**9 Jan.**Toti Daskalopoulos

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**16 Jan.**Lorenzo Foscolo

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**23 Jan.**Alice Rizzardo

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**30 Jan.**Eleonora Di Nezza

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**6 Feb.**

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**13 Feb.**Soheyla Feyzbakhsh

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**20 Feb.**Lara Bossinger

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**27 Feb.**Alexandr Buryak

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**6 Mar.**Konstantin Ardakov

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**13 Mar.**Dylan Allegretti

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Anna Barbieri

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**18 Mar.**Paul Biran

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### KCL/UCL Geometry seminar, Autumn 2018

The seminar will meet on Wednesdays between 15:00-16:00 at S0.12 Strand Building, King's College London.

Julian Scheuer

**26 Sep.**Brent Pym*Title: Holonomic Poisson manifolds**Abstract: Poisson brackets can exhibit very complicated singularities, and as a result it is typically quite difficult to determine their possible deformations. For instance, the deformation space is usually infinite-dimensional and highly singular. I will describe joint work with Travis Schedler, in which we introduce a natural new nondegeneracy condition for Poisson brackets, called holonomicity. It ensures strong finiteness properties for the relevant deformation complex, making the deformation spaces computable in terms of topological invariants such as intersection cohomology. As an application, we establish the deformation-invariance of some of Feigin--Odesskii's Poisson structures on moduli spaces of bundles over elliptic curves.***3 Oct.**Matias Delgadino*Title: Alexandrov's Theorem Revisited**Abstract: We will show that, among sets of finite perimeter, balls are the only volume-constrained critical points of the perimeter functional. We will utilize heavily the Heintze-Karcher inequality, which relates the volume of a open set with the integral of the reciprocal of the mean curvature on the boundary of the set. This is joint work with Francesco Maggi.*Julian Scheuer

**4-5 pm***Title: Alexandrov-Fenchel inequalities for convex free boundary hypersurfaces in a ball**Abstract: (joint w/ Guofang Wang (Freiburg) and Chao Xia (Xiamen)) The relative isoperimetric problem for hypersurfaces with free boundary on the unit sphere is understood to a very broad extent. It's higher order versions are the so-called Alexandrov-Fenchel inequalities, which are defined via higher order curvature integrals. For closed and convex hypersurfaces of the Euclidean space, these inequalities are well-known from convex geometry. In a recent work (with Guofang Wang and Chao Xia) we proved some new Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary on the unit sphere. In this talk we introduce the geometric quantities, that seem to be the right generalisations of the classical quermassintegrals to hypersurfaces with boundary. The proof of the inequalities will build on a locally constrained inverse harmonic mean curvature flow with free boundary.***10 Oct.**Kim Moore*Title: Deformation theory of Cayley submanifolds**Abstract: Calibrated submanifolds are a special class of volume minimising submanifolds that arise naturally in manifolds with special holonomy. In this talk, I will introduce Cayley submanifolds, four-dimensional calibrated submanifolds that live in manifolds with holonomy Spin(7), describe their relationship to complex and special Lagrangian submanifolds and explain how we can use elliptic PDE theory to study moduli spaces of Cayley submanifolds.***17 Oct.**Victor Przyjalkowski*Title: Katzarkov-Kontsevich-Pantev conjectures**Abstract: The initial mirror symmetry conjecture claims that any Calabi-Yau threefold has a pair Calabi-Yau threefold whose Hodge diamond is given from a Hodge diamond of the initial Calabi-Yau by rotating by 90 degrees. We discuss a generalization of this phenomenon on the Fano case. That is, following Katzarkov, Kontsevich, and Pantev, we define Hodge numbers for Landau-Ginzburg models. We discuss proofs of the conjectures in dimensions 2 and 3. A crucial ingredient for the threefold case is Harder's result that express the Hodge-type numbers in terms of geometry of Landau-Ginzburg models.***24 Oct.**Daniel Huybrechts*Title: The Hodge conjecture for products of K3 surfaces**Abstract: In the talk I will explain how far we are from proving the Hodge conjecture for products of two K3 surfaces. The known results use derived categories of twisted K3 surfaces and are best phrased in terms of motives. In comparison almost nothing is known in the analogous situation of products of cubic fourfolds.***31 Oct.**Hiro Tanaka*Title: Broken lines and associative algebras**Abstract: After a biased review of Morse theory, I will talk about a stack that naturally arises when contemplating Morse theory on a point. Surprisingly, this stack has the following property (our main theorem): A factorizable sheaf on this stack is the same thing as an associative algebra. As we will explain, this equivalence gives a gateway into enriching Lagrangian Floer theory over spectra, and hence into more powerful invariants in symplectic geometry. This is joint work with Jacob Lurie.***7 Nov.**Toby Dyckerhoff*Title: Towards categorified homological algebra**Abstract: Various recent developments, in particular in the context of topological Fukaya categories, seem to be glimpses of an emerging theory of categorified homotopical and homological algebra. The increasing number of meaningful examples and constructions make it desirable to develop such a theory systematically. In this talk, we discuss some progress towards this goal.***14 Nov.**Mark Gross*Title: Intrinsic Mirror Symmetry**Abstract: I will talk about joint work with B. Siebert which constructs an algebro-geometric analogue of degree zero symplectic cohomology ring for a log Calabi-Yau pair (X,D). Conjecturally, this ring is the ring of regular functions on the mirror of X\D, and hence leads to a vast generalization of the Gross-Hacking-Keel mirror construction for log Calabi-Yau surfaces. The method also applies to maximally unipotent degenerations of Calabi-Yau manifolds and hence should give a general mirror construction.***21 Nov.**Guido De Philippis*Title: Boundary regularity for solutions of the Plateau problem**Abstract: Plateau problem consists in finding the surface of minima area spanning a given boundary. Since the beginning of the 50’s the study of this problem led to the development of fundamental tools in Geometric Analysis and in the Calculus of Variations. Aim of the talk is give an overview of the problem and of the technques used to solve it. In the end I will also present some recent results concerning boundary regularity obtained in collaboration with C. De Lellis, J. Hirsch and A. Massaccesi.***28 Nov.**Andrea Malchiodi*Title: Prescribing Gaussian and Geodesic curvature on surfaces with boundary**Abstract: We consider the classical problem of finding conformal metrics on a surface such that both the Gaussian and the geodesic curvatures are assigned functions. We use variational methods and blow-up analysis to find existence of solutions under suitable assumptions. A peculiar aspect of the problem is that there are blow-up profiles with infinite volume that have to be excluded. In order to do this, we classify their stability properties and employ holomorphic vector fields. This is joint work with R. Lopez-Soriano and D. Ruiz.***5 Dec.**Roberta Maccheroni*Title: The Space of Totally Real Submanifolds: Geometric Properties and Applications**Abstract: In the first part of the talk we will describe some geometric properties of the (infinite-dimensional) space of totally real submanifolds of a complex manifold. After introducing a notion of "geodesics" in this space, we will show that the subspace of "special totally real submanifolds" in a Calabi-Yau manifold is a totally geodesic subset. In the second part we will study a complex analytic property of minimal Lagrangian submanifolds of a Kahler manifold: the existence of holomorphic discs fillings. It will be fundamental to consider them as particular totally real submanifolds, so as to use the theory of the J-volume functional developed by Lotay and Pacini.*### KCL/UCL Geometry seminar, Summer 2018

