## Geometry TeaGeometry Tea is the junior geometry seminar series in Cambridge covering all aspects of geometry. Talks are usually given by PhD students or postdocs and take place on Fridays in MR13 at 3pm at the DPMMS, followed by tea and biscuits in the Pavilion E common room. If you would like to give a talk or invite a speaker please get in touch. Expenses can be reimbursed by the department for external speakers, roughly on the level of travel from London. ## Current Year | 2016-17 | 2015-16 | 2014-15 | 2013-14## Easter Term 201827/04: David Reutter (Oxford). Hopf algebras and 3-categories It is well known that commutative Frobenius algebras can be represented as oriented surfaces, using the graphical calculus of dualizable objects in monoidal 2-categories. I will build on related ideas to show that certain Hopf algebras arise naturally in a similar way, by requiring a single 3-morphism in a 3-category to be invertible. This gives a purely geometrical proof of Mueger's version of Tannakian reconstruction of Hopf algebras from fusion categories equipped with a fiber functor. This talk is intended as an informal introduction to several topological ideas in categorical algebra, I will not assume any previous knowledge of the subject. 4/05: Michal Buran (DPMMS). Alternating quotients of right-angled Artin groups Right-angled Artin groups generalize abelian and free groups. The only relations in RAAGs is that some generators commute. We get right-angled Coxeter group as a quotient of a RAAG by killing a square of each generator. The Cayley complex associated to RAAG or RACG is a square complex and we can exploit its geometry to get an algebraic statement about these groups. Namely, we get that RAAG or RACG either is a direct product, or it surjects on alternating groups in many ways. 11/05: Yuto Yamamoto (University of Tokyo). Periods of tropical K3 hypersurfaces Let $\Delta$ be a smooth reflexive polytope in dimension 3 and $W$ be a tropical polynomial whose Newton polytope is the polar dual of $\Delta$. One can construct a $2$-sphere equipped with an integral affine structure with singularities by contracting the tropical K3 hypersurface defined by $W$. We write the complement of the singularity as $i \colon B_0 \hookrightarrow B$, and the local system of integral tangent vectors on $B_0$ as $T$. In the talk, we will give a primitive embedding of the Picard group $\mathrm{Pic} X$ of the toric variety $X$ associated with the normal fan of $\Delta$ into $H^1(B, i_\ast T)$, and compute the radiance obstruction of $B$, which sits in the image of $\mathrm{Pic} X$. 18/05: Tom Brown (DPMMS). Symplectizations and foliations in embedded contact homology In this talk I will discuss symplectizations of manifolds and how we can construct foliations of holomorphic curves which contribute to the ECH differential. If time permits I will discuss how this relates to my research. 25/05: Dexter Chua (Cambridge). When is a manifold a fiber bundle over the circle? I will discuss how you can make a manifold a fiber bundle over the circle. ## Lent Term 20182/2: Luca Pol (Sheffield). A global approach to equivariant homotopy theory In equivariant homotopy theory there are some phenomena that exist not just for a specific group but rather for a family of groups. An instance of these are certain equivariant cohomology theories: K-theory, cobordism, cohomotopy groups etc... The idea of global stable homotopy theory is to study these equivariant phenomena simultaneously without choosing a specific group. There are many way to formalize this idea, in this talk we will describe Schwede's approach via orthogonal spectra. 9/2: Albert Wood (LSGNT). Isoperimetric Inequalities and Geometric Flows Isoperimetric inequalities quantify the maximum amount of a field one can enclose, given a certain amount of fence. The answer to this problem on a nice flat field is solving by arranging the fence into a circle, but as fellow Northeners will know, fields can get quite bumpy, which demands that we pursue the subject of isoperimetric inequalities on more general manifolds. In this talk I will introduce the subjects of geometric flows (in particular Mean Curvature flow) and isoperimetric inequalities on manifolds, and delve a little into how each can help us with studying the other. 16/2: Nils Prigge (DPMMS). An outline of how manifolds relate to algebraic K-theory In this talk, I will motivate and discuss some aspects of the parametrized h-cobordism theorem, which is a deep connection between automorphisms of manifolds and algebraic K-theory. I will not assume any knowledge of algebraic K-theory. 23/2: Angel Gonzalez Prieto (UCM Madrid). Hodge structures through Topological Quantum Field Theory Topological Quantum Field Theories are powerful categorical tools that provide deep insight into the behaviour of topological invariants under gluing. In this talk, we will review some properties of TQFT , including duality and classification problems, as well as their lax counterparts and how to construct them. Using this procedure, we will construct a lax monoidal TQFT that computes the mixed Hodge structure on the cohomology of representation varieties. For that, Saito’s mixed Hodge modules will play an important role as quantizations of Hodge structures. Joint work with M. Logares and V. Muñoz. 2/3: Tomas Zeman (Oxford). Operads with homological stability and infinite loop space structures In a recent paper, Basterra, Bobkova, Ponto, Tillmann and Yeakel defined operads with homological stability (OHS) and showed that algebras over an OHS group-complete to infinite loop spaces. This can in particular be used to put a new infinite loop space structure on stable moduli spaces of high-dimensional manifolds in the sense of Galatius and Randal-Williams, which are known to be infinite loop spaces by a different method. To complicate matters further, I shall introduce a mild strengthening of the OHS condition and construct yet another infinite loop space structure on these stable moduli spaces. This structure turns out to be equivalent to that constructed by Basterra et al. It is believed that the infinite loop space structure due to Galatius--Randal-Williams is also equivalent to these two structures. 