## Information

The KCL/UCL Junior Geometry Seminar is a joint seminar of King's College London and University College London. Speakers present topics from Algebraic Geometry, Differential Geometry, Topology, Geometric Analysis, Geometric Group Theory, and related topics.

The target audience is young researchers—in particular PhD students—from all London universities. The atmosphere is friendly and informal, most talks are accessible to a wide audience, and snacks are provided.

To receive updates about upcoming talks you can subscribe to our mailing list by emailing "subscribe" to maths-juniorgeom-subscribe@ucl.ac.uk. For inquiries please contact Kristina Kubiliūtė (kristina.kubiliute.15@ucl.ac.uk) and Daniel Platt (daniel.platt.17@ucl.ac.uk).

## Schedule

Thu 04.10.18 17.30-18.30 Location: Imperial College, Sherfield Building, CDT space, Teaching Room 3 (402) |
Title: Ouverture: the art of being a blow-up Speaker: Mirko Mauri Abstract: The main challenge of modern geometry is to describe or classify high dimensional entities. This poses the serious problem of how to visualize such objects. Although they rarely provides rigorous demonstrations, drawings and pictures are essential in building our intuition. In this opening seminar we will discuss some techniques to visualize complex algebraic varieties. In particular, we will focus on a construction of crucial interest both in algebraic geometry and symplectic topology: the blow-up. We will "see" most of its properties through pictures. Keywords: blow-up, real trace, algebraic geometry, symplectic topology, tropicalization |

Thu 11.10.18 |
No seminar |

Thu 18.10.18 17.00-18.00 Location: UCL Maths 707 |
Title: Lie groups, singularities and Springer theory Speaker: Dougal Davis Abstract: Complex simple Lie groups are groups of matrices that behave in a similar fashion to the group SL_n(C) of n x n complex matrices of determinant 1. They are classified up to isogeny by a short list of simple graphs, called Dynkin diagrams. In an apparently unrelated theorem, the very same graphs also classify a particular class of singularities on complex algebraic surfaces, called du Val singularities. In this talk, I will briefly explain these two classification theorems. I will then explain how this mysterious combinatorial coincidence is realised directly as a special case of Springer theory, which gives a general method for producing singularities (and resolutions of them) from Lie groups. Time permitting, I will also explain some of the deeper aspects of this correspondence, such as the representation theory hidden within the singularities and resolutions of Springer theory. Keywords: [Lie theory], [singularities], [algebraic geometry], [representation theory] |

Thu 25.10.18 17.00-18.00 Location: UCL Maths D103 |
Title: Introduction to Intersection Theory Speaker: Navid Nabijou Abstract: The cohomology of a smooth algebraic variety X comes equipped with a cup product. If we are given two smooth subvarieties of X, then each of them defines a Poincare dual cohomology class. The cup product of these classes can then be interpreted (under Poincare duality) as a “transverse intersection” of the subvarieties we started with. This makes sense even if the original varieties did not happen to intersect transversely. This notion - of transverse or “generic” intersection - is ubiquitous in algebraic geometry. It forms the cornerstone of modern enumerative geometry, and is central to fields as diverse as birational geometry and the theory of motives. For various reasons, the usual cohomology ring is not well-suited for the aforementioned applications. To start with, it does not behave well for singular varieties, or for varieties defined over fields other than the complex numbers. More crucially, it does not provide a way to construct so-called “refined intersection products”. In this talk I will explain, by means of examples, what the term “refined intersection product” means, and why constructing one is so important and useful. I will then explain the algebro-geometric construction of Fulton-MacPherson, and give some example applications. Keywords: algebraic geometry, intersection theory, refined intersection, Chow groups |

