# Hodge Club

The Hodge Club is the seminar for Hodge Institute graduate students and postdocs. That means we're interested in Algebra, Geometry, Topology, Number Theory, and all possible combinations and derivations of the four. Before the 2016/17 academic year, the Hodge Club was known as the Geometry Club.

We meet every Friday at 3:15pm in JCMB 4312 where we take it in turn to present a topic of interest to the rest of the group. Talks tend to be fairly informal and provide excellent practice for conference talks in front of a friendly audience. You can find our current schedule and a historical list of talks below.

Future events are circulated on our mailing list and advertised on the Graduate School calendar. See instructions below on how to join our mailing list.

The Hodge Club is organised by Tim Weelinck and Matt Booth.

### Current schedule of talks for 2016/17:

##### Semester 2

January 20, 2:45pm | Tim Weelinck | Characteristic Classes for Beginners |

Abstract: Characteristic classes provide a cohomological measure to `how twisted` a vector bundle is. By now characteristic classes are widely recognised to be very important objects, in and outside of geometry/topology. Applications include the topological classification of line bundles, obstruction theory, non-embedding theorems, bordism questions and (of course) mathematical physics (e.g. instanton theory). Unfortunately this subject often doesn't get the attention it deserves in pre-PhD education, whereas PhD students are expected to know what they are. To remedy this we will provide a gentle introduction to characteristic classes. Gentle means focussing on examples and intuition. By the end of the talk everyone (!) will be able to compute the Euler classes of all their favourite (orientable) bundles. | ||

January 27, 2:45pm | Simon Crawford | Singularity categories and why they're useful |

Abstract: Singularity categories were introduced by Buchweitz in '86, and one of their uses nowadays is to study the "singularities" of noncommutative rings which are "singular". In this talk, I will focus my attention on a category which is equivalent to the singularity category, and make more precise the notion of a singular ring. I will also explicitly calculate some singularity categories in some nice cases. For those of you going to my Maximals talk on Tuesday, this talk can be viewed as the pre-seminar. | ||

February 3, 2:45pm | Tim Weelinck | Chern-Weil Theory |

Abstract: Chern classes are characteristic classes associated to complex vector bundles. They are ubiquitous throughout mathematics: appearing in obstruction theory, the Atiyah-Singer index theorem, classification of line bundles, they provide a bridge from K-theory to cohomology, have important analogues in algebraic geometry, etc, etc. Besides being important, Chern classes have the excellent property of being computable. In fact, there is a simple algorithm to compute Chern classes. This algorithm is what is called Chern-Weil theory. In this talk I will describe the algorithm, and apply it in various examples. | ||

February 10, 2:45pm | Juliet Cooke | Complete Segal spaces and what they are good for |

Abstract: What is the connection between categories and topological spaces? In this talk I shall discuss an answer to this question which involves nerves and the classifying space of a category, Segal spaces and completeness. I shall end by extending our considerations to higher categories. | ||

February 17, 3:15pm | Sjoerd Beentjes | Crepant resolutions and Donaldson-Thomas invariants |

Abstract: The crepant resolution conjecture is a conjecture in enumerative geometry, the geometry of counting things. It originates from string theory and is concerned with counting curves on three-dimensional Calabi-Yau varieties and resolutions of singularities. In the first half of the talk, I'll sketch the setting of the conjecture by giving a concrete example and explaining how we count curves using Donaldson-Thomas invariants. This is the "pretalk" for the second half in which I'll discuss the statement of the conjecture and look at a number of techniques that could help one prove it. The second half will be presented at a conference, so any comments and feedback will be much appreciated. | ||

March 3, 3:15pm | Roberto Fringuelli | Compactifications of reductive groups and moduli space of bundles. |

Abstract: Constructing a (suitable) compactification of an algebraic group is a classical problem. Finding such compactifications in the case the group is abelian, or with trivial center, has let to important constructions. In the first case, we have the theory of toric varieties. In the second one, we have the De Concini-Procesi wonderful compactification of a reductive group with trivial center. In this talk, we present the Martens-Thaddeus compactification of reductive groups as moduli space of principal bundles on chains of projective lines, which includes the examples above as special cases. | ||

March 10, 3:15pm | Ruth Reynolds | Point Modules and a Noncommutative Projective Variety |

Abstract: In commutative algebra, many things can be learnt about the algebra by looking at associated geometric objects; for instance taking the proj of a graded algebra. In this talk we describe two methods to associate a geometric object to a noncommutative algebra : point modules and a noncommutative projective variety. The former provide many good examples that are easy to calculate and the latter is a generalisation of the proj construction. | ||

March 17, 3:15pm | Trang Nguyen | Introduction to derived algebraic geometry |

Abstract: In this talk, we give an informal introduction to derived algebraic geometry. We introduce the derived versions of schemes, algebraic stacks, and discuss some examples (derived critical locus, derived loop space). We will also discuss the HKR theorem at the end if time permits. | ||

March 31, 3:15pm | Matt Booth | Threefold Flops and the Contraction Algebra |

Abstract: The minimal model program is an attempt to classify varieties up to birational equivalence. More precisely, it looks to find `nice' representatives in each birational equivalence class. In dimensions 1 and 2, minimal models are (more or less) unique up to isomorphism, but in dimension 3 this fails. It turns out that any minimal models of a given threefold are linked by special codimension two surgery operations called flops. Classically, it's quite hard to determine when a curve in a threefold flops, outside of simply computing all of the geometry. I'll describe a new homological invariant, the contraction algebra of Donovan and Wemyss, that detects when a curve is floppable. | ||