9/3: Emily Maw (LSGNT). Symplectic topology of some surface singularities When you smooth a singularity in a symplectic manifold, you introduce some extra topology in its place: a Lagrangian vanishing cycle. Hence questions about degenerations of algebraic surfaces can be rephrased in terms of Lagrangian embeddings of 2D cell complexes. We will meet certain cell complexes called "pinwheels" (vanishing cycles of Wahl singularities), whose Lagrangian embeddings in CP^2 are classified by so-called Markov numbers (by work of Evans-Smith). I will talk a bit about my work on extending this to other surfaces, and give an idea of the proof in the case of P^1 x P^1, which uses holomorphic curve techniques. The talk should be accessible to all, regardless of symplectic background (or lack thereof!). ## Michaelmas Term 201713/10: Renee Hoekzema (Oxford). Introduction to the Farrell-Jones conjecture The Farrell-Jones conjecture is an important conjecture at the intersection of topology and geometric group theory. It concerns the K- and L-theory of groups and has some interesting corollaries, for example the Borel conjecture. This talk will be a very basic introduction to the statement of the conjecture and some of its implications. It will not be assumed that people are familiar with K- and L-theory. 20/10: Jack Smith (UCL). The Fukaya-Morse algebra of a manifold Given a closed smooth manifold (and an appropriate Morse function and metric) you can define the Morse cochain complex, whose cohomology is isomorphic to that of the usual singular cochain complex. You can also define a product on the Morse complex, which induces the familiar cup product on cohomology, but in general it fails to be associative at chain level and does not encode all of the structure contained in the singular complex (e.g. Massey products). I will describe how an idea of Fukaya leads naturally to the notion of an A-infinity algebra, which is the correct weakening of the notion of associativity, and a way to build the structure of such an algebra on the Morse complex so that it captures (essentially) all of the information of the singular complex. If time permits I will also discuss how to quantise (i.e. deform) this algebra in certain ways. 27/10: Christian Lund (DPMMS). Finding distinguished metrics on manifolds A smooth manifold admits many different Riemannian structures. It is reasonable to ask if one of these could be considered distinguished or best? This is no idle question as the moduli space of Riemannian structures in general is infinite dimensional. In this talk we will explore this question and possible answers. While we do not have an affirmative answer in general, then we can identify a family of distinguished structures for which the moduli space is in fact finite dimensional.
7/11:
PhD in Geometry (4pm, MR5)
A short series of talks (~5 minutes each) by all current geometry PhDs, aimed at Part III students interested in doing a PhD in geometry at Cambridge. The (tentative) programme is as follows: 10/11: Brunella Torricelli (DPMMS). Introduction to Lagrangian Floer theory Given a symplectic manifold M, a Lagrangian submanifold is a maximal (co-)isotropic submanifold L in M. Lagrangian submanifolds play a central role in symplectic topology mainly because they provide a fruitful approach to study symmetries of symplectic manifolds. The aim of the talk is to introduce this tool -Lagrangian Floer theory- starting from the Arnold's conjecture and hopefully I will get some time to talk about the Lagrangians I deal with in my research. 17/11: Ben Morley (DPMMS). Gromov-Witten Invariants and Quantum Cohomology I plan to talk about one algebraic approach to counting curves in a variety, namely Gromov-Witten theory. Then there will be some discussion of the quantum cohomology ring and the other extra structure hidden in these invariants. If time permits I will try and mention how this theory relates to the Floer theory described last week. 24/11: Adam Baranowski (DPMMS). Detecting the Thurston norm via twisted Reidemeister torsion Given a knot we can consider two classical invariants associated to it; its genus and Alexander polynomial. The degree of the latter gives a lower bound for the genus. However, the Alexander polynomial often fails to detect the genus of a knot. This motivates a generalization of it to a twisted invariant that also depends on a choice of a representation of the knot group. I will discuss relations between such an invariant and the genus. If time permits I will discuss a further generalization to the case of links and its relation with the Thurston norm on a link complement in the 3-sphere 1/12: Jordan Williamson (Sheffield). Morita Theory in Stable Homotopy Theory Morita theory was developed in the 1950s as a tool for studying rings by studying their categories of modules. Since then, reincarnations of Morita theory for abelian categories, derived categories and stable model categories have been developed. We will outline the classical version of Morita theory, the extension to the world of stable homotopy theory, and then use this extension to show how this result can be powerful in the search for algebraic models of spectra. ## Easter Term 201728/4: Fritz Hiesmayr (Cambridge). The Allen-Cahn construction of minimal hypersurfaces The existence of embedded minimal hypersurfaces is a question that eluded geometric analysts for the better part of the twentieth century; it was eventually settled in 1981 via the so-called Almgren-Pitts construction. This construction on min-max arguments for a suitably enlarged space of hypersurfaces, and relies on machinery from geometric measure theory. My talk will instead focus on a more recent approach to the problem, where one constructs minimal hypersurfaces that arise from the level sets of solutions to an elliptic PDE, called the Allen-Cahn equation. I will present an overview of this theory that requires no prior knowledge of geometric analysis. 