Thu 01.11.18 17.00-18.00 Location: UCL Maths 707 |
Title: An Introduction to Geometric Flows Speaker: Albert Wood Abstract: In this talk, I’d like to show you why geometric flows are exciting and relevant. To do this, I plan on showing you three explicit examples of popular geometric flows (Heat flow, Mean Curvature flow, and Ricci flow), and prove a variety of things with them (existence of geodesics, the Hodge theorem, and the Isoperimetric Inequality). Along the way I’ll provide flow-y art to accompany the mathematics, to accentuate the beauty of the subject! Keywords: Link to video (video file is corrupted and cannot be viewed at the moment) |

Thu 08.11.18 17.00-18.00 Location: UCL Maths 707 |
Title: Introduction to Geometric Group Theory with applications to Accessibility Speaker: David Sheard Abstract: Geometric group theory aims to study groups using tools from geometry and topology, more precisely to study finitely generated groups by looking at the way they act on geometric and topological spaces. In this talk I shall introduce some of the aims of geometric group theory, and construct a well-known space associated to a finitely presented group. I will talk about algebraic splittings of groups in the sense of van Kampen's Theorem, and finish by sketching the proof of Dunwoody's theorem that finitely presented groups admit only a finite number of such splittings over finite subgroups. Keywords: geometric group theory; fundamental group, finitely presented group, accessibility of groups; simplicial complex; group action |

Thu 15.11.18 17.00-18.00 Location: UCL Maths D103 |
Title: Introduction to Derived Categories Speaker: Vladimir Eremichev Abstract: Derived categories were discovered by Grothendieck and Verdier in 1960s and have since then became an ubiquitous tool in homological algebra and its applications to algebraic geometry, number theory, and other parts of mathematics. This talk will be a gentle introduction to the subject, based on lots of examples and very few proofs. We will start by overviewing the basics of categories, concentrating on our main examples: modules over a ring and coherent sheaves over a variety. We will then construct the corresponding derived categories and see how they greatly simplify many `scary' concepts, like Tors, Exts, spectral sequences and so on. If time permits, we will also explore how the geometry of a variety is encoded within its derived category. No prior knowledge of categories or derived functors is necessary! Keywords: derived categories, derived functors, duality, cohomology, spectral sequence, birational geometry, canonical bundle, six functors |

Thu 22.11.18 17.00-18.00 Location: UCL Maths 707 |
Title: Exceptional Holonomy: The Groups G2 and Spin(7) Speaker: Fabian Lehmann Abstract: The holonomy group of a Riemannian manifold describes parallel transport of vectors around closed loops. It is closely related to the curvature of the metric. In 1953 Berger gave a classification of all groups which can be the holonomy group of irreducible and nonsymmetric Riemannian metrics. Two of the groups in the classification, G2 and Spin(7), have been termed the exceptional holonomy groups as metrics with these holonomy groups are particularly hard to find. I will give a thorough introduction to these two groups. Keywords: #Differential Geometry #Riemannian Geometry #Holonomy #G2 #Spin(7) #Quaternions #Octonions |

Thu 29.11.18 17.00-18.00 Location: UCL Maths D103 |
Title: A Horizontal Introduction to Stacks Speaker: Damián Gvirtz Abstract: Algebraic stacks are an enlargement of the category of schemes. They most naturally appear whenever the problem of "too many automorphisms" prevents the existence of moduli spaces, e.g. for parameter spaces of curves. Instead of engaging in a hand-waving story, my talk will focus on the technical machinery behind stacks so that when the time comes you can say "quasi-separated Deligne-Mumford stack" with confidence. Keywords: fine moduli space, Grothendieck topology, 2-category, descent theory |

Thu 06.12.18 17.00-18.00 Location: UCL Maths 707 |
Title: Picard-Lefschetz fibrations Speaker: Angela Wu Abstract: Picard-Lefschetz Theory (the complex analogue of Morse Theory) was first invented for complex surfaces by Emile Picard, then generalized to higher dimensions by Solomon Lefschetz (who by the way had no hands). It studies Lefschetz fibrations: smooth proper maps from smooth oriented 2n-manifolds to the complex numbers which have finitely many isolated and non-degenerate critical points. Just like in Morse Theory, interesting things happen to the fibres of these maps near the critical values that we can use to deduce interesting things about our original manifold. And what's more, it turns out that Picard-Lefschetz Theory is, in some sense, essentially a symplectic phenomenon. In this talk, I'll introduce you to Lefschetz fibrations and show you some reasons why they are so very useful in Symplectic Geometry. Keywords: symplectic geometry, Lefschetz fibration, monodromy, Dehn twist, vanishing cycle |

