UCL Junior Geometry Seminar

We are a group of students at UCL (and also Imperial, KCL, and Cambridge) who gather around **usually at 5pm on Tuesdays **at **Room 500, Department of Mathematics at UCL** to have a friendly seminar in which we learn various aspects of Geometry and improve our presentation skills. Anyone interested in our seminar is welcome to join, irrespective of background (indeed we do vary a lot in background). Please send an email to Tobias (tobias.sodoge.13_AT_ucl.ac.uk) to be on our email list, if you'd like to join us.

**Upcoming talks:**

*Speaker: *TBA,

*Time:* TBA,

*Venue:* TBA,

*Title*: TBA.

**Past talks:**

*Speaker: *Yoshi Hashimoto (UCL),

*Time:* 5pm, Tuesday 24/Mar/2015,

*Venue:* Room 500, Department of Mathematics, UCL, 25 Gordon Street,

*Title*: Teichmüller space from the Riemannian point of view.

*Abstract:* The moduli space of Riemann surfaces, parametrising all complex structures up to biholomorphism, is a fundamentally important object in Geometry and Theoretical Physics. Teichmüller space can be regarded as the universal cover of the moduli space, and also plays an important role. This talk aims to construct it by using the methods of Riemannian geometry and elliptic PDE theory, following the lecture notes of Tromba. We shall see that various techniques that arise in moduli theory, such as gauge theory, appear here with less technical complexities. This talk hopefully serves as a leisurely introduction to these methods (and a refreshment for the connoisseurs), assuming no prior knowledge of PDE theory and only little knowledge of Riemannian geometry.

Reference: Tromba's lecture notes.

*Speaker: *Dmitry Tonkonog (Cambridge),

*Time:* 5pm, Tuesday 17/Mar/2015,

*Venue:* Room 500, Department of Mathematics, UCL, 25 Gordon Street,

*Title: * Elliptic relation for Floer cohomology.

*Abstract:* For any symplectomorphism of a symplectic manifold, one defines a graded vector space called its Floer cohomology. Two commuting symplectomorphisms give rise to actions on Floer cohomologies of each other, and the elliptic relation says that the supertraces of these two actions are equal. I will explain a sketch proof of the elliptic relation, and its application to the study of symplectic Dehn twists.

*Speaker: *Marco Marengon (Imperial),

*Time:* 5pm, Tuesday 3/Mar/2015,

*Venue:* Room 500, Department of Mathematics, UCL, 25 Gordon Street,

*Title:* An introduction to sutured Floer homology.

*Abstract:* Sutured Floer homology is a homological theory for balanced sutured manifolds that has been developed as a generalisation of Heegaard Floer homology for closed oriented connected 3-manifolds (which is in turn motivated by Lagrangian Floer homology). In the talk I will give an introduction to sutured manifolds and sutured Floer homology, trying to see how it relates to the invariants that are contained in the Heegaard Floer homology package. If there's enough time, I'll show some applications of sutured Floer homology to few topological problems.

*Speaker: *Francesca Carocci (LSGNT),

*Time:* 5pm, Tuesday 24/Feb/2015,

*Venue:* Room 500, Department of Mathematics, UCL, 25 Gordon Street,

*Title: *Deformation theory and Differential Graded Lie Algebras (DGLAs ): an example.

*Abstract: *

"In characteristic 0 each deformation problem is controlled by a suitable DGLA, uniquely determined up to quasi-isomorphism" (Deligne).

Following the above principle, about 30 years ago, people started studying a completly new approach to deformation theory ; the aim of the talk is to explain and motivate the "DGLA approach" to deformation theory. We won't go through the general theory, but we will try to study in detail a single example. We will see the DGLA which controls infinitesimal deformations of a complex manifold and give some applications of the DGLA description.

Tuesday 17/Feb/2015 - **no seminar** on the 17th of February due to the reading week at UCL.