The seminar will meet on Wednesdays between 15:00-16:00 at the room D103 in the same building as the Department of Mathematics at UCL (25 Gordon Street).

Ziyu Zhang

**25 Apr.**Laura Schaposnik*Title: Higgs bundles, branes, and applications**Abstract: Higgs bundles are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle, and their moduli spaces carry a natural Hyperkahler structure, through which one can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes). Notably, these A and B-branes have gained significant attention in string theory. We shall begin the talk by first introducing Higgs bundles for complex Lie groups and the associated Hitchin fibration through which one can realize Langlands duality. We shall then look at natural constructions of families of subspaces which give different types of branes, and relate these spaces to the study of 3-manifolds, surface group representations and mirror symmetry.***2 May.**Marta Mazzocco*Title: Dualities in the q-Askey scheme and degenerate DAHA**Abstract: The Askey-Wilson polynomials are a four-parameter family of orthogonal symmetric Laurent polynomials R_n[z] which are eigenfunctions of a second-order q-difference operator L, and of a second-order difference operator in the variable n with eigenvalue z + z^{-1}. Then L and multiplication by z+z^{-1} generate the Askey-Wilson (Zhedanov) algebra. A nice property of the Askey-Wilson polynomials is that the variables z and n occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the non-symmetric case and in the underlying algebraic structures: the Askey-Wilson algebra and the double affine Hecke algebra (DAHA). In this paper we follow the degeneration of the Askey-Wilson polynomials until two arrows down and in four diferent situations: for the orthogonal polynomials themselves, for the degenerate Askey-Wilson algebras, for the non-symmetric polynomials and for the (degenerate) DAHA and its representations.***9 May.**Mariel Sáez*Title: On the uniqueness of graphical mean curvature flow**Abstract: In this talk I will discuss recent work with P. Daskalopoulos on sufficient conditions to prove uniqueness of complete graphs evolving by mean curvature flow. It is interesting to remark that the behaviour of solutions to mean curvature flow differs from the heat equation, where non-uniqueness may occur even for smooth initial conditions if the behaviour at infinity is not prescribed for all times.***16 May.**Norman Zergänge*Title: Convergence of Riemannian 4-manifolds with L^2-curvature bounds**Abstract: A key challenge in Riemannian geometry is to find ``best'' metrics on compact manifolds. To construct such metrics explicitly one is interested to know if approximation sequences contain subsequences that converge in some sense to a limit manifold. In this talk I will present convergence results of sequences of closed Riemannian 4-manifolds with almost vanishing L^2-norm of a curvature tensor and a non-collapsing bound on the volume of small balls. For instance I consider a sequence of closed Riemannian 4-manifolds, whose L^2-norm of the Riemannian curvature tensor is uniformly bounded from above, and whose L^2-norm of the traceless Ricci-tensor tends to zero. Here, under the assumption of a uniform non-collapsing bound, which is very close to the euclidean situation, and a uniform diameter bound, I show that there exists a subsequence which converges in the Gromov-Hausdorff sense to an Einstein manifold. To prove these results, I use Jeffrey Streets' L^2-curvature flow. In particular, I use his ``tubular averaging technique'' in order to prove fine distance estimates of this flow which only depend on significant geometric bounds.***23 May.**Travis Schedler*Title: Symplectic resolutions of multiplicative quiver varieties**Abstract: In joint work with Bellamy, I classified quiver varieties which admit symplectic resolutions of singularities, as well as character varieties of closed surfaces. I will discuss recent work with Tirelli which generalizes this to multiplicative quiver varieties, which includes analogues of character varieties for surfaces with punctures and prescribed monodromy conditions. This is closely related to the multiplicative preprojective algebra and its conjectural Calabi-Yau property, as I will explain.***30 May.**Diego Conti*Title: Indefinite homogeneous Einstein metrics**Abstract: The structure of left-invariant Einstein Riemannian metrics on solvable Lie groups is well understood thanks to work of Lauret and Heber. The indefinite case presents several differences; in particular, it is possible to construct unimodular solvable Lie groups with a left-invariant, indefinite Einstein metric. I will describe a method based on combinatorics and linear algebra for the systematic construction of metrics of this type on a nilpotent Lie group. I will also illustrate a necessary condition for the existence of such metrics that is useful for classification purposes. An obstruction is based on a formula for the Ricci operator which can be interpreted as a symplectic moment map for a GL(n,R) action. This is joint work with Federico A. Rossi***6 Jun.**Thomas Madsen**4:00-5:00 pm Imperial, in Huxley 140***Title: Toric geometry of G_2-manifolds**Abstract: G_2-manifolds are 7-dimensional and come with a Ricci-flat metric. When looking for examples, it is natural to include symmetry assumptions. This talk will focus on the case of torus symmetry and explain that one specific rank of the torus is of particular interest. I will then discuss what we currently know about these toric $G_2$-manifolds. If time permits, I shall also touch on the topic of toric Spin(7)-manifolds. The talk is based on joint work with Andrew Swann.***13 Jun.**Valentino Tosatti**1:45-2:45 pm***Title: Smooth collapsing of Ricci-flat metrics**Abstract: Consider a compact Calabi-Yau manifold with a holomorphic fibration onto a lower-dimensional space, and consider a family of Ricci-flat Kahler metrics on the total space whose Kahler class is degenerating to the pullback of a class from the base. In earlier work I proved that the metric collapse, away from the singular fibers, to a limiting metric on the base, in the locally uniform topology (and smoothly if the fibers are tori). I will describe new estimates that prove a uniform Holder bound in general, and bounds for all derivatives when the smooth fibers are isomorphic to each other. This is joint work with H.-J. Hein.*Ziyu Zhang