April 7, 3:15pm | Jenny August | The Homological Minimal Model Program |

Abstract: Last week, Matt told us about the Minimal Model Program where the aim is to find some “nicest approximation” of a given variety. Each surface has a unique minimal model but for 3-folds this is not generally the case and finding all the minimal models geometrically is often very challenging. In this talk, we see how we can attach algebraic objects to each minimal model which somehow store some of the geometric data. Further, using tools from cluster tilting theory, we obtain an algorithm to produce all minimal models from a given one. | ||

April 14, 3:15pm | Chunyi Li | Brill-Noether theory on curves |

Abstract: Brill-Noether theory aims to describe all the ways in which a curve (smooth over C) may be mapped to projective space P^r. Equivalently, this is to study the space of line bundles (on C) that carry r+1 global sections. I’ll introduce this topic via examples of curves with small genus. I’ll describe the main classical results of Brill-Noether theory, as well as some recent result via the method of stability conditions. |

##### Semester 1

October 7, 2:45pm | Matt Booth | Stable Homotopy Theory |

Abstract: Stable homotopy theory is an important area of modern algebraic topology. It was observed back in the 1930s that the process of iteratively taking suspensions of a space 'stabilises' in some sense. The first real result in this direction was the Freudenthal suspension theorem. In this talk I'll start from the very basics of homotopy theory, and then discuss what I really mean by stabilisation. I'll talk about the modern viewpoint of spectra, and try to explain how spectra are related to generalised (co)homology theories. I'll start from the basics - very little knowledge of algebraic topology will be necessary! | ||

October 14, 2:45pm | Igor Krylov | Birational geometry of del Pezzo fibrations of degree 2 |

Abstract: I will give an introduction to birational geometry and classification of algebraic varieties. I will explain the role of canonical bundle in these problems. I will define Mori fiber spaces, give examples in dimension 2 and 3 and state results on del Pezzo fibrations: subclass of Mori fiber spaces. | ||

October 21, 2:45pm | Jenny August | Introduction to Tilting Theory |

Abstract: This talk aims to give an overview of tilting theory; an area of mathematics which grew out of an attempt to generalise a result determining when two algebras have equivalent module categories. I will start by talking about this motivation and then introduce the key concepts of tilting modules and torsion pairs which together make up what is now called classical tilting theory. I will then discuss some of the many ways this has been generalised further and hopefully finish with a result that shows, although these generalisations may seem completely different, they are actually all strongly connected. | ||

October 28, 2:45pm | Graham Manuell | Monsky's theorem on square equidissection |

Abstract: An equidissection of a polygon is a partition of the polygon into triangles of equal area. It is easy to find equidissections of squares into an even number of triangles, but odd equidissections prove to be more elusive. Indeed, Monsky's theorem states that no odd equidissections of squares exist. Surprisingly, the proof makes use of results from combinatorics and valuation theory. In particular, the p-adic numbers feature in an essential way. I will discuss this proof and the necessary background.
| ||

November 4, 2:45pm | Juliet Cooke | The Symplectic Structure of Character Varieties |

Abstract: This talk is an introduction to character varieties, and how to give a symplectic structure to them. A character variety is roughly a space of representations of the fundamental group of a space, and is intimately tied to the classification of Riemann surfaces via moduli theory; in particular, the Teichmuller space of a Riemann surface and the moduli space of stable vector bundles. | ||

November 11, 2:45pm | Matt Booth | Stable Homotopy Theory, Part 2 |

Abstract: I'll start with a bit more on the Freudenthal suspension theorem (since there was a bit of confusion last time) before asking how we can 'stabilise' topological spaces, which leads us directly into the world of spectra. I'll give a bunch of examples of spectra, and explain how a spectrum is more or less the same thing as a generalised cohomology theory. | ||

November 18, 2:45pm | Carlos Zapata-Carratala | "A mathematician and a physicist walk into a bar..." - The Mathematical Foundations of Modern Physical Theories. |

Abstract: This talk aims to give an account of modern physical theories in a conceptually clear and mathematically rigorous way. More specifically, a unifying mathematical framework for phase space models (essentially, models that try to describe the time evolution of a physical system) will be presented. In the eyes of the speaker, this framework appears remarkably elegant since, with a relatively short list of axioms based in clear physical intuitions, we can regard the dynamical content of theories as seemingly different as classical mechanics, classical field theories and quantum mechanics as natural examples of the same mathematical theory. Then we discuss how relativity goes beyond this phase space description to account for the existence of different observers. Similar to the case of phase space models, relativity, understood as space-time physics, is then framed again in a very elegant mathematical theory. The mathematical content of the talk will draw concepts from differential geometry, more concretely, Poisson and pseudo-Riemannian geometry. | ||

November 25, 2:45pm | Veronika Breunhölder | The problem with quantum gravity |

Abstract: Quantum gravity is often considered the holy grail of modern theoretical physics. What is rarely talked about is why it is actually so hard to find such a theory. The reason for that is that naively quantising gravity, one gets something called a “non-renormalisable” quantum field theory. In my talk I will try to explain what that means - no prior knowledge of QFTs required. |

### Historical schedules

Geometry club 2015/16

Geometry club 2014/15

Geometry club 2013/14

Geometry club 2012/13

You can also visit the old Geometry club website for more historical schedules

### Mailing list

Announcements are handled by the mailing list hodgeclub. To subscribe, send a message to sympa at mlist.is.ed.ac.uk with the following content:

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