5/5: Nils Prigge (DPMMS). A theorem of Kontsevich on graph complexes and some applications in topology There are several interesting results in somewhat different aspects of algebraic topology that involve chain complexes that are defined via graphs and certain graph operations. For example, Watanabe produced non-trivial elements in \(\pi^{\mathbb{Q}}_*(\text{Diff}_{\partial}(D^n))\) beyond the known stable range, and Berglund and Madsen showed that they appear in the stable cohomology of certain moduli spaces. A good interpretation of the emergence of these graph complexes is via (cyclic) operads. I want to discuss this definition and some of the results by Kontsevich on their homology that relate them to some problems in topology. 12/5: Nina Otter (Oxford) The persistent homology of data Topological data analysis (TDA) is a field that lies at the intersection of data analysis, algebraic topology, computational geometry, computer science, and statistics. The main goal of TDA is to use ideas and results from geometry and topology to develop tools for studying qualitative features of data. One of the most successful methods in TDA is persistent homology (PH), a method that stems from algebraic topology, and has been used in a variety of applications from different fields, including robotics, materials science, biology, and finance. PH allows to study qualitative features of data across different values of a parameter, which one can think of as scales of resolution, and provides a summary of how long individual features persist across the different scales of resolution. In many applications, data depend not only on one, but several parameters, and to apply PH to such data one therefore needs to study the evolution of qualitative features across several parameters. While the theory of 1-parameter persistent homology is well understood, the theory of multi-parameter PH is hard, and it presents one of the biggest challenges of TDA. In this talk I will first give an introduction to persistent homology; I will then discuss some applications, and the theoretical challenges in the multi-parameter case. No prior knowledge on the subject is assumed. This talk is based on joint work with Heather Harrington, Henry Schenck, and Ulrike Tillmann. 19/5: Tom Brown (DPMMS). Heegaard Floer and Embedded Contact homology, an introduction Heegaard Floer homology, HF, is a 3-manifold invariant defined combinatorially in terms of the Heegaard diagram for the manifold. There is also a version for knots in 3-manifolds called HFK. Embedded contact homology, ECH, is a dynamic invariant defined in terms of periodic orbits of the contact Reeb vector field, and by controlling orbits near the knot a knot version, ECK, can be obtained in this setting as well. Colin, Ghiggini and Honda recently proved that the 3-manifold invariants HF and ECH are isomorphic, and it is conjectured that HFK and ECK are isomorphic. With the help of pictures and examples, I will talk about these two theories, discuss their isomorphism and, if time permits, talk about my research concerning a surgery formula in ECK analogous to a well-known formula on HFK. 26/5: Charlotte Kirchhoff-Lukat (DPMMS). Lagrangian branes and symplectic methods in generalised complex geometry Generalised complex geometry (introduced by Hitchin and Gualtieri in the early 2000s) interpolates between ordinary complex and symplectic geometry. Stable generalised complex manifolds (first introduced by Cavalcanti and Gualtieri in 2015) provide a class of examples of generalised complex manifolds that admits neither a symplectic nor a complex structure. Their generalised complex structure is, up to gauge equivalence, fully determined by a Poisson structure which is symplectic everywhere except on a real codimension 2 submanifold. I will give an introduction on how to apply symplectic techniques to this class of manifolds, and their natural submanifolds, generalised complex branes, in particular a new class of Lagrangian branes with boundary, and outline how we hope to use these to define a Fukaya category for certain types of stable generalised complex manifold. 2/6: Danica Kosanovic (MPIM Bonn). Topological Incarnations of the Arf invariant The Arf invariant is an invariant of nonsingular quadratic forms over \( \mathbb{F}_2 \). Although seemingly too simple, it appears in several contexts in geometric topology (as a concordance invariant of knots, in the Freedman-Kirby generalisation of Rokhlin theorem, as Kervaire invariant), which also hint its relation to the theory of 4-dimensional manifolds. In this introductory talk I will give an overview of spin and \(Pin^{\pm}\)-structures and associated cobordism rings in low dimensions, present a geometric interpretation of the Arf invariant and relate it to the Rokhlin invariant of homology 3-spheres. 9/6: Marco Marengon (DPMMS). An introduction to Spin\(^c \) structures Spin\(^c\) structures, which are a kind of complexification of Spin structures, turn out to be very useful in low-dimensional topology, especially when studying Heegaard Floer homology of 3-manifolds. However, the fact that there are several definitions of Spin\(^c\) structure, along with the fact that the proof of their equivalence is spread out on a number of Russian papers, can make it hard to understand them. After giving some motivations for Spin\(^c\) structures, I’ll try to go through some of the most popular definitions of them and to give an idea of why they are all equivalent. 