**3-4 pm***Title: Formality conjecture and moduli spaces of sheaves on K3 surfaces**Abstract: The formality conjecture on K3 surfaces, formulated by Kaledin and Lehn, states that on a complex projective K3 surface, the differential graded algebra RHom(F,F) is formal for any coherent sheaf F polystable with respect to an ample line bundle. In this talk, I will explain how to combine techniques from twistor spaces, dg categories and Fourier-Mukai transforms to prove this conjecture and its generalization to derived objects. Based on joint work with Nero Budur.***18 Jul.**Anton Petrunin**3-4pm KCL, room 5.20, Strand***Title: Bipolar comparison**Abstract: I will define a new type of metric comparisons and describe its relation to optimal transport, Alexandrov comparison and quotients of Hilbert space. This is a joint project with Nina Lebedeva and Vladimir Zolotov.*### KCL/UCL Geometry seminar, Winter 2018

The seminar will meet on Wednesdays between 15:00-16:00 at the room D103 in the same building as the Department of Mathematics at UCL (25 Gordon Street).

**10 Jan.**Dmitry Tonkonog*Title: Landau-Ginzburg + Gromov-Witten**Abstract: Since the work of Vianna, it is known that most Fano varieties contain an abundance of monotone Lagrangian tori up to Hamiltonian isotopy, distinguished by a disk counting invariant called the Landau-Ginzburg potential. Moreover, these potentials reveal beautiful connections to cluster algebra and mirror symmetry. I will report on recent progress towards a complete classification of the LG potentials of all tori in a given Fano variety, partially done in collaboration with James Pascaleff, and on the some interesting observations one makes on the way. For example, I will introduce new string topology operations on a smooth manifold, of gravitational descendant nature, which augment the Chas-Sullivan L-infinity algebra.***17 Jan.**Momchil Konstantinov*Title: Higher rank local systems for monotone Lagrangians**Abstract: Lagrangian Floer homology is a tool for studying intersections of Lagrangian submanifolds of symplectic manifolds. It is defined by ``counting'' pseudoholomorphic discs with boundary on these submanifolds. It can be enriched by using local coefficients to record some homotopy data about the boundaries of these disks. In this talk I will explain how one can do this under the monotonicity assumption and when the Lagrangians are equipped with local systems of rank higher than one. The presence of holomorphic discs of Maslov index 2 poses a potential obstruction to such an extension. However, for an appropriate choice of local systems the obstruction might vanish and, if not, one can always restrict to some natural unobstructed subcomplexes. I will showcase these constructions with some explicit calculations for the Chiang Lagrangian in CP^3. Its Floer theory was computed by Evans and Lekili, who also pointed out that standard Floer homology cannot tell us whether the Chiang Lagrangian and RP^3 can be disjoined by a Hamiltonian isotopy. We will see how using a rank 2 local system in this example allows us to show that these two Lagrangians are in fact non-displaceable. Time permitting I will also discuss an invariant of a single Lagrangian called monodromy Floer cohomology and its relation to the the other Floer invariants.***24 Jan.**Costante Bellettini*Title: Compactness questions for triholomorphic maps**Abstract: A triholomorphic map u between hyperKahler manifolds solves the "quaternion del-bar" equation du=I du i + J du j + K du k. Such a map turns out, under suitable assumptions, to be stationary harmonic. We focus on compactness issues regarding the quantization of the Dirichlet energy and the structure of the blow-up set. We can relax the assumptions on the manifolds, in particular we can take the domain to be merely "almost hyper-Hermitian": this more general setting leads to the weaker notion of"almost-stationarity", without however affecting our compactness results and it leads e.g. to gauge-theoretic applications. This is a joint work with G. Tian (Princeton and Beijing).***31 Jan.**Dmitri Panov*Title: Spherical metrics with conical singularities: existence and non-existence**Abstract: I'll speak about my joint work with Gabirele Mondello. We are interested in the following two questions. 1) For which n-tuples of angles there exists a genus g surface with metric of curvature 1 and conical singularities of these given angles? 2) When such a metric exists in each conformal class? I'll report on our solution of 1) and on our recent progress in 2) that sheds some light on the difficulty of this question.***7 Feb.**Pierrick Bousseau*Title: Quantum mirrors of log Calabi-Yau surfaces and higher genus curve counting.**Abstract: Gross-Hacking-Keel have given a construction of mirror families of log Calabi-Yau surfaces in terms of counts of rational curves. I will explain how to deform this construction by counts of higher genus curves to get non-commutative deformations of these mirror families. The proof of the consistency of this deformed construction relies on a recent tropical correspondence theorem.***14 Feb.**Dougal Davis*Title: A Chevalley isomorphism for the stack of principal bundles on an elliptic curve**Abstract: Let G be a simply connected simple algebraic group. The classical Chevalley isomorphism identifies the invariant theory quotient of the Lie algebra of G by the adjoint G-action with an explicit quotient of an affine space by a finite group W. The natural morphism from the Lie algebra of G to its quotient turns out to be a flat family of affine varieties with some singular fibres, and the pullback along this finite quotient admits a simultaneous resolution of singularities, called the Grothendieck-Springer resolution, which is of both geometric and representation-theoretic interest. In this talk, I will explain this classical situation in some detail before explaining a very closely related theorem for the stack Bun_G of principal G-bundles on an elliptic curve. I will also give some (mostly known) applications of this theorem, including computations of the Picard group of Bun_G and the coarse moduli space of semistable G-bundles, and a realisation of Bun_G as the total space of a family of singular stacks with an analogue of the Grothendieck-Springer resolution.***21 Feb.**Raf Bocklandt*Title: Local quivers and Morita theory for matrix factorizations**Abstract: We will discuss how techniques in Mirror symmetry developed by Cho, Hong and Lau can lead to a Morita theory for matrix factorizations and tie this to the notion of local quivers in representation theory.***28 Feb.**Peter Hintz*Title: Stability of Minkowski Space and Asymptotics of the Metric**Abstract: I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a suitable compactification of R^4 to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity; I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. This talk is based on joint work with András Vasy.***7 Mar.**Tyler Kelly*Title: Open Mirror Symmetry for the LG model x^r**Abstract: I will describe open B-model invariants in the context of Saito-Givental theory that mirror open r-spin invariants constructed by Buryak, Clader, and Tessler. This is joint work with Gross and Tessler.***14 Mar.**Marco Freibert*Title: The homogeneous spinor flow**Abstract: Recent years have seen different proposals for geometric flows of G_2-structures on compact 7d manifolds whose critical points are given by the torsion-free ones. One proposal by Weiß and Witt was to study the negative L^2-gradient flow of a Dirichlet-type energy functional for G_2. Regarding a G_2-structure as a pair consisting of a Riemannian metric g and a unit spinor field, this gradient flow has been generalized by Ammann-Weiß-Witt to the spinor flow for such pairs (g,phi) in arbitrary dimensions. The critical points of the spinor flow in dimensions higher than 2 are pairs consisting of a Ricci-flat metric g of special holonomy and a parallel spinor field. In my talk, I will describe the spinor flow in detail in general, as well as on compact homogeneous spaces, and indicate how the same formulas allow one to define it on non-compact homogeneous spaces. I will present some first results on the homogeneous spinor flow and associated solitons on certain low dimensional spaces. This is joint work with Hartmut Weiß and Lothar Schiemanowski (both at Kiel).***21 Mar.**Joel Fine*Title: Examples of compact Einstein manifolds with negative curvature**Abstract: I will describe joint work with Bruno Premoselli, giving new examples of compact Einstein manifolds. They are apparently the first such examples with negative curvature which are not simply quotients of homogeneous spaces. The manifolds which carry the metrics were constructed by Gromov and Thurston in the 80s. They found a sequence of hyperbolic 4-manifolds M_k which admit branched covers X_k, ramified along surfaces S_k, for which the normal injectivity radius of S_k tends to infinity with k. I will explain that for all sufficiently large k, X_k carries an Einstein metric of negative curvature (and yet no locally homogeneous metric). The first step is to construct an approximate solution to Einstein’s equations on X_k, using a model metric near the branch locus and the pull-back of the hyperbolic metric from M_k. The larger the normal injectivity radius, the smaller the error which therefore tends to zero as k tends to infinity. The second step is to perturb these approximate solutions to genuine ones, for all large k, via a parameter dependent implicit function theorem. The analysis involved turns out to be quite delicate. The talk will hopefully be accessible to all geometers; in particular I will assume no prior knowledge of Einstein metrics.*### KCL/UCL Geometry seminar, Fall 2017

