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 B3.01, Cruciform 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: *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.

**Members:** Senja Barthel, Thomas Begley, Grego Benincasa, Bjorn Berntson, Michael Betancourt, Sam Brown, Alex Cioba, Martin de Borbon, Fyodor Gainullin, Yoshi Hashimoto, Thomas Hockenhull, Julian Hodgeson, Maryam Anabelle Santiago Jamil, Elana Kalashnikov, Niko Laaksonen, Sam Livingstone, Marco Marengon, Kim Moore, Agustin Moreno, Jamil Nadim, Navid Nabijou, Raul Sanchez Galan, Tobias Sodoge, Chris Sutton, Apimook Watcharangkool.

Following more senior members kindly support the Seminar: Dr Jacqui Espina, Dr Jonny Evans, Dr Jason Lotay, Dr Oldrich Spacil.

**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.