23/6: Ronja Kuhne (Warwick). Train tracks, curves and efficient position Train tracks were introduced by Thurston in the late 1970s as a combinatorial tool for studying surface diffeomorphisms. After giving relevant background material and elaborating on the interplay between train tracks and curves on surfaces, I am going to define the notion of efficient position of curves with respect to train tracks. Efficient position can be understood as some kind of general position for curves on surfaces with respect to train tracks and I intend to address the question of its existence as well as discuss possible applications. ## Lent Term 201727/1: Mihajlo Cekić (DPMMS). Calderon problem for Yang-Mills connections We will consider the problem of identifying the connection up to gauge equivalence from the associated Dirichlet-to-Neumann map in the case of Yang-Mills connections. I will sketch the proof in the smooth case for line bundles. The approach is based on picking a special gauge in which the Yang-Mills equations become elliptic and using a unique continuation principle for elliptic systems for identification near the boundary. Along the way, I will try to explain how pseudodifferential operator symbol calculus plays its role in the proof. 3/2: James Gundry (DAMTP). Connections in Twistor Theory Twistor theory solves the self-dual 4D Einstein equations in a single-step procedure by employing complex geometry. In this talk I will review this construction and emphasise the lesser-known role played by affine connections in twistor theory. An understanding of the direct construction of such connections allows us to describe a new version of twistor theory for Newton-Cartan manifolds, in which the connections are not metric. 10/2: Daniel Lütgehetmann (FU Berlin). Configuration Spaces of Graphs The configuration space of a finite number of particles in a topological space is an object of interest in many areas of mathematics, in particular if the ambient space is a manifold. While the geometry of configuration spaces of manifolds is understood quite well, the case of particles in general simplicial complexes remains rather mysterious, even for 1-dimensional complexes. In this talk, I will describe an efficient combinatorial model for configurations of particles in a finite graph, which was first defined by Jacek Światkowski. Afterwards, I will sketch the other techniques we used to prove torsion-freeness and representation stability of those configuration spaces' homology. 24/2: Marco Golla (Uppsala). Heegaard Floer correction terms, semigroups, and plane cuspidal curves What does a low-dimensional topologist think when he sees a curve in the complex projective plane? What do semigroups have to do with Heegaard Floer homology? Come and find out the answer to these and other exciting questions! 3/3: Raúl Sánchez Galán (UCL). Monopoles in \(\mathbb{R}^3\) We will start defining and giving an overview of monopoles. Then I will explain the calculation of the virtual dimension of the moduli space of SU(2) monopoles on asymptotically conic 3-manifolds done by C. Kottke and its generalisation to SU(n). Finally I will discuss monopoles with singularities. 10/3: Manuel Krannich (University of Copenhagen). Moduli Spaces of Manifolds Initiated by the solution of the Mumford conjecture by Madsen and Weiss shedding light on the cohomology of the moduli space of smooth complex algebraic curves, tremendous progress in our understanding of manifold bundles and moduli spaces of manifolds has recently been made. In this talk, I will make the adventurous attempt to give an overview of this impressive development in geometric topology during the past two decades. 17/3: Paul Wedrich (Imperial). Skein algebra, 3-manifolds and categorification The Jones polynomial and its cousins are invariants of knots and links in the 3-sphere, which are determined by local so-called skein relations. This allows a simple definition of an invariant of oriented 3-manifolds M: the space of all framed links in M modulo the skein relations. Of particular interest are these invariants for thickened surfaces, in which case they carry an algebra structure and act on the invariants of 3-manifolds co-bounding the surface. They are also related to character varieties, quantum Teichmueller spaces and feature in several important conjectures in quantum topology. After surveying this area, I will talk about positive bases for skein algebras that were found by D. Thurston, and how they might be related to Khovanov's categorification of the Jones polynomial and its desired extension to a 4-dimensional TQFT. ## Michaelmas Term 201614/10: Tom Hockenhull (Imperial). The Alexander polynomial, its categorification and gluing The Alexander polynomial is a classical (1923) invariant for knots and links, and can be defined in a number of different, elementary ways. Link Floer homology is a less classical invariant for knots and links, and 'categorifies' the Alexander polynomial. I will try to give one definition of the Alexander polynomial, and hint at how one defines knot Floer homology and sees that it categorifies the Alexander polynomial. Time permitting, I will discuss how we might 'glue' things together in both settings. 21/10: Dr Kenny Wong (Cambridge, DAMPT). String theory on K3 surfaces This will be a gentle and non-technical introduction to moonshine - a mysterious relationship between modular forms, finite groups and K3 surfaces. 28/10: Lawrence Barrott (Cambridge). Being Vladimir Berkovich To any complex algebraic variety one can associate a complex analytic variety. This analytification preserves an incredible amount of information about the original variety. Over normed fields it is not clear how to produce a good theory of analytification, the naive definitions often product totally disconnected spaces. I will talk about one construction, due to V. Berkovich, which preserves all the information one could wish for. 4/11: Katie Vokes (Warwick). Geometry of mapping class groups The mapping class group of an orientable surface is the group of isotopy classes of orientation-preserving homeomorphisms of the surface. Given an infinite group such as the mapping class group, we are often interested in studying the large-scale geometry of this group. I will describe what is meant by the geometry of a group and outline some geometric properties of mapping class groups and how we can study them.