The seminar will meet on Wednesdays between 15:00-16:00 at K-1.56 at KCL. (-1.56 is on floor

David Treumann

**minus**1).**27 Sept.**Xavier Cabré*Title: Nonlocal minimal cones and surfaces with constant nonlocal mean curvature**Abstract: The talk will be concerned with hypersurfaces of R^n with zero, or constant, nonlocal mean curvature. This is the equation associated to critical points of the fractional s-perimeter. We prove that half spaces are the only stable s-minimal cones in R^3 for s sufficiently close to 1. We will then turn to the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in R^n with constant mean curvature. Finally, we will describe results establishing the existence of periodic Delaunay-type cylinders in R^n, as well as periodic lattices made of near-spheres, with constant nonlocal mean curvature.***4 Oct.**Ana Rita Pires*Title: Infinite staircases in symplectic embedding capacity functions**Abstract: McDuff and Schlenk studied an embedding capacity function, which describes when a 4-dimensional ellipsoid can symplectically embed into a 4-ball. The graph of this function includes an infinite staircase related to the odd index Fibonacci numbers. Infinite staircases have been shown to exist also in the graphs of the embedding capacity functions when the target manifold is a polydisk or the ellipsoid E(2,3). I will describe how we use ECH capacities, lattice point counts and Ehrhart theory to show that infinite staircases exist for these and a few other target manifolds, as well as to conjecture that these are the only such target manifolds. This is a joint work with Cristofaro-Gardiner, Holm and Mandini.***11 Oct.**Ruadhaí Dervan*Title: Stable maps in higher dimensions**Abstract: Kontsevich's version of Gromov-Witten theory rests on the notion of a "stable map" from a curve to a variety. I will discuss a notion of stability for maps between arbitrary varieties, which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. I will discuss some examples, and also an analogue of the Yau-Tian-Donaldson conjecture in this setting, which relates stability to the existence of certain canonical Kähler metrics. This is joint work with Julius Ross.***18 Oct.**Wahei Hara*Title: Deformation of tilting-type derived equivalences for crepant resolutions**Abstract: We say that an exact equivalence between the derived categories of two algebraic varieties is tilting-type if it is constructed by using tilting bundles. In this talk I would like to discuss the behavior of tilting-type equivalences for crepant resolutions under deformations. As an application, we discuss the derived equivalence for stratified Mukai flops and stratified Atiyah flops in terms of tilting bundles.***25 Oct.**Alexander Polishchuk*Title: A-infinity structures associated to elliptic curves and Eisenstein-Kronecker series.**Abstract: The A-infinity structures in the title, i.e., higher products satisfying certain higher associativity constraint, arise in connection with homological mirror symmetry for elliptic curves. The main result that I will present is the expression of the structure constants of these higher products in terms of some classical functions considered by Eisenstein and Kronecker.***1 Nov.**Adam Sawicki*Title: Convexity of the momentum map and quantum entanglement**Abstract: Symplectic and algebraic geometry tools have proven to be very useful for the description of quantum correlations. They not only provide a mathematically consistent way of phrasing these problems but also offer an insight which is not available with linear algebra approach. In this talk I will discuss ideas and concepts standing behind these methods. First I will review the connection between symmetries of symplectic manifolds and the momentum map. Then I will study the situation when the considered symplectic manifold is the complex projective space of the multi-particle Hilbert space. In particular I will discuss connections with the Kirwan-Ness stratification and the Brion’s convexity theorem that lead to the concept of the entanglement polytope. Entanglement polytopes have been recently proposed as a way of witnessing multipartite entanglement classes using single particle information. I will present first asymptotic results concerning feasibility of this approach for large number of qubits.***8 Nov.**Peter Samuelson*Title: Hall algebras and the Fukaya category**Abstract: The Hall algebra is an invariant of an abelian (or triangulated) category C whose multiplication comes from "counting extensions in C." Recently, Burban and Schiffmann defined the "elliptic Hall algebra" using coherent sheaves over an elliptic curve, and this algebra has found applications in knot theory, mathematical physics, combinatorics, and more. In this talk we discuss some background and then give a conjectural description of the Hall algebra of the Fukaya category of a topological surface. This is partially motivated by an isomorphism between the elliptic Hall algebra and the skein algebra of the torus, which we also discuss. (Joint works with H. Morton and with B. Cooper.)***15 Nov.**Tristan Rivière*Title: Variational construction of minimal surfaces**Abstract:*David Treumann

