UCL Junior Geometry Seminar

We are a group of students at UCL (and also Imperial, KCL, and Cambridge) who gather around **usually at 4pm 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 Alex (a.cioba.12_AT_ucl.ac.uk) or Yoshi (yoshinori.hashimoto.12_AT_ucl.ac.uk) to be on our email list, if you'd like to join us.

**Upcoming talks:**

**NB: **We are currently running a reading session based on Milnor-Stasheff's Characteristic Classes. We are always open to suggestions, so please do let us know of any topic that you would like to learn (or talk) about. Any volunteer to give a talk will be highly appreciated.

*Speaker: *TBA,

*Time:* TBA,

*Venue:* TBA,

*Title:* TBA.

**Past talks:**

*Speaker: *Tobias Sodoge,

*Time:* 4pm, Tuesday 10/Jun/2014,

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

*Title:* Topics in Lagrangian Floer Theory.

*Abstract:* In this talk we will take a closer look at certain aspects of the construction of Floer homology and discuss many examples along the way. I will be presenting core ideas more than going into too much depth. (So don’t be afraid of showing up even if you don't understand much of what follows, the talk should still be accessible for you)

We will first discuss action functionals on path and loop spaces and see how this gives rise to a PDE, whose space of solutions (the moduli space of J-holomorphic strips) is the key object of interest in the construction of Floer homology.

Then we will discuss the techniques (most importantly Fredholm operators, Sobolev spaces and elliptic regularity) needed to establish that the moduli space of J-holomorphic strips is (under suitable assumptions) a smooth, finite dimensional manifold of the “right” dimension. Finally I hope to give a sketch of the proof of this result.

References: Gerig's notes, Salamon's lectures, Audin-Damian's book, Oh's monograph.

Tuesday 3/Jun/2014 - **no seminar** on the 3rd of June due to the clash with MOST 2014 at KCL.

*Speaker: *Alex Cioba,

*Time:* 4pm, Tuesday 27/May/2014,

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

*Title: *Introduction to Lagrangian Floer Homology.

*Abstract:* We take a look at one of the outreaching constructions in Floer homology - roughly, infinite dimensional Morse theory. It arises naturally as a type of intersection theory for Lagrangians meant to be invariant under Hamiltonian isotopy. It has connections to other Floer theories, such as Symplectic Floer Homology and Heegaard - Floer Homology, and leads to the development of Fukaya categories and the formulation of Mirror Symmetry, a solution to the Arnold conjecture, and is at the moment the most potent tool for distinguishing Lagrangians under Hamiltonian isotopy. There will be examples and pretty pictures, mostly in the second part of the talk.

*Speaker: *Yoshi Hashimoto,

*Time:* 4pm, Tuesday 20/May/2014,

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

*Title:* Approximately holomorphic sections in symplectic geometry.

*Abstract:* The theory of divisors and line bundles is undoubtedly one of the most important machineries in algebraic geometry, and it is interesting to ask how much of this can be transplanted in symplectic geometry. An important progress in this direction is the theory of approximately holomorphic sections, developed by Donaldson, who applied them to prove several foundational results in symplectic geometry, e.g. existence of a symplectic hypersurface (which seems to be called "Donaldson hypersurface" by symplectic geometers). This talk aims to give a reasonably detailed account on the construction of approximately holomorphic sections, by assuming only the minimum knowledge on symplectic geometry. I am likely to talk for a bit more than an hour.

References: Donaldson's original paper; see also Auroux's paper for the transversality statement which I won't be able to have time to talk about.

Tuesday 13/May/2014 - **no seminar** on the 13th of May due to the clash with the Brussels-London geometry seminar.

*Speaker: *Apimook Watcharangkool,

*Time:* 3pm, Tuesday 18/Mar/2014,

*Venue:* Room 707, Department of Mathematics, UCL, 25 Gordon Street,

*Title:** *Introduction to Non-commutative geometry (2).