*Speaker: *Andrea Tobia Ricolfi (University of Stavanger, Imperial),

*Time:* 5pm, Tuesday 10/Feb/2015,

*Venue:* Room 500, Department of Mathematics, UCL, 25 Gordon Street,

*Title: *Localization in Donaldson-Thomas theory.

*Abstract*: Curve counting invariants are not defined when the moduli space of curves is not compact. But when a torus acts (nicely) on the moduli space, one can use localization techniques to *define* invariants: as a reminiscence of Atiyah and Bott's famous localization formula, the *virtual* localization formula proved by Graber-Pandharipande tells us that only the torus-fixed loci will contribute to the invariant. We will see some explicit computations in the land of Donaldson-Thomas invariants, and if time permits we will explore the Gromov-Witten/Donaldson-Thomas correspondence of the local **P**^1.

*Speaker: *Navid Nabijou (Imperial),

*Time:* 5pm, Tuesday 3/Feb/2015,

*Venue:* Room 500, Department of Mathematics, UCL, 25 Gordon Street,

*Title: *Homotopy Groups of the Spheres.

*Abstract*: Although the fundamental group is a staple of any algebraic topology course, its natural higher-dimensional analogues receive far less coverage. This is in spite of the fact that the study of such groups has motivated (and indeed continues to motivate) a considerable chunk of modern algebraic topology. This talk provides an introduction to the study of these higher homotopy groups. We motivate our development of the theory via the surprisingly deep example of the homotopy groups of the n-spheres.

*Speaker: *Yoshi Hashimoto (UCL),

*Time:* 5pm, Tuesday 27/Jan/2015,

*Venue:* Room 500, Department of Mathematics, UCL, 25 Gordon Street,

*Title: *Introduction to Kaehler geometry.

*Abstract:* A smooth projective variety is a fascinating object in which algebraic geometry, differential geometry, and complex analysis play an equally important role. Moreover, these seemingly unrelated theories interact with each other at a deep level, often in an unexpected way. The final aim of this talk is to give an accessible introduction to one such interaction, namely the connection between canonical metrics (such as Kaehler-Einstein metrics) and algebro-geometric "stability". We will also discuss this problem from the viewpoint of geometric quantisation, following the approach of Donaldson. We will start from the very basics, including the definition of Kaehler metrics, so no prior knowledge on differential or algebraic geometry will be assumed.

References: survey by Fine, Szekelyhidi, and Thomas.

*Speaker: *Agustin Moreno (UCL),

*Time:* 5pm, Tuesday 9/Dec/2014,

*Venue:* B3.01, Cruciform, UCL,

*Title: *Arnold's conjecture and Floer Theory; Floer's approach.

*Abstract:* We give an overview account of Floer's classical approach to Arnold's conjecture, a very celebrated result. Lying in the crossbridge of symplectic geometry, Hamiltonian dynamics and topology, and involving algebraic and analytical techniques, this has motivated many ideas still present and exploited in today's research. The strategy consists in extending the techniques from the even more classical finite dimensional Morse theory to a suitable infinite dimensional setting, on which a (co)homology theory can be defined (i.e Floer (co)homology), and which coincides with the usual Morse homology under suitable conditions. Time permitting, we will illustrate how we can use both Lagrangian intersection Floer (co)homology, and Hamiltonian Floer (co)homology to tackle the same problem.

*Speaker: *Lars Sektnan (Imperial),

*Time:* 5pm, Tuesday 2/Dec/2014,

*Venue:* B3.01, Cruciform, UCL,

*Title: *CscK metrics and blow-ups on toric manifolds.

*Abstract:* Whether or not a constant scalar curvature Kähler (cscK) metric exists in a given Kähler class on a Kähler manifold is one of the most interesting questions in Kähler geometry. Our aim is to show in the toric setting that there is an obstruction to the existence of such a metric, and use this to show that CP^2 blown up in a point does not admit a cscK metric. We will explain what blow-ups are, how one can describe toric manifolds through polytopes and what blow-ups correspond to in this situation, before introducing a functional that must be 0 whenever the toric manifold admits a cscK metric. By showing this functional is not 0 for the blow-up of CP^2 in a fixed point, we will show that this Kähler manifold cannot admit a cscK metric.