**4-5 pm at S-3.20***Title: Lagrangian surfaces and Deligne-Lusztig varieties*

*Abstract:*Define an F-field on a manifold M to be a local system of algebraically closed fields of characteristic p. You can study local systems of vector spaces over this local system of fields: on a 3-manifold, they're rigid, and the rank one local systems are counted by the Alexander polynomial. On a surface, they come in positive-dimensional moduli (perfect of characteristic p), but they are more stable than ordinary local systems in the GIT sense. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On S^1 x R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it.**22 Nov.**Alex Kite*Title: Derived equivalences and the stringy Kähler moduli space**Abstract: After motivating derived categories and equivalences between them, I will describe a conjecture by physicists as to how some such equivalences are supposed to fit together according to the topology of the "stringy Kähler moduli space". Specifically, I will explain how this works in the case when you start with a torus action on a vector space (with trivial determinant) and consider all the possible Calabi-Yau GIT quotients. My motivating example will be various braid group actions on the derived category of the resolution of the A_2 surface singularity. There will be a nominal nod to mirror symmetry.***29 Nov.**Michael Wemyss*Title: Monodromy, flops and deformations**Abstract: The aim of the talk is to explain how the topology of (simplicial) hyperplane arrangements, through the work of Deligne, enshrines various derived categories in algebraic geometry with an action of the associated fundamental group. This group action encodes, roughly, the "birational geometry" piece of the derived category. The equivalences corresponding to monodromy in this action arise from objects which are not strictly spherical, but are after (noncommutative) deformation. Towards the end of the talk I will explain a technique that shows that the action is faithful. Parts are joint with Will Donovan, and parts with Yuki Hirano.***6 Dec.**Nick Lindsay*Title: Symplectic Fano manifolds with a Hamiltonian circle action**Abstract: A symplectic Fano manifold is a compact symplectic manifold where the first Chern class is a positive multiple of the cohomology class of the symplectic form. In dimension 4, these manifolds are necessarily symplectomorphic to del Pezzo surfaces, by results of Ohta, Ono and others. In dimensions 12 and above, some non-Kähler examples where found by Fine and Panov. They also conjectured that 6-dimensional symplectic Fano manifolds with a Hamiltonian circle action are symplectomorphic to Fano 3-folds. In this talk, I will discuss a joint work with Dmitri Panov, where we show that 6-dimensional symplectic Fano manifolds with a Hamiltonian circle action are simply connected. This may be interpreted as positive evidence towards the aforementioned conjecture.***13 Dec.**Luciano Mari*Title: A survey on the spectrum of Riemannian manifolds and their minimal submanifolds**Abstract: In this talk, I will describe some classical and recent results about the spectrum of the Laplace-Beltrami operator on Riemannian manifolds, focusing on noncompact ones. After an overview of the interplay between curvature and spectrum in the intrinsic case, I will then consider minimal immersions in Euclidean or hyperbolic space, and show some new criteria to ensure that the Laplace-Beltrami operator has purely discrete (respectively, purely essential) spectrum, addressing a question posed by S.T. Yau. The geometric conditions involve the Hausdorff dimension of the limit set of and the behaviour at infinity of the density function. This is based on joint works with G.Pacelli Bessa, L.P. Jorge, J.F. Montenegro, B.P. Lima, F.B. Vieira.*### KCL/UCL Geometry seminar, Summer 2017

The seminar will meet on Wednesdays between 15:00-16:00 at the Harrie Massey lecture theatre, 25 Gordon Street UCL.

Christine Breiner

**26 Apr.**Alessandro Carlotto*Title: Effective index estimates for free boundary minimal hypersurfaces and applications**Abstract: I will present a general method to obtain universal and effective index estimates for free boundary minimal hypersurfaces inside a Riemannian manifold with boundary, given an isometric embedding of the latter in some (possibly high-dimensional) Euclidean space. This approach turns out to be very powerful and provide striking results even in the special case of Euclidean domains: among other things, we prove a lower bound for the Morse index of a free boundary minimal surface, inside a weakly mean convex domain, which is linear both with respect to the genus and the number of boundary components. Applications to strong compactness theorems (without a priori area bounds required), to the explicit analysis of known examples (due to Fraser-Schoen, to Folha-Pecard-Zolotareva and to Ketover) and to novel classification theorems will also be mentioned. This is joint work with Lucas Ambrozio and Benjamin Sharp.*Christine Breiner