Thu 13.12.18 17.00-18.00 Location: UCL Maths D103 |
Title: A journey towards Mirror Symmetry Speaker: Enrica Mazzon Abstract: The main characters of this talk are Calabi-Yau manifolds. After showing their relevance in the study of manifolds, I will explain their relation to Mirror Symmetry, a fast-moving research area at the boundary between mathematics and theoretical physics. Originated from observations in string theory, Mirror Symmetry suggests that Calabi-Yau manifolds should "come in pairs": I will explain what this means geometrically. Keywords: Calabi-Yau, quintic 3-fold, Mirror Symmetry, SYZ conjecture |

Thu 10.01.19 17.00-18.00 Location: Bush House South East Wing 1.05 |
Title: Nearly Kähler six-manifolds with two-torus symmetry Speaker: Giovanni Russo Abstract: We consider nearly Kähler six-manifolds with a two-torus symmetry. We construct a multi-moment map for the torus action and describe its general properties. At regular values, the action is necessarily free on the level sets and determines the geometry of three-dimensional quotients. Finally we give a result on an inverse construction producing nearly Kähler six-manifolds from three-dimensional data. Keywords: nearly Kähler, multi-moment maps, two-torus symmetry |

Thu 17.01.19 17.00-18.00 Location: Bush House South East Wing 1.05 |
Title: Classification of simple singularities Speaker: Erik Paemurru Abstract: We consider complex algebraic varieties and analytic spaces. We say two singularities are "equivalent" if they are biholomorphic in a neighbourhood of the singular points. A singularity of the hypersurface f = 0 is "simple" if we only get finitely many equivalence classes of singularities when changing the coefficients of f slightly. I will give a proof of the classical result that simple singularities are precisely the ADE-singularities, which are Du Val singularities in the surface case. Keywords: Milnor number, right-equivalence, contact-equivalence, splitting lemma |

Thu 24.01.19 17.00-18.00 Location: Bush House South East Wing 1.05 |
Title: Introduction to Twistor Theory Speaker: Benjamin Aslan Abstract: In this talk, we will learn in which way twistor theory builds a bridge between Riemannian and complex geometry. More precisely, every even-dimensional Riemannian manifold M can be equipped with a twistor space which parametrises certain almost complex structures on M. When M is four-dimensional, the twistor space can itself be equipped with two canoncial almost structures. We will learn how properties about these structures translate into properties of the Riemannian structure of M. Keywords: |

Thu 31.01.19 17.00-18.00 Location: Bush House South East Wing 1.05 |
Title: Many reasons to fear the Grothendieck ring of varieties Speaker: Luigi Lunardon Abstract: The Grothendieck ring of varieties is an interesting and mysterious ring. In this talk, we introduce it and try to understand something more about its properties. We see how some of its algebraic properties are closely related to many interesting geometric problems. For instance, if the class of the affine line was not a zero-divisor in this ring, it was possible to conclude that the cubic fourfold was rational. Moreover, if this was not enough, we present further geometric reasons to motivate the interest in it. After we have risen your hope that this may actually prove some conjectures, we show that the class of the affine line is a zero-divisor in the Grothendieck ring. Hopefully, even after this shocking revelation, your life will continue as usual; but you will know why this ring is something to be scared of. Keywords: Grothendieck ring of varieties, class of the affine line, zero divisor, stable birationality, Calabi-Yau |