*Speaker: *Tobias Sodoge (UCL),

*Time:* 5pm, Tuesday 25/Nov/2014,

*Venue:* B3.01, Cruciform, UCL,

*Title: *Introduction to Symplectic Geometry (part 2).

*Abstract:* After recapping briefly what I said last week I will cover the remaining parts of the talk: Give more examples of submanifolds, explain some important constructions in symplectic geometry and talk about the connections to other areas. Finally I will give a brief overview of Floer theory.

References: Introduction to Symplectic Topology , McDuff & Salamon

*Speaker: *Tobias Sodoge (UCL),

*Time:* 5pm, Tuesday 18/Nov/2014,

*Venue:* B3.01, Cruciform, UCL,

*Title: *Introduction to Symplectic Geometry.

*Abstract:* Symplectic Geometry is a relatively new and exciting field, which has relations to several other branches of mathematics. I will first explain the origins and basics of the theory and then give a brief overview of the connections to other areas of mathematics. I will then focus on the most prominent tool in symplectic geometry: Floer (co)homology. As this is an introduction, no prior knowledge of symplectic geometry will be assumed!

References: Introduction to Symplectic Topology , McDuff & Salamon

*Speaker: *Francesca Tripaldi (KCL),

*Time:* 5pm, Tuesday 11/Nov/2014,

*Venue:* B3.01, Cruciform, UCL,

*Title: *Intersection Theory mod 2 (continued).

*Abstract:* An introduction to transversality and its properties. Definition of intersection theory mod 2 and its first corollaries.

Reference: Differential Topology by Guillemin and Pollack.

Tuesday 4/Nov/2014 - **no seminar** on the 4th of November due to the room availability.

*Speaker: *Francesca Tripaldi (KCL),

*Time:* 5pm, Tuesday 28/Oct/2014,

*Venue:* B1.06, Cruciform, UCL,

*Title: *Intersection Theory mod 2.

*Abstract:* An introduction to transversality and its properties. Definition of intersection theory mod 2 and its first corollaries.

Reference: Differential Topology by Guillemin and Pollack.

*Speaker: *Yoshi Hashimoto (UCL),

*Time:* 5pm, Tuesday 21/Oct/2014,

*Venue:* B1.06, Cruciform, UCL,

*Title: *Reading Session on the characteristic classes 2.

*Abstract: *We begin the reading session by a short talk on the basics of Chern-Weil theory, which is a differential-geometric theory of writing the de Rham representative of characteristic classes in terms of the curvature. This will be followed by a discussion, in which we ask questions, discuss specific examples, talk about the applications of the theory, etc. They may come from the problem sheet.

References: Milnor-Stasheff's Characteristic Classes, textbook of Robbin-Salamon, textbook of Kobayashi, notes typeset by the speaker some years ago for a different purpose.

*Speaker: *Yoshi Hashimoto (UCL),

*Time:* 5pm, Tuesday 14/Oct/2014,

*Venue:* B1.06, Cruciform, UCL,

*Title: *Reading Session on the characteristic classes 1.

*Abstract: *We begin the reading session by a short talk on the basics of Chern classes, with a focus on the splitting principle and Hirzebruch-Riemann-Roch theorem. This will be followed by a discussion, in which we ask questions, discuss specific examples, talk about the applications of the theory, etc.

References: Milnor-Stasheff's Characteristic Classes, Chapter IV of Bott-Tu's textbook.

**Seminar archive:** 2012-2013, 2013-2014.

**Added**: This website is maintained by Yoshi Hashimoto. I welcome your comments and suggestions for our website, particularly at this formative stage. Please send them to yoshinori.hashimoto.12_AT_ucl.ac.uk.