*(Room 500 in Dept of Maths between 16:30-17.30)**Title: Harmonic maps into metric spaces with upper curvature bounds**Abstract: We consider harmonic maps from Riemannian manifolds to metric spaces with upper curvature bounds in the sense of Alexandrov. We will present a Sacks-Uhlenbeck type result for such maps, which we prove by demonstrating that the tools harmonic replacement can be extended to this setting. This work is joint with Fraser, Huang, Mese, Sargent, Zhang.***03 May.**Michael Atiyah*Title: The Magic Square and the Kervaire Invariant**Abstract: The Freudenthal-Tits magic square arises by using the 4 classical division algebras over R for both rows and columns. I will explain how this square sheds new light on the famous topological problem of the Kervaire Invariant.***10 May.**Jon Woolf*Title: Stability conditions: phases and masses**Abstract: Each triangulated category T has an associated complex manifold Stab(T) of stability conditions. In general Stab(T) is difficult to compute, and only a few examples are understood in detail. I will discuss an emerging picture of how properties of the phase distribution of a stability condition in Stab(T) relate to the complexity of the category T. In the simplest case, roughly when all stability conditions in Stab(T) have discrete phase distributions, this can be used to show that Stab(T) is contractible. This is joint work with Yu Qiu. I will also outline an approach to partially compactifying Stab(T) by allowing the masses of objects to vanish. Conjecturally, one obtains a space stratified by components of Stab(T/N) for various thick subcategories N of T. This is joint work with Nathan Broomhead, David Pauksztello, and David Ploog.***17 May.**Fritz Hiesmayr*Title: Index and spectrum of minimal hypersurfaces arising from the Allen--Cahn construction**Abstract: The Allen--Cahn method of constructing minimal hypersurfaces has recently produced a new proof of the classical theorem that any closed Riemannian manifold of dimension at least 3 contains a minimal hypersurface embedded away from a singular set of codimension 7. In my talk I will first give an overview of the Allen--Cahn construction, and then present my work on the variational properties of two-sided hypersurfaces arising from it. Specifically, I will show that the Morse index of these Allen--Cahn hypersurfaces can be bounded above, and the spectrum of their Jacobi operator below, by the corresponding quantities for the Allen--Cahn functional.***24 May.**Colin Diemer*Title: Compactifications of group actions and derived equivalences**Abstract: A conjecture of Bondal and Orlov says that any flop between algebraic varieties induces an equivalence of the corresponding derived categories. Much progress has been made on this over the years (by Bridgeland, Kawamata, and others), but it remains largely open. On the other hand, flops and other birational maps can be studied via geometric invariant theory quotients, or even so-called birational cobordisms coming from C^* actions. This talk will discuss ways to build functors of a geometric origin by partially closing up group actions which gives a unified way of thinking about the Bondal-Orlov and related conjectures. The speaker will attempt to keep the talk geometrically minded and will assume minimal background on derived categories.***31 May.**Weiyi Zhang*Title: J-holomorphic subvarieties in symplectic 4-manifolds**Abstract: In this talk, we discuss the J-holomorphic subvarieties in a 4-dimensional symplectic manifold. A J-holomorphic subvariety is a finite collection of pairs of an irreducible J-holomorphic curve and a positive integer. We will start by showing that a subvariety of a complex rational surface in an exceptional rational curve class could have higher genus components. On the other hand, such exotic phenomenon won't happen for any tamed almost complex structures on ruled surfaces. We will then show that the moduli space of subvarieties in a sphere class resembles the linear system structures in algebraic geometry***07 Jun.**Anne Lonjou*Title: Hyperbolic spaces and the Cremona group**Abstract: The Cremona group is the group of birational transformations of the projective plane. It acts on a hyperbolic space which is an infinite dimensional version of the hyperboloid model of H^n. This action is the main recent tool to study the Cremona group. After defining it, we will study its fundamental domain, and describe some graphs naturally associated with this construction. Finally we will discuss which of these graphs are Gromov-hyperbolic.*### KCL/UCL Geometry seminar, Winter 2017

The seminar will meet on Wednesdays between 15:00-16:00 at the Gavin de Beer Lecture Theatre in the Anatomy Building at UCL.

Mariel Saez

**11 Jan.**Dan Pomerleano*Title: Two or infinity**Abstract: I will sketch a proof that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion ﬁrst Chern class, has either two or inﬁnitely many simple Reeb orbits. Key ingredients in the proof are the isomorphism between embedded contact homology and Seiberg-Witten Floer cohomology as proven by Taubes, an identity recovering the contact volume from the lengths of certain Reeb orbit sets, and the theory of global surfaces of section as developed by Hofer-Wysocki-Zehnder. This is joint work with Daniel Cristofaro-Gardiner and Michael Hutchings.***18 Jan.**Jack Smith*Title: Symmetries in monotone Lagrangian Floer theory**Abstract: Lagrangian Floer cohomology groups are nowadays viewed as morphism spaces in Fukaya categories. Despite much recent progress in understanding the abstract properties of these categories, and the rich algebraic structures which Floer groups inherit from them, it is still extremely hard to make explicit computations, and even to determine whether or not a given object is zero. In this talk I'll describe work in progress on new methods for constraining and calculating self-Floer cohomology groups of monotone Lagrangians which possess certain kinds of discrete or continuous symmetry, based on the closed-open string map and the Oh spectral sequence. This leads to examples of monotone Lagrangians which are invisible to basic Floer theory but become non-zero objects in an appropriately bulk deformed category.***25 Jan.**Paolo Cascini*Title: Effective birationality for Fano varieties**Abstract: Very recently there has been lots of progress towards the study of the anti-pluricanonical morphism for singular Fano varieties, mostly due to Birkar. The aim of this talk is to give an overview of some of these results, from different points of view. Joint work with J. McKernan.***01 Feb.**Andras Vasy*Title: The local inverse problem for the geodesic X-ray transform on tensors and boundary rigidity**Abstract: In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, I discuss the geodesic X-ray transform on a Riemannian manifold with boundary. The geodesic X-ray transform on functions associates to a function its integral along geodesic curves, so for instance in domains in Euclidean space along straight lines. The X-ray transform on symmetric tensors is similar, but one integrates the tensor contracted with the tangent vector of the geodesics. I will explain how, under a convexity assumption on the boundary, one can invert the local geodesic X-ray transform on functions, i.e. determine the function from its X-ray transform, in a stable manner. I will also explain how the analogous result can be achieved on one forms and 2-tensors up to the natural obstacle, namely potential tensors (forms which are differentials of functions, respectively tensors which are symmetric gradients of one-forms). Here the local transform means that one would like to recover a function (or tensor) in a suitable neighborhood of a point on the boundary of the manifold given its integral along geodesic segments that stay in this neighborhood (i.e. with both endpoints on the boundary of the manifold). Our method relies on microlocal analysis, in a form that was introduced by Melrose. I will then also explain how, under the assumption of the existence of a strictly convex family of hypersurfaces foliating the manifold, this gives immediately the solution of the global inverse problem by a stable `layer stripping' type construction. Finally, I will discuss the relationship with, and implications for, the boundary rigidity problem, i.e. determining a Riemannian metric from the restriction of its distance function to the boundary.*Mariel Saez