*Abstract:* The objective of non-commutative geometry(NCG) is to find the link between the spectrum of operator algebra and the geometrical space. The motivation of the study came from Gelfand-Naimark Theorem which will be the first topic of this talk. Then, I will give the definition of Spectral Triple and I will demonstrate(for commutative case) how this triple characterised the geometry. After that, I will give the example of non-commutative geometry and then say a few words about the (Fredholm) index of this spectral triple.

References: Updated version of Apimook's notes.

*Time:* 3pm, Tuesday 11/Mar/2014,

*Venue:* Room 707, Department of Mathematics, UCL, 25 Gordon Street,

*Title:* Discussion on the lecture given by Prof Donaldson.

*Abstract:* We shall discuss the topics treated in Prof Donaldson's lectures on the index theory. What we exactly do remains open, but most likely (1) we ask questions to each other on the materials that we didn't understand during the lecture, (2) a person with relevant background explains and supplements the necessary background knowledge for the course. Having said that, what we do will be most influenced by the participants' demands, and we welcome any suggestions from anyone interested in our discussion.

*Speaker: *Raul Sanchez Galan,

*Time:* 3pm, Tuesday 4/Mar/2014,

*Venue:* Room 707, Department of Mathematics, UCL, 25 Gordon Street,

*Title:* Reading session on Characteristic classes.

*Abstract: *I will quickly revise chapter 3 of Milnor's book and then proceed onto chapter 4 on Stiefel-Whitney classes.

References: Milnor-Stasheff's Characteristic Classes.

Tuesday 25/Feb/2014 - **no seminar** on the 25th of February due to the clash with the Brussels-London geometry seminar.

*Time:* 3pm, Tuesday 18/Feb/2014,

*Venue:* Room 707, Department of Mathematics, UCL, 25 Gordon Street,

*Title:* Discussion on the lecture given by Prof Donaldson.

*Abstract:* We shall discuss the topics treated in Prof Donaldson's lectures on the index theory. What we exactly do remains open, but most likely (1) we ask questions to each other on the materials that we didn't understand during the lecture, (2) a person with relevant background explains and supplements the necessary background knowledge for the course. Having said that, what we do will be most influenced by the participants' demands, and we welcome any suggestions from anyone interested in our discussion.

*Speaker: *Apimook Watcharangkool,

*Time:* 3pm, Tuesday 11/Feb/2014,

*Venue:* Room 707, Department of Mathematics, UCL, 25 Gordon Street,

*Title: *Introduction to Non-commutative geometry (1).

*Abstract:* The objective of non-commutative geometry(NCG) is to find the link between the spectrum of operator algebra and the geometrical space. The motivation of the study came from Gelfand-Naimark Theorem which will be the first topic of this talk. Then, I will give the definition of Spectral Triple and I will demonstrate(for commutative case) how this triple characterised the geometry. After that, I will give the example of non-commutative geometry and then say a few words about the (Fredholm) index of this spectral triple.

References: Apimook's notes.

*Time:* 3pm, Tuesday 4/Feb/2014,

*Venue:* Room 707, Department of Mathematics, UCL, 25 Gordon Street,

*Title:* Discussion on the lecture given by Prof Donaldson.

*Abstract:* We shall discuss the topics treated in Prof Donaldson's lectures on the index theory. What we exactly do remains open, but most likely (1) we ask questions to each other on the materials that we didn't understand during the lecture, (2) a person with relevant background explains and supplements the necessary background knowledge for the course. Having said that, what we do will be most influenced by the participants' demands, and we welcome any suggestions from anyone interested in our discussion.

*Speaker: *Yoshi Hashimoto,

*Time:* 3pm, Tuesday 28/Jan/2014,

*Venue:* Room 707, Department of Mathematics, UCL, 25 Gordon Street,

*Title:* Pragmatic introduction to Chern classes.