Thu 07.02.19 17.00-18.00 Location: Bush House South East Wing 1.05 |
Title: Intersections of secant varieties to algebraic curves Speaker: Mara Ungureanu Abstract: For a smooth projective curve, the varieties parametrising its secant planes are among the most studied objects in classical enumerative geometry. In order to better understand their geometry, which in turn describes the extrinsic properties of the curve, one is lead to the study of Brill-Noether theory. This allows us to translate such extrinsic geometry problems in terms of objects belonging to the intrinsic geometry of the curve, namely subvarieties of its symmetric product. In this talk we shall introduce some basic notions of Brill-Noether theory, define secant varieties to a curve embedded in projective space and study some unexpected properties of their geometry that arise as non-transversality of intersections inside the symmetric product of the curve. Keywords: enumerative geometry, algebraic curves, symmetric product, secant planes, Brill-Noether theory |

Thu 14.02.19 17.00-18.00 Location: Bush House South East Wing 1.05 |
Title: The Topology of 4-Manifolds with Finite Fundamental Group Speaker: Johnny Nicholson Abstract: In the 1980s Freedman and Quinn showed that techniques from high-dimensional topology, such as surgery theory and the s-cobordism theorem, can be made to work for topological 4-manifolds. Whilst these techniques can be used to classify simply-connected topological 4-manifolds, similar results in the non-simply connected case are few and far between. This is largely due to the mystery surrounding the second homotopy group \pi_2(M) which has the structure of a \Z[\pi_1(M)] module. In this talk, we will give an overview of two approaches to the classification of 4-manifolds using techniques from high-dimensional topology. The first approach is to start by classifying up to homotopy and then to use classical surgery to classify up to homeomorphism within each homotopy type. Whilst a clean homotopy classification can often be obtained, the second step throws up obstructions which can only be computed for a small class of groups. The second approach, which also works for smooth 4-manifolds, is to first classify up to connected sum with S^2 \times S^2 using Kreck’s modified surgery and then to deal with the cancellation of the (S^2 \times S^2)-summands separately. In the 1990s, Hambleton and Kreck explored the connection between this cancellation problem and the problem of cancellation of a free summand in a \Z[\pi_1(M)] module and the cancellation of a wedge with S^2 in a finite two-complex. This led to a complete classification of topological 4-manifolds with cyclic or odd-order fundamental group. I will report on recent progress made on the cancellation problem for modules and complexes, and will discuss a current project to extend the known examples of non-cancellation for manifolds, i.e. examples of 4-manifolds M and N for which M \# (S^2 \times S^2) = N \# (S^2 \times S^2) but for which M and N are not even homotopic. Keywords: Manifolds, CW-complexes, modules, surgery theory, group homology, cancellation |

Thu 21.02.19 17.00-18.00 Location: Bush House South East Wing 1.05 |
Title: Ricci Flow in Milnor Frames Speaker: Syafiq Johar Abstract: In this talk, we are going to talk about the Type I singularity on 4-dimensional manifolds foliated by homogeneous S3 evolving under the Ricci flow. We review the study on rotationally symmetric manifolds done by Angenent and Isenberg as well as by Isenberg, Knopf and Sesum. In the latter, a global frame for the tangent bundle, called the Milnor frame, was used to set up the problem. We shall look at the symmetries of the manifold, derived from Lie groups and its ansatz metrics, and this global tangent bundle frame developed by Milnor and Bianchi. Numerical simulations of the Ricci flow on these manifolds are done, following the work by Garfinkle and Isenberg, providing insight and conjectures for the main problem. Some analytic results will be proven for the manifolds S1×S3 and S4 using maximum principles from parabolic PDE theory and some sufficiency conditions for a neckpinch singularity will be provided. Finally, a problem from general relativity with similar metric symmetries but endowed on a manifold with different topology, the Taub-Bolt and Taub-Nut metrics, will be discussed. Keywords: Ricci flow, Milnor frames, maximum principle, parabolic system of PDEs |