*Title: Fractional Laplacians and extension problems: the higher rank case**Abstract: The aim of this talk is to define conformal operators that arise from an extension problem of co-dimension two. To this end we interpret and extend results of representation theory from a purely analytic point of view. In the first part of the talk I will give definitions and interpretations of the fractional Laplacian and the conformal fractional Laplacian in the general framework of representation theory on symmetric spaces and also from the point of view of scattering operators in conformal geometry. In the second part of the talk I will show constructions of boundary operators with good conformal properties that generalise the fractional Laplacian in $\mathbb R^n$ using an extension problem in which the boundary is of co-dimension two. Then we extend these results to more general manifolds that are not necessarily symmetric spaces.***08 Feb.**Melanie Rupflin*Title: Sharp eigenvalue estimates on degenerating surfaces**Abstract: We consider the first non-zero eigenvalue $\lambda_1$ of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and show that the gradient of $\lambda_1$ is given essentially explicitly in terms of the dual of the differential of the degenerating length coordinate. As a corollary we obtain estimates which establish that $\lambda_1$ essentially only depends on this length coordinates with error estimates that are sharp for surfaces of genus at least 3.***15 Feb.**Nicolo Sibilla*Title: Topological Fukaya category and homological mirror symmetry**Abstract: The topological Fukaya category is a combinatorial model of the Fukaya category of exact symplectic manifolds which was first proposed by Kontsevich. In this talk I will explain recent work (joint with J. Pascaleff) on gluing techniques for the topological Fukaya category that are closely related to Viterbo functoriality. I will emphasize applications to homological mirror symmetry for three-dimensional CY Landau-Ginzburg models.***22 Feb.**Antonio De Rosa*Title: Allard's rectifiability theorem for anisotropic energies and Plateau problem**Abstract: We present our recent extension of Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand. In particular, we identify a necessary and sufficient condition on the integrand to obtain the rectifiability of every d-dimensional varifold with locally bounded first variation and positive d-dimensional density. In codimension one, this condition is shown to be equivalent to the strict convexity of the integrand with respect to the tangent plane. We can apply this result to the minimization of anisotropic energies among families of d-rectifiable closed subsets of R^n, closed under Lipschitz deformations (in any dimension and codimension). Easy corollaries of this compactness result are the solutions to three formulations of the Plateau's problem: one introduced by Reifenberg, one proposed by Harrison and Pugh and another one studied by Guy David. Finally we can apply the rectifiability theorem to the energy minimization in classes of varifolds and to a compactness result of integral varifolds in the anisotropic setting.***1 Mar.**Ian Grojnowski*Title: From exceptional groups to Del Pezzo surfaces, and simultaneous log resolutions**Abstract:***8 Mar.**Paolo Ghiggini*Title: The wrapped Fukaya category of a Weinstein manifold is generated by the cocores of the critical handles.**Abstract: A Weinstein manifold is an open symplectic manifold admitting a handle decomposition adapted to the symplectic structure. It turns out that the handles of such a decomposition have index at most half of the dimension. When the index is half the dimension, they are called critical handles and their cocores are Lagrangian discs. In a joint work with Baptiste Chantraine, Georgios Dimitroglou Rizell and Roman Golovko, we decompose any object in the wrapped Fukaya category of a Weinstein manifold as a twisted complex built from the cocores of the critical handles in a Weinstein handle decomposition. The main tools used are the Floer homology theories of exact Lagrangian immersions, of exact Lagrangian cobordisms in the SFT sense (i.e. between Legendrians), as well as relations between these theories.***15 Mar.**Alix Deruelle*Title: Kaehler Ricci expanders coming out of metric cones**Abstract: Existence of expanding self-similarities of a given evolution equation create an ambiguity in the continuation of the flow after it reached a first time singularity. We will survey various methods to prove the existence and uniqueness of such solutions in the context of the Kähler Ricci flow. (This is joint work with Ronan Conlon)***22 Mar.**Vidit Nanda*Title: Applied topology: from classifying proteins to classifying spaces**Abstract: Recent applications of algebraic-topological methods to scientific and engineering contexts have produced intriguing results: from the discovery of a new type of breast cancer, to the detection of coverage in mobile sensor networks and accurate prediction of protein compressibility directly from crystallography data. This talk will survey the underlying methodology of topological data analysis, describe its primary computational challenges, and show how a discrete version of Morse theory renders enormous computations tractable. Nothing is asked of the audience beyond a basic understanding of linear algebra and a burning desire to chase gradient trajectories between critical points.*### KCL/UCL Geometry seminar, Fall 2016

The seminar will meet on Wednesdays at KCL between 15:00-16:00. The room is S4.23 in the Strand Building, KCL (Strand Campus).

