. In addition to being objects of intrinsic interest, Alexandrov spaces play an important role in Riemannian geometry, for example, in Perelman's proof of the Poincaré Conjecture. In this talk, I will discuss the topology and geometry of 3-dimensional Alexandrov spaces, focusing on extensions of basic results in 3-manifold topology (such as the prime decomposition theorem) to general three-dimensional Alexandrov spaces. This is joint work with Luis Atzin Franco Reyna, José Carlos Gómez-Larrañaga, Luis Guijarro, and Wolfgang Heil.

Speaker: Jeffrey D. Streets (University of California, Irvine) (22:00-23:00 Beijing Time !)

Title: Formal structure of scalar curvature in generalized Kahler geometry

Abstract: The Fujiki-Donaldson moment map formulation of scalar curvature problem, and the attendant Mabuchi-Semmes-Donaldson geometry of a Kahler class, play a central role in addressing the existence and uniqueness of constant scalar curvature Kahler metrics. Generalized Kahler (GK) geometry is a natural extension of Kahler geometry arising from Hitchin’s generalized geometry program and mathematical physics, and forms a particularly well-structured extension of Kahler geometry. Recently Goto defined a notion of scalar curvature in GK geometry as the moment map of a particular Hamiltonian action on the space of generalized Kahler structures. In this talk I will describe joint work with Vestislav Apostolov and Yury Ustinovskiy where we give an explicit description of the scalar curvature, and define a natural generalization of the Mabuchi-Semmes-Donaldson metric, leading to a Calabi-Lichnerowicz-Matsushima obstruction, generalizations of Futaki’s invariants, and a conditional uniqueness result.

Speaker: Gonçalo Oliveira (Instituto Superior Técnicon Lisbon)

Title: Einstein metrics from Derdziński duality

Abstract: (joint work with Rosa Sena-Dias) A theorem of Derdziński from the 1980's establishes that certain Einstein metrics are conformal to Bach-flat extremal Kahler metrics. Using this result Rosa Sena-Dias and I classified conformally Kähler, U(2)-invariant, Einstein metrics on the total space of O(−m). This yields infinitely many 1-parameter families of metrics exhibiting several different behaviors including asymptotically hyperbolic metrics (more specifically of Poincaré type), ALF metrics, and metrics which compactify to a Hirzebruch surface with a cone singularity along the ''divisor at infinity''. As an application of these results, we find many interesting phenomena. For instance, we exhibit the Taub-bolt Ricci-flat ALF metric as a limit of cone angle Einstein metrics on the blow up of CP2 at a point (in the limit when the cone angle converges to zero). We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat Kähler metric and cannot be obtained by applying Derdziński's theorem.

Speaker: Gabriel Katz (MIT)

Title: Recovering contact forms from boundary data

Abstract: Let $X$ be a compact connected smooth manifold with boundary. The paper deals with contact $1$-forms $\beta$ on $X$, whose Reeb vector fields $v_\b$ admit Lyapunov functions $f$. We prove that any odd-dimensional $X$ admits such a contact form.

We tackle the question: how to recover $X$ and $\beta$ from the appropriate data along the boundary $\partial X$? We describe such boundary data and prove that they allow for a reconstruction of the pair $(X, \beta)$, up to a diffeomorphism of $X$. We use the term ``holography" for the reconstruction. We say that objects or structures inside $X$ are {\it holographic}, if they can be reconstructed from their $v_\b$-flow induced ``shadows" on the boundary $\partial X$.

For a given $\beta$, we study the contact vector fields $u$ on $X$ that preserve it invariant. Integrating $u$, we get a $1$-parameter family of contactomorphisms $\{\Phi^t(u)\}_{t \in \mathbb R}$ which maps Reeb trajectories to Reeb trajectories. This leads to estimates from below of the number of $\Phi^t(u)$-fixed Reeb trajectories.

We also introduce numerical invariants that measure how ``wrinkled" the boundary $\partial X$ is with respect to the $v_\beta$-flow and study their holographic properties under the contact forms preserving embeddings of equidimensional contact manifolds with boundary. We get some ``non-squeezing results" about such contact embedding, which are reminiscent of Gromov's non-squeezing theorem in symplectic geometry.

Speaker: Jacopo Stoppa (Scuola Internazionale Superiore di Studi Avanzati)

Title: K-stability and large complex structure limits

Abstract: According to mirror symmetry, the geometry of a given Fano manifold endowed with some extra data, including an arbitrary Kähler class, should be reflected in a mirror Landau-Ginzburg model, i.e. a noncompact complex manifold endowed with a nonconstant holomorphic function. On the other hand, a fundamental notion for constructing moduli of Fano manifolds is K-polystability, i.e. positivity of the Donaldson-Futaki invariants for nonproduct test-configurations. In this talk I will introduce the problem of characterising K-polystable Kähler classes on a Fano in terms of their mirror Landau-Ginzburg models. I will then discuss some first concrete results in the case of slope stability for del Pezzo surfaces. The computations involve the particular “large complex structure limit” of the Landau-Ginzburg model corresponding to scaling the Kähler class on the Fano, which acts trivially on K-polystability.