7/11:
PhD in Geometry (4pm, MR5)
A short series of talks (~5 minutes each) by all current geometry PhDs, aimed at Part III students interested in doing a PhD in geometry at Cambridge. The programme is as follows: 11/11: Andrea Tirelli (Imperial). Symplectic resolutions of Higgs Bundles Symplectic resolutions have been defined by A. Okounkov as the “Lie algebras of the 21ˢᵗ century”. Indeed, their study uses methods from geometry, algebra and representation theory. In the first part of the talk I will give the definition of a symplectic resolution, mention some properties and describe some examples, such as the Springer resolution and Nakajima Quiver varieties. In the second part of the talk I will describe how a recent result of Schedler and Bellamy concerning symplectic resolutions of character varieties can be used to obtain a nice result about the existence of symplectic resolution in the case of the moduli space of Higgs bundles on a smooth projective curve. 18/11: Nick Lindsay (King's College London). Hamiltonian group actions on symplectic 6-manifolds I will begin with a quick intro to Hamiltonian group actions in symplectic geometry, in particular symplectic 6-manifolds with a Hamiltonian circle action. By results of Tolman and McDuff there are only 4 examples with second Betti number equal to 1. I will discuss a conjectural description of Hamiltonian circle actions on 6-manifolds that are “Fano” in the sense that the cohomology class of the symplectic form is a positive multiple of the first Chern class.
25/11: (2.30pm, MR5) Joint with the Algebra Kinderseminar 2/12: Nils Prigge (Cambridge). Classifying spaces of categories and topological monoids There is an interesting application of methods from simplicial homotopy theory to category theory. In particular, one can construct a classifying space for every small category and it coincides with the classifying space of a group that is represented as a category with one object. I want to discuss this construction and then focus on the application to topological monoids. ## Easter Term 201610/6: Bernadette Stolz (Oxford). Application of Persistent Homology to Biological Networks Computational topology is a set of algorithmic methods developed to understand topological invariants such as loops and holes in high-dimensional data sets. In particular, a method know as persistent homology has been used to understand such shapes and their persistence in point clouds and networks. It has only been applied in biological contexts in recent years. In network science, most tools focus solely on local properties based on pairwise connections, the topological tools reveal more global features. I apply persistent homology to biological networks to see which properties these tools can uncover, which might be invisible to existing methods. In my talk I will show the use of three different methods from Computational Topology, so called filtrations: a filtration by weights, a weight rank clique filtration and a Vietoris-Rips filtration to analyse networks. The example networks I apply these tools to consist of fMRI data from neuroscientific experiments on human motor-learning and the study of schizophrenia, a mathematical oscillator model (the Kuramoto model), as well as imaging data from tumour blood vessel networks. In all cases I will show how these tools reveal insights into the biology or dynamics of the studied problems. 3/6: Giuseppe Papallo (Cambridge, DAMTP). Causality in Lovelock theories of gravity In Einstein's theory of General Relativity (GR), gravity is described in terms of curvature of the spacetime, a four-dimensional Pseudo-Riemannian manifold. The field equations, which relate the curvature of spacetime and its matter content, form a system of quasilinear second-order PDEs in the metric. Interestingly, Lovelock showed that GR is the unique geometric theory of gravity that we can write down in four dimensions, such that the field equations are second order and energy is locally conserved. However, this 'uniqueness' does not persist in higher dimensions. In fact there exist more general theories satisfying these assumptions, the so-called Lovelock theories of gravity. These theories differ from GR in that their field equations are fully non-linear and these non-linearities give rise to several interesting phenomena which don't appear in GR, such as superluminal propagation of signals or formation of shocks. I will discuss some general properties of Lovelock theories, their causal structure and the (im)possibility of constructing time machines. 27/5: Kim Moore (Cambridge). Calibrated Geometry The seminal paper of Harvey and Lawson 'Calibrated Geometries' was published in the early 1980s. Motivated by Kähler manifolds (a certain class of complex manifold) they define what it means to be a calibrated submanifold, which has the desirable property of being volume minimising in its homology class. With links to special holonomy and complex geometry and also such physical theories as string theory and M-theory, the field has received much attention from mathematicians over the years. In this talk I hope to introduce you to calibrated geometry, with motivating examples. Time permitting, I will go on to discuss some highlights of work in the field, and perhaps talk briefly about my own research. 20/5: Tim Talbot (Cambridge). Gluing asymptotically cylindrical Calabi-Yau threefolds: or, how seven dimensions are easier than six Given two asymptotically cylindrical Calabi-Yau threefolds, with matching asymptotics, we may ask whether they can be glued to give a Calabi-Yau threefold. I will explain this problem and why it is not as straightforward as we might like, and also how it can be done, using an extra dimension to simplify the problem. 