**28 Sep.**Atsuhira Nagano*Title:*Periods of toric K3 hypersurfaces and Hilbert modular surfaces*Abstract:*K3 surfaces are 2-dimensional Calabi-Yau varieties. The deformations of complex structures of K3 surfaces are controlled by its periods (Torelli’s Theorem) and the moduli spaces of K3 surfaces are given by quotient spaces of Hermitian symmetric domains of type IV. In this talk, the speaker will present results of K3 surfaces given by hypersurfaces of 3-dimensional toric varieties. We can precisely study periods of such K3 surfaces via techniques of toric varieties. Especially, we will obtain an explicit and non-trivial example of K3 surfaces whose moduli space coincides with a Hilbert modular surface.**05 Oct.**Andrea Mondino*Title:*Isoperimetric inequalities in non-smooth spaces with Ricci curvature bounded below.*Abstract:*In the first part of the talk I will recall the notion of Ricci curvature lower bounds in non-smooth spaces via optimal transport introduced by Lott-Sturm-Villani, then I will discuss some basic properties of such spaces and finally I will discuss the non-smooth extension of the celebrated Levy-Gromov isoperimetric inequality.**12 Oct.**Konstantin Shramov*Title:*Hilbert schemes and automorphisms of Fano threefolds*Abstract:*I will survey results concerning automorphism groups of smooth Fano threefolds of Picard rank 1. In particular, I will discuss the action of automorphism groups on Hilbert schemes of lines and conics, and show that the automorphism group of a smooth Fano threefold of Picard rank 1 is finite except for a handful of explicitly described cases. The talk is based on a joint work with Alexander Kuznetsov and Yuri Prokhorov.**19 Oct.**Vlad Moraru*Title:*A Rigidity Result for Area-Minimizing Cylinders*Abstract:*The study of stable minimal surfaces in Riemannian 3-manifolds with non-negative scalar curvature has a rich history. Recently, in a joint work with O. Chodosh and M. Eichmair, we proved that a Riemannian 3-manifold with non-negative scalar curvature, containing an area-minimizing cylinder, is flat. This result confirms a conjecture by Fischer-Colbrie-Schoen and Cai-Galloway.**26 Oct.**Mircea Petrache*Title:*The space of weak connections in general dimensions*Abstract:*We make a parallel between the study of connections over principal G-bundles with low regularity assumptions, and that of Nonlinear Sobolev maps, in both cases using natural geometric energies. In higher dimensions (>2 for harmonic maps or >4 in nonabelian Yang-Mills theory) vortices of nontrivial degree are not energetically prohibited, and indeed even energy-minimizers are known to form topological singularities. However while for maps these singularities are forming "holes in the graph" of the functions and there is no change in the coordinates in which the graphs have to be studied, for connections on bundles the topology of the bundles "follows" by Chern-Weil theory the structure/degree of the singularities of the connections, therefore in general (as the singularities can form a dense set) there may exist no good classical local coordinates. In joint works with Tristan Riviere we tackle this problem by using singular trivializations which "translate" changes in the bundle's topology into "jumps" of the trivialization maps, so that at first sight "any bundle becomes trivial". The new feature of this class of "weak connections on singular bundles" is that existence of minimizers always follows naturally. I will describe the approximation and slicing methods which then furnish a link to the classical Sobolev bundle regularity theory by Uhlenbeck later extended by Isobe, Meyer, Riviere, Tao, Tian. For minimizers these tools still allow to recover the classical smooth bundle structure outside a codimension-5 singular set.**02 Nov.**Melanie Rupflin (CANCELLED)*Title:*Fine properties of degenerating hyperbolic surfaces and their tangent space and applications*Abstract:*The tangent space of the set of hyperbolic surfaces splits into directions generated by pull-back with diffeomorphisms and a finite dimensional horizontal space, which can be characterized as real part of the space of holomorphic quadratic differentials. In this talk we will discuss the properties of this horizontal space, obtaining precise information in situations where one or more geodesics in the domain collapse. We furthermore discuss some applications of these result, such as the analysis of geometric flows on surfaces.**09 Nov.**Igor Burban*Title:*Fourier-Mukai transform on Weierstrass cubics and commuting differential operators*Abstract:*Any commutative subalgebra A of the algebra of ordinary differential operators admits a natural geometric invariant consisting of an irreducible (possibly singular) projective curve C (called spectral curve), a distinguished smooth point p of C and a semi-stable torsion free sheaf \mathcal{F} on C with vanishing cohomology (called spectral sheaf). In the case the rank of A is one (meaning that A contains a pair of differential operators of mutually prime orders), the algebra A can be recovered from its spectral datum (C, p, \mathcal{F}) (Krichever correspondence). All commutative subalgebras of ordinary differential operators of genus one and rank two were classified in the 80ies by Krichever & Novikov and Grünbaum. It is a natural problem to describe the spectral sheaf of such an algebra. This problem was solved by Previato and Wilson in the case the spectral curve is elliptic. However, the case of a singular spectral curve remained open. In my talk (based on a joint work with Alexander Zheglov arXiv:1602.08694) I shall explain how Fourier-Mukai transforms appear naturally in the theory of commuting differential operators, allowing to describe the spectral sheaf of a genus one commutative subalgebra.**16 Nov.**Ben Sharp*Title:*Minimal hypersurfaces with bounded index*Abstract:*An embedded hypersurface in a Riemannian manifold is said to be minimal if it is a critical point with respect to the induced area. The index of a minimal hypersurface (roughly speaking) tells us how many ways one can locally deform the surface to decrease area (so that strict local area-minimisers have index zero). We will give an overview of recent works linking the index, topology and geometry of closed and embedded minimal hypersurfaces.**23 Nov.**Johan Martens*Title:*Quantum representations and higher-rank Prym varieties*Abstract:*Riemann’s moduli space of curves can naturally be equipped with a range of bundles, whose fibres are spaces of non-abelian theta functions or, equivalently, spaces of conformal blocks. These bundles come naturally equipped with flat projective connections, in many ways mirroring an old story for (abelian) theta functions, who were classically known to satisfy a heat-equation. In some aspects however the non-abelian theta functions behave quite differently, most clearly exhibited when considering the projective representations of the mapping class group they give rise to. For a few sporadic, low-level versions this difference brakes down though, a phenomenon best understood through strange duality. In this talk we will describe the situation for rank 4, where the situation gets clarified by thinking about higher-rank Prym varieties. This is joint ongoing work with T. Baier, M. Bolognesi and C. Pauly.**30 Nov.**Victor Sanmartin-Lopez*Title:*Isoparametric hypersurfaces in anti-de Sitter space and complex hyperbolic space*Abstract:*In this talk we will present different families of isoparametric hypersurfaces in the complex hyperbolic space. Moreover, using their relation with some families of isoparametric hypersurfaces in the anti De Sitter space we deduce a classification in the complex hyperbolic space. Finally, we will see some additional results for these kind of hypersurfaces in the anti De Sitter space.**07 Dec.**Vladimir Fock*Title:*Cluster integrable systems*Abstract:*Cluster integrable systems can be viewed from five rather different points of view. 1. As a double Bruhat cell of an affine Lie-Poisson group; 2. As a space of pairs (planar algebraic curve, line bundle on it); 3. As the space of Abelian connection on a bipartite graph on a torus; 4. As a Hilbert scheme of points on an algebraic torus. 5. As a collection of flags in an infinite space invariant under the action of two commuting operators. We will see the relation between all these descriptions and discuss its consequences, quantization and possible generalizations.**14 Dec.**Daniel Alvarez-Gavela*Title:*The simplification of singularities of Lagrangian and Legendrian fronts*Abstract:*There are many situations in symplectic and contact topology where one would like a Lagrangian or Legendrian submanifold to have a front whose singularities are as simple as possible. Examples include Family Floer Homology and Legendrian Contact Homology as well as combinatorial and sheaf-theoretic setups. However, generically the singularities of Lagrangian and Legendrian fronts are terrible and there exists an obvious homotopy-theoretic obstruction to simplifying them. We will present a full h-principle (C^0-close, relative, parametric) which states that if this obstruction vanishes, then the simplification can indeed be achieved by means of an ambient Hamiltonian isotopy. The main ingredients in the proof are an improved version of the Holonomic Approximation Lemma and the observation that Lagrangian and Legendrian submanifolds can be locally wrinkled, building on the ideas of the Wrinkled Embeddings Theorem.