Thu 28.02.19 17.00-18.00 Location: Bush House South East Wing 1.05 |
Title: The spaces of stability conditions of the Kronecker quiver Speaker: Caitlin McAuley Abstract: It is well known that the space of stability conditions of a triangulated category is a complex manifold. In fact, mirror symmetry predicts that this space carries a richer geometric structure: that of a Frobenius manifold. From a quiver, one can construct a sequence of triangulated categories which are indexed by the integers. It is then natural to study the stability manifolds of these categories, and in particular to consider any changes to the manifolds as the integer indexing the triangulated category varies. We will study this construction for the Kronecker quiver, and discuss how the results provide evidence for a Frobenius structure on these stability manifolds Keywords: stability conditions, quiver representations, triangulated categories, Frobenius manifolds |

Thu 07.03.19 17.00-18.00 Location: Bush House South East Wing 1.05 |
Title: What's so special about the Lagrangian angle? Speaker: Christopher Evans Abstract: The mean curvature of a submanifold is a notoriously irritating quantity to compute. For Lagrangian submanifolds of Calabi-Yau manifolds however, there is a better way. In this talk, we will introduce a function called the Lagrangian angle and see how it determines the mean curvature. We'll also try and generalise our discussion to non-flat manifolds and see some of the problems therein. Keywords: |

Thu 14.03.19 17.00-18.00 Location: Bush House South East Wing 1.05 |
Title: Contraction Algebras and the Homological Minimal Model Program Speaker: Jenny August Abstract: Contraction algebras are a class of finite dimensional algebras introduced by Donovan and Wemyss as a tool to study minimal models of 3-folds or more generally, flopping contractions. In this talk, I will give an introduction to these algebras, including the key conjecture in this area which states that the derived category of such an algebra completely controls the associated geometry. I will then go on to give evidence towards this conjecture by providing a complete description of the derived equivalence class of these algebras. Keywords: derived category, flopping contractions, tilting complexes |

Thu 21.03.19 17.00-18.00 Location: Bush House South East Wing 1.05 |
Title: Seminar cancelled Speaker: Abstract: Keywords: |

Thu 25.04.19 17.00-18.00 Location: UCL, Gordon Street 25, Room: Maths 505 |
Title: Geometric Structure of Moduli Spaces of Morse-Smale Flows and Manifolds with Analytical Corners Speaker: Yixuan Wang Abstract: In 1857 Riemann observed that an isomorphism class of compact Riemann surfaces of genus g has (3g-3) parameters, which he named "die Moduln dieser Klasse" (the moduli of the class). The concept of moduli as parameters finds itself in geometry, where understanding the parameter space of objects and the variation of an object are often times vital. These moduli problems generate novel geometric structures, too. In this talk we shall see an illustrated study of moduli spaces of generic Morse-Smale flows, with a brief introduction to theories of manifolds with strange corners (so-called analytical corners in particular), and a new result on linearizing certain vector fields (analytical Morse's Lemma). Keywords: moduli problem, Morse theory, manifold with corners, analytical Morse's Lemma |

Thu 02.05.19 17.00-18.00 Location: UCL, Gordon Street 25, Room: Maths 505 |
Title: A taste of categorical geometry Speaker: Bradley Doyle Abstract: I will aim to give a brief overview of two connected ideas related to algebraic geometry and category theory. First I will give an informal description of a derived category and introduce semiorthogonal decompositions. I will give a few examples as well as mentioning some of the main results. Then I will talk about categorical or noncommutative resolutions. Kuznetsov showed a connection between categorical resolutions and semiorthogonal decompositions which I will explain. Time permitting I will finish by mentioning noncommutative resolutions for quotients by reductive groups. Keywords: algebraic geometry, category theory, derived category, semiorthogonal decomposition, noncommutative resolution |