13/5: Zhi Jin (Cambridge). Gromov-Witten invariants and point constraints This talk will start with an introduction to the Gromov-Witten invariants in algebraic geometry. Then the calculation of the invariants with point constraints will be illustrated by an example before we generalize the invariants and the calculation to logarithmic geometry. 6/5: Ben Barrett (Cambridge). JSJ decompositions of groups The JSJ decomposition of an object in a class of a topological spaces or groups is a maximal canonical way of cutting it up into simpler pieces. These were introduced in the study of 3-manifolds and have since been applied in more general situations. In this talk we shall first look at some general definitions and results before specialising to the case of hyperbolic groups and the Bowditch JSJ decomposition. 19/4: Claudius Zibrowius (Cambridge). On the wrapped Fukaya category of the 4-punctured sphere The starting point of this talk will be a non-glueable categorification of certain tangle invariants generalising the Alexander polynomial of knots and links. After discussing their construction and some properties, we will specialise to 2-stranded tangles and explore what a glueing formula should look like in this case. As an application, we will see why mutation of a \((-2,3)\)-pretzel tangle does not change \(\delta\)-graded knot Floer homology. This is joint work with Jake Rasmussen. ## Lent Term 201611/3: Alexander Betts (Oxford). Introduction to anabelian geometry The etale fundamental group of a scheme is a profinite group which simultaneously generalises the notion of the fundamental group of a topological space and the Galois group of a field. As a result, the etale fundamental group sees much of the Diophantine geometry of a scheme, in a sense made precise by Grothendieck's anabelian conjectures. We will introduce the notion of the etale fundamental group, and its relationship to the Diophantine geometry of curves over number fields. Time permitting, we may also introduce a suitable linearised variant, the de Rham fundamental group, as well as describing how one relativises the definition to S-schemes. 4/3: Mihajlo Cekic (Cambridge). Calderon problem for connections We consider the problem of determining the connection on a vector bundle from the knowledge of the associated Dirichlet-to-Neumann map. This problem admits a natural gauge symmetry, namely automorphisms fixing the boundary. In this talk we will discuss the approach based on Limiting Carleman Weights, that allow us to construct the Complex Geometric Optics solutions (CGO). We will shortly describe how to construct the CGOs from the Gaussian Beams, which are approximate eigenfunctions of the Laplacian that concentrate along geodesics. A reconstruction procedure will be discussed in the case of line bundles and also the interaction between the unique continuation principle and the holonomy with the Cauchy data. 26/2: Anna Barbieri (University of Pavia). Frobenius manifolds and isomonodromic families of connections A Frobenius structure on a manifold is a structure which makes the tangent bundle into an algebra, with a multiplication related to a flat bilinear pairing (metric). I will give an introduction to complex Frobenius manifolds, and we will see their relation with isomonodromic families of meromorphic connections over the projective line. 19/2: Christian Lund (Cambridge). Einstein structures on manifolds and orbifolds In this talk we study the moduli space of Einstein structures on compact manifolds. We will see how the deformation theory of Einstein metrics on compact Kähler manifolds allows us to get a better understanding of this moduli space. Time permitting, we will also tough upon the generalization of this to orbifolds. 12/2: Georgios Dimitroglou Rizell (Cambridge). An introduction to knot contact homology We give an exposition of the geometric and algebraic setup of knot contact homology. This is a powerful knot invariant due to Ekholm-Etnyre-Ng-Sullivan which is based upon the symplectic geometry of cotangent bundles, and defined using pseudoholomorphic curve counts. It is known that this invariant contains the A-polynomial of a knot and that it distinguishes the unknot inside threespace. 5/2: Charlotte Kirchhoff-Lukat (Cambridge, DAMTP). Generalised Geometry and Dorfman Brackets I will give an introduction to the generalised geometry introduced by Nigel Hitchin and his collaborators, and outline how it is applied in theoretical physics to describe closed string theory in compact spacetimes. The second part of the talk introduces extended generalised geometries and some of my own work on Dorfman brackets for extended generalised tangent bundles. 29/1: Dmitry Tonkonog (Cambridge). Laurent phenomenon and symplectic cohomology The function \(x+y+1/xy\) stays Laurent under a sequence of certain birational changes of coordinates called cluster tranformations. We will look at this phenomenon from the point of view of symplectic cohomology, observing along the way several other (classical) applications of the theory. 22/1: Jack Smith (Cambridge). The Auroux-Kontsevich-Seidel criterion The Auroux-Kontsevich-Seidel criterion is a constraint on the self-Floer cohomology of monotone Lagrangians, which is important both conceptually and in calculations. It follows from the construction of a ring homomorphism, the closed-open string map, from the quantum cohomology of the ambient symplectic manifold into Floer cohomology. In this talk I'll try to give a non-technical introduction to these ideas. 