*Abstract:* This talk aims to give a rapid introduction to Chern classes, where we skip many foundational issues (which will probably be discussed in the later sessions of the Characteristic classes reading seminar) and focus on how to "use" them. The hope is that this talk may give an "overview" of what we will cover later in the Characteristic classes reading sessions. More specifically, after explaining the very useful splitting principle, we will discuss some issues related to the index theorem and how they prove interesting results. In particular, we wish to discuss Hirzebruch-Riemann-Roch formula and Rokhlin's theorem, but there may not be enough time to cover all these.

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

*Speaker: *Raul Sanchez Galan,

*Time:* 3.30pm (note the late starting time), Wednesday 11/Dec/2013,

*Venue:* B10, 18 Gordon Square,

*Title:* Reading session on Characteristic Classes.

*Abstract:* Raul will start off the reading session by explaining the first three chapters of the Milnor-Stasheff's Characteristic Classes.

References: Milnor-Stasheff's Characteristic Classes, Appendix of Kobayashi-Nomizu's textbook, Griffiths-Harris' Principles of Algebraic Geometry.

Wednesday 4/Dec/2013 - **no seminar** on the 4th of December due to the clash with the open day at the UCL.

*Speaker: *Dr Henry Wilton,

*Time:* 3pm, Wednesday 27/Nov/2013,

*Venue:* B10, 18 Gordon Square,

*Title:* Geometry & Topology of 3-manifolds.

References: Survey papers by Aschenbrenner-Friedl-Wilton, Scott.

*Speaker: *Jamil Nadim,

*Time:* 3pm, Wednesday 13/Nov/2013,

*Venue:* B10, 18 Gordon Square,

*Title:* Serre-Leray Spectral Sequence.

*Abstract:* The theory will be accompanied by many examples of computing the homology and cohomology of various spaces.

References: Jamil's notes.

*Speaker: *Yoshi Hashimoto ,

*Time:* 3pm, Wednesday 30/Oct/2013,

*Venue:* B10, 18 Gordon Square,

*Title:* Spin^c-structures and Seiberg-Witten theory.

*Abstract:* Seiberg-Witten equation first appeared as an equation for the magnetic monopoles in some sort of string theory. Mathematicians (notably Taubes) soon realised that it can be applied to the study of differential topology in dimension 4, replacing and often simplifying the (then already well-established) Donaldson theory of SU(2) instantons. The aim of this talk is to define the Seiberg-Witten invariant, applications of which can capture extremely subtle smooth structure of 4-manifolds. We will start from the basic definition of Spin(4) and no prior knowledge on differential geometry will be assumed. On the other hand, we will not touch on the global analytic aspects of the moduli space, and the applications to 4-manifold topology will not be discussed (but perhaps mentioned).

References: Moore's lecture notes, Morgan's textbook, Lawson-Michelsohn's textbook.

*Speaker: *Alex Cioba,

*Time:* 3pm, Wednesday 23/Oct/2013,

*Venue:* B10, 18 Gordon Square,

*Title:* Pseudoholomorphic curves (2).

*Speaker: *Alex Cioba,

*Time:* 3pm, Wednesday 16/Oct/2013,

*Venue:* B10, 18 Gordon Square,

*Title:* Pseudoholomorphic curves (1).

*Abstract:* We will be doing some of the basics. Our final goal (probably in a second talk) will be a celebrated theorem of Gromov, namely the uniqueness of minimal symplectic fillings for the standard contact 3-sphere. A very similar result was proved by McDuff, Eliashberg, and Floer. Its statement can be found here (Theorem 1.5). We won't be following the exposition in this paper, but it is still relevant as background.

References: Paper by McDuff.

**Seminar archive:** 2012-2013

**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, Niko Laaksonen, Sam Livingstone, Marco Marengon, Kim Moore, Jamil Nadim, Raul Sanchez Galan, Tobias Sodoge, Apimook Watcharangkool.

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

**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.