Thu 09.05.19 17.00-18.00 Location: UCL, Gordon Street 25, Room: Maths 505 |
Title: Homological stability for Artin monoids Speaker: Rachael Boyd Abstract: Many sequences of groups satisfy a phenomenon known as homological stability. In my talk, I will report on recent work proving a homological stability result for sequences of Artin monoids, which are monoids related to Artin and Coxeter groups. From this, one can conclude homological stability for the corresponding sequences of Artin groups, assuming a well-known conjecture in geometric group theory called the K(\pi,1)-conjecture. This extends the known cases of homological stability for the braid groups and other classical examples. I will give a gentle introduction to Coxeter and Artin groups, homological stability and the K(\pi,1)-conjecture, before stating my results. Keywords: Homology, group homology, classifying space |

Thu 16.05.19 17.00-18.00 Location: UCL, Gordon Street 25, Room: Maths 505 |
Title: Introduction to Geometric Deep Learning Speaker: Mehdi Bahri Abstract: Machine Learning and Deep Learning have had a profound impact on artificial intelligence. Most of the research so far has focused on Euclidean data (images, sound, text), and even though impressive results have been achieved on many applications, important types of data cannot be processed with traditional machine learning algorithms due to their non-Euclidean structure: graphs (e.g., molecules, social networks), and manifolds (e.g., 3D shapes). This talk introduces the emerging field of Geometric Deep Learning, a novel branch of machine learning aimed at developing non-Euclidean extensions of Deep Learning algorithms. We will present motivating examples, review past and recent approaches, and touch on the future challenges and research directions in the field. Keywords: Convolution, Convolutional Neural Networks, Graphs, Manifolds |

Thu 23.05.19 17.00-18.00 Location: UCL, Gordon Street 25, Room: Maths 505 |
Title: Homological Mirror Symmetry for Invertible Polynomials Speaker: Matthew Habermann Abstract: In this talk I will describe the statement and significance of homological mirror symmetry for Landau-Ginzburg models. Specifically, I will focus on invertible polynomials, which give rise to a (perhaps more-than-usual) mysterious relationship between the algebraic geometry of singularities, and the symplectic geometry of Lefschetz fibrations. I will then describe recent advances in the subject through examples, emphasising the novelty of the techniques used. Keywords: A_\infty structures, moduli space, quiver, Fukaya-Seidel category, matrix factorisation |

Thu 30.05.19 17.00-18.00 Location: UCL, Gordon Street 25, Room: Maths 505 |
Title: Generic vanishing theorems and applications Speaker: Fabio Bernasconi Abstract: Green and Lazarsfeld discovered a Kodaira type vanishing theorem for general topologically trivial line bundles on abelian varieties over the complex numbers. This has been further refined in work by Hacon, Pareschi, Popa and Schnell by introducing the notion of GV sheaf. In this talk, I would like to explain the statement of the generic vanishing theorem and their formulation via derived category machinery. The goal would be to give a sketch of a proof by Hacon and Chen on the birational characterisation of abelian varieties. Keywords: generic vanishing, Fourier-Mukai transforms, abelian varieties |

Thu 06.06.19 17.00-18.00 Location: UCL, Gordon Street 25, Room: Maths 505 |
Title: Deep Learning on graphs: a journey from continuous manifolds to discrete networks Speaker: Michaël Defferrard Abstract: This talk will discuss learning and processing with structured data. Structured data appears in many applications: the pixels of an image are structured by a grid, weather measurements are structured by the Earth's geometry, the atoms of a molecule are structured by chemical forces, economic and demographic indicators are structured by transportation networks. Graphs will be our preferred representation of structure. We will start with the modeling of sampled manifolds by graphs, and see how the graph Laplacian converges to the Laplace-Beltrami. As an example, we'll consider the classification of cosmological observations on the sphere. We'll then move on to purely discrete domains, where the absence of a strong theory leaves freedom in the design of computational operations. After the reinterpretation of our Laplacian-based tool in a more general framework, the passing and aggregation of information across vertices, we will see some alternative formulations. As an example we'll consider the classification of scientific papers given their content and relations, defined by a citation network. Keywords: |

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