15/1: Nina Friedrich (Cambridge). Homological Stability of Moduli Spaces of High Dimensional Manifolds We start the talk with a short introduction to homological stability. For the example of moduli spaces of high dimensional manifolds we will see a way to relate this geometric setting to an algebraic one. Using a generalisation on the algebraic site we can then improve the homological stability theorem, which in the version proven by Galatius--Randal-Williams just holds for simply-connected manifolds, to a much larger class of manifolds. ## Michaelmas Term 20154/12: Ruadhai Dervan (Cambridge). Symplectic reduction and stability of points Somewhat surprisingly, there is a close link between the process of taking quotients in symplectic and algebraic geometry. This talk will discuss the Kempf-Ness Theorem, which relates the two constructions. I will not assume any previous knowledge of symplectic or algebraic geometry. 27/11: Dominic Wallis (Bath University). In quest for a non-formal \(G_2\)-manifold Does a manifold with special holonomy have to be formal? Answering the question to the affirmative has so far eluded mathematicians (to the author's knowledge). Thus the quest is on to find a counter example. Basics of formality and holonomy will be covered briefly. We examine the results that give us clues for where (mainly not) to look, such as Delign et al that Kähler manifolds, which are precisely those with holonomy \(U(n)\), are formal. We introduce the technology available to us to conduct a meaningful search. That is, our constructor of manifolds with holonomy \(G_2\), the twisted connected sum; and our formality detector, the Bianchi-Massey product. We conclude with our results to date. 20/11: Marco Marengon (Imperial). Cobordism maps in link Floer homology and reduced Khovanov homology Link Floer homology (HFL) and Khovanov homology (Kh) are two link invariants that have been extensively studied during the last 15 years, during which several interactions between the Khovanov theory and the Floer theory were proved or conjectured. In this talk we prove that a reduced version of Khovanov homology can be computed in terms of HFL groups and cobordism maps, adding this result to the list of somewhat mysterious connections between the two theories. 13/11: Alejandro Betancourt (Oxford). An introduction to Ricci solitons Ricci solitons are generalizations of Einstein metrics. They were introduced by R. Hamilton in the late 80's to study the behaviour of the Ricci flow. In this talk I will discuss some of the basic notions around Ricci solitons and see how they are used to understand the Ricci flow. In particular we will analyse the long time behaviour of the flow and study its singularities via blow-up methods. 6/11: Pierre Haas (Cambridge, DAMTP). "O Volvox, how dost thou turn thyself inside out?" Deformations of cell sheets pervade early animal development, but they arise from an intricate interplay of cell shape changes, cell migration, cell intercalation, and cell spanision. We explore this interplay of geometry and elasticity in what is perhaps the simplest instance of cell sheet folding: the "inversion" process in the green alga Volvox, whose embryos, in a process hypothesised to be driven by cell shape changes alone, must turn themselves inside out to complete their development. An elastic model, in which cell shape changes correspond to local variations of intrinsic curvature and stretches of an elastic shell, reproduces the dynamics and sheds light on the underlying mechanics of inversion. 30/10: Nicolau Sarquis (Imperial College). Geodesic networks in the Sphere We will discuss how to find closed geodesic networks in the sphere that are critical points (of a mass functional) with higher index. We explain what is the width of a manifold and try to compute it for the round sphere. Furthermore, we give a counterexample to a conjecture regarding the index and nullity of closed geodesic networks in ellipsoids. 23/10: Giulio Codogni (University of Pavia and University Roma Tre). Periods in the classical and super setting This talk will be mainly an introduction to the theory of periods in classical complex geometry. I will discuss both the global and local period map; the main examples will be the period of a curve and the period of a K3 surface. If time permits, and you are interested in, I will say a few words about super-geometry and the generalization of periods in this new set up. 16/10: Thomas Prince (Imperial College). Hodge theory for curves Hodge structures enrich the cohomology of a variety with a filtration dependent on the complex structure, providing a very rich geometric invariant and an essential tool in many areas of modern algebraic geometry. By studying hyperelliptic curves we can introduce a number of important tools and techniques very geometrically and very explicitly. We will cover topics from a subset of: Period domains and the Siegel upper half space, Picard-Fuchs equations, Torelli theorems and the Abel-Jacobi map, degenerations of Hodge structures and mixed Hodge structures from log poles. In particular I hope this will be accessible for beginning graduate students but also of general interest. 