Speaker: Mario Garcia Fernandez (Universidad Autónoma de Madrid)

Title: Futaki invariants and harmonic metrics for the Hull-Strominger system

Abstract: The Hull-Strominger system has been proposed as a geometrization tool for understanding the moduli space of Calabi-Yau threefolds with topology change (‘Reid’s Fantasy’). In this talk we will introduce obstructions to the existence of solutions for these equations which combine infinite-dimensional moment maps, ‘harmonic metrics’, and a holomorphic version of generalized geometry. We will discuss the implications of these new obstructions in relation to a conjecture by S.-T. Yau for the Hull-Strominger system. Based on joint work with Raúl Gonzalez Molina, in arXiv:2303.05274 and arXiv:2301.08236.

Speaker: James Bonifacio (University of Mississippi)

Title: Spectral bounds on hyperbolic 3-manifolds

Abstract: I will discuss some new bounds on the spectra of Laplacian operators on hyperbolic 3-manifolds. One example of such a bound is that the spectral gap of the Laplace-Beltrami operator on a closed orientable hyperbolic 3-manifold must be less than 47.32, or less than 31.57 if the first Betti number is positive. The bounds are derived using two approaches, both of which employ linear programming techniques: 1) the Selberg trace formula, and 2) identities derived from the associativity of spectral decompositions. The second approach is inspired by an area of physics called the conformal bootstrap. This talk is based on work done in collaboration with Dalimil Mazáč and Sridip Pal.

Speaker: Javier Martínez-Aguinaga (Universidad Complutense de Madrid)

Title: $h$-Principle for maximal growth distributions

Abstract: The existence and classification problem for maximal growth distributions on smooth manifolds has garnered much interest in the mathematical community in recent years. Prototypical examples of maximal growth distributions are contact structures on $3$-dimensional manifolds and Engel distributions on $4$-dimensional manifolds.

The existence and classification of maximal growth distributions on open manifolds follows from Gromov’s $h$-Principle for open manifolds. Nonetheless, not so much was known for the case of closed manifolds and several open questions have been posed in the literature throughout the years about this problem.

In our recent work [1] we show that maximal growth distributions of rank $> 2$ abide by a full h-principle in all dimensions, providing thus a classification result from a homotopic viewpoint. As a consequence we answer in the positive, for $k > 2$, the long-standing open question posed by M. Kazarian and B. Shapiro more than 25 years ago of whether any parallelizable manifold admits a $k$-rank distribution of maximal growth.

[1]. Martínez-Aguinaga, Javier. Existence and classification of maximal growth distributions . Preprint. arXiv:2308.10762

Speaker: Timothée Bénard (Warwick Mathematic Institute)

Title: Limit theorems on nilpotent Lie groups.

Abstract: I will present limit theorems for random walks on nilpotent Lie groups, obtained in a recent work with Emmanuel Breuillard. Most works on the topic assumed the step distribution of the walk to be centered in the abelianization of the group. Our main contribution is to authorize non-centered step distributions. In this case, new phenomena emerge: the large-scale geometry of the walk depends on the increment average, and the limit measure in the central limit theorem may not have full support in the ambient group.

Speaker: Kajal Das (Indian Statistical Institute)

Title: On super-rigidity of Gromov's random monster group （Slides)

Abstract: In my talk, I will speak on the super-rigidity of Gromov's random monster group. It is a finitely generated random group $\Gamma_\alpha$ ( $\alpha$ is in a probability space $\mathcal{A}$) constructed using an expander graph by M. Gromov in 2000. It provides a counterexample to the Baum-Connes conjecture for groups with coefficients in commutative $C^*$-algebra. It is already known that it has global fixed point property for isometric affine action on $L^p$ spaces for $1< p<\infty$ (in particular, Property (T) ) for a.e. $\alpha$ due to Gromov and Naor-Silberman. It is also hyperbolically rigid, i.e., any isometric action of the group on a Gromov-hyperbolic space is elementary for a.e. $\alpha$ (due to Gruber-Sisto-Tessera). In this talk, I will discuss the following type of super-rigidity problem (motivated by Margulis super-rigidity theorem): for which countable group $G$, any collection of homomorphisms $\phi_\alpha:\Gamma_\alpha\rightarrow G$ have a finite image for a.e. $\alpha$? This question was first addressed for linear groups by Naor-Silberman. The super-rigidity follows immediately from the literature for groups with a-$L^p$-menablity and K-amenable groups. In this talk, we will show that $\Gamma_\alpha$ has super-rigidity with respect to the following groups $G$: mapping class group $MCG(S_{g,p})$, braid group $B_n$, automorphism group of a free group $Aut(F_n)$, outer automorphism group of a free group $Out(F_N)$ . We will also show a stability result of the class of groups $G$.