9/10: Matthias Ohst (Cambridge). Deformations of asymptotically cylindrical Cayley submanifolds Cayley submanifolds of \(\mathbb{R}^8\) were introduced by Harvey and Lawson as an instance of calibrated submanifolds, extending the volume-minimising properties of complex submanifolds in Kähler manifolds. More generally, Cayley submanifolds are 4-dimensional submanifolds which may be defined in an 8-manifold \(M\) equipped with a certain differential 4-form invariant at each point under the spin representation of \(Spin(7)\). If this 4-form is closed, then the holonomy of \(M\) is contained in \(Spin(7)\) and Cayley submanifolds are calibrated minimal submanifolds. In this talk I will present an extension of McLean's deformation theory of closed Cayley submanifolds to the setting of asymptotically cylindrical Cayley submanifolds. ## Easter Term 201512/6: Zhi Jin (Cambridge). Logarithmic geometry. -no abstract available- 5/6: Nina Friedrich (Cambridge). Introduction to homological stability. -no abstract available- 29/5: Diletta Martinelli (Imperial). Around the Cone Conjecture. -no abstract available- 22/5: Anton Isopoussu (Cambridge). Cones and fibrations in the theory of K-stability. -no abstract available- 15/5: Tom Brown (Cambridge). An introduction to Floer simple manifolds, taut foliations and a possible connection between them. -no abstract available- 8/5: Jack Smith (Cambridge). Axial discs on homogeneous Lagrangians. -no abstract available- 1/5: Ben Barrett (Cambridge). Boundaries of hyperbolic groups. -no abstract available- ## Lent Term 201513/3: Mihajlo Cekic (Cambridge). Limiting Carleman weights and related inverse problems. -no abstract available- 6/3: Jonathan Grant (Durham). The Alexander polynomial as a Reshetikhin-Turaev invariant. -no abstract available- 27/2: Alexandru Cioba (UCL). Topological methods in contact dynamics. -no abstract available- 20/2: Christian Lund (Cambridge). The moduli space of Ricci flat Kähler metrics. -no abstract available- 13/2: Ruadhai Dervan (Cambridge). Stability of twisted constant scalar curvature Kähler metrics. -no abstract available- 6/2: Paul Wedrich (Cambridge). Deformations of link homologies. -no abstract available- 30/1: Lawrence Barrott (Cambridge). Gromov-Witten invariants in algebraic geometry. -no abstract available- 23/1: Carmelo di Natale (Cambridge). Grassmannians and period mappings in derived algebraic geometry. -no abstract available- 16/1: Joe Waldron (Cambridge). The LMMP for threefolds in positive characteristic. -no abstract available- ## Michaelmas Term 201413/12: Claudius Zibrowius (Cambridge). Alexander polynomials for tangles. -no abstract available- 5/12: Lars Sektnan (Imperial). The non-existence of a Kähler-Einstein metric on \(\mathbf{CP}^2 \# (- \mathbf{CP}^2)\) via toric geometry. -no abstract available- 28/11: Anton Isopoussu (Cambridge). Analytic and algebraic aspects of moduli problems. -no abstract available- 21/11: Marco Marengon (Imperial). An introduction to sutured Floer homology. -no abstract available- 14/11: Renato Vianna (Cambridge). Distinguishing Clifford and Chekanov Lagrangian tori in \(\mathbf{C}^2\) and \(\mathbf{CP}^2\) via count of J-holomorphic discs. -no abstract available-
7/11: Joint with Junior Algebra seminar. 31/10: Yoshinori Hashimoto (UCL). Gromov-Hausdorff limits of smooth projective varieties. -no abstract available- 24/10: Peter Overholser (KU Leuven). Tropical invariants through mirror symmetry. -no abstract available- 17/10: Senja Barthel (Imperial). An introduction to spatial graph theory. -no abstract available- 10/10: Ruadhai Dervan (Cambridge). An introduction to K-stability. -no abstract available- ## Easter Term 201427/6: Claudius Zibrowius (Cambridge). Introduction to Bordered Floer Homology. -no abstract available- 13/6: Tom Lovering (Harvard University). A Hitchhiker's guide to Shimura Varieties. -no abstract available- 30/5: Diletta Martinelli (Imperial College). What is a flip? -no abstract available- 23?5: Joe Keir (DAMTP, Cambridge). Geometric Analysis in General Relativity. -no abstract available- 16/5: Michael Kasa (UCSD). Log Geometry and Stable Maps. -no abstract available- 9/5: Julien Meyer (Universite Libre de Bruxelles). CscK-Metrics and projective embeddings. -no abstract available- 2/5: Thomas Prince (Imperial College). From Lagrangian tori to Helices via string theory on Gorenstein toric singularities -no abstract available- ## Lent Term 201414/3: Jakob Blaavand (University of Oxford). What a Higgs bundle is - and why you should care. -no abstract available- 7/3: Christian Lund (Cambridge). Growth properties of manifolds with curvature. -no abstract available- 21/2: Anton Isopoussu (Cambridge). Constant scalar curvature metrics on projective bundles. -no abstract available- 14/2: Dmitry Tonkonog (Cambridge). Affine varieties versus cotangent bundles from a symplectic viewpoint. -no abstract available- 7/2: Ruadhai Dervan (Cambridge). Peaked sections and algebraic approximations of Kaehler metrics. -no abstract available- 31/1: Paul Wedrich (Cambridge). Computing categorified colored \(\mathfrak{sl}(N)\) quantum invariants of rational tangles. -no abstract available- 24/1: David Witt-Nystrom (Cambridge). Okounkov bodies and the moment map problem. -no abstract available- 17/1: Julian Holstein (Cambridge). Morita Cohomology. -no abstract available- ## Michaelmas Term 201313/12: Giulio Codogni (Cambridge). Surface singularities, representation theory and QFT. -no abstract available- 6/12: John Rizkallah (Cambridge). Affine Group Schemes (special joint junior geometry-algebra seminar). -no abstract available- 29/11: Joe Waldron (Cambridge). The cone theorem and base point freeness. -no abstract available- 22/11: Carmelo di Natale (Cambridge). DGLAs in Deformation Theory. -no abstract available- 15/11: Yoshi Hashimoto (University College London). Approximating constant scalar curvature Kähler metrics and Chow stability. -no abstract available- 8/11: Robert Pirisi (Scuola Normale Superiore). Arithmethic Stiefel-Whitney classes in algebraic geometry. -no abstract available- 1/11: Gabriele Benedetti (Cambridge). The dynamics of magnetic fields on \(S^2\). -no abstract available- 25/10: Andrea Fennelli (Imperial College). The fibres of Mori fibre spaces. -no abstract available- 18/10: John Ottem (Cambridge). Hassett's work on cubic fourfolds. -no abstract available- 11/10: Anton Isopoussu (Cambridge). Stability of flag varieties. -no abstract available- 4/10: Ruadhai Dervan (Cambridge). Multiplier ideal sheaves and Kähler-Einstein metrics. -no abstract available- |

last updated on 20th May 2018