Talks will take place fortnightly during terms. Each talk is 2 hours: the first hour is expository, with the goal of explaining a fundamental idea or computation, and is aimed at HDR students. The second hour is a research talk. You may come for one or both hours.
Schedule: Term 3
Wednesday, September 24, 1-3pm:
Location: Red Centre 3085
Speaker: Daniel Chan (UNSW)
Expository talk: Localisation and quotient categories.
Abstract: In algebraic geometry, much of the geometry can be reduced to algebra. In this talk, we look at the geometric notion of restricting to open subsets from an algebraic perspective. This "localisation" procedure involves inverting elements and gives rise to the notion of a quotient category. With this technology, we obtain a purely algebraic way of studying projective geometry which can be generalised to the noncommutative context. More precisely, quasi-coherent sheaves on projective varieties can now be studied via their Serre modules over the homogeneous coordinate ring.
Research Talk: Non-commutative projective lines and elliptic curves.
Abstract: Noncommutative algebraic geometry is based partly on Grothendieck's philosophy that to do geometry on an algebraic variety, we need only study the category of quasi-coherent sheaves on it. In this talk, we begin with a brief introduction of noncommutative projective geometry, where we look at Artin-Zhang-Polischchuk's theory of categories which can be considered categories of quasi-coherent sheaves on some noncommutative projective scheme. In particular, there is a notion of a homogeneous coordinate ring. We then give some simple examples of noncommutative analogues of projective lines and elliptic curves and explore how they are related to Ringel's notion of species and commutative elliptic curves.
This is a report on joint work with Adam Nyman.
Wednesday, October 8, 1-3pm:
Location: Red Centre 3085
Speaker: Caleb Ji (UNSW)
Expository talk: An overview of Hodge theory
Abstract: The cohomology of smooth projective varieties over the complex numbers posseses a rich structure given by Hodge theory. This structure can be extended to arbitrary complex varieties by the mixed Hodge theory of Deligne. In this talk we will review this classical theory and discuss variations of Hodge structure following work of Griffiths, Schmid, Steenbrink, and others. Time permitting, we will also touch on the non-abelian Hodge theory of Simpson as well as analogues in characteristic p and p-adic Hodge theory.
Research talk: Relative versions of the Du Bois complex
Abstract: When X is a singular complex variety, its Du Bois complex can be used in place of its de Rham complex to obtain the Hodge filtration in its mixed Hodge structure. The Du Bois complex has found numerous applications to studying singularities and vanishing theorems, and a well-behaved relative version, generalizing the relative de Rham complex, may have such applications as well. In this talk I will describe work in progress with Kovács and Taji towards constructing and using such a complex in various settings.
Wednesday, October 22, 1-3pm:
Location: Red Centre 4082
Speaker: Jie Du (UNSW)
Expository talk: Finite dimensional algebras and quantum groups
Abstract: There are two important classes of H-algebras --- Hecke algebras and (Ringel-)Hall algebras, which provide both geometric and algebraic approaches to quantum groups. In early 90s, only quantum linear groups can be constructed using Hecke algebras of symmetric groups, while the Hall algebras associated with Dynkin quivers with automorphisms provide a realization for the positive part of the corresponding quantum groups. We recorded the two beautiful theories in book form as part of mutually enriching interactions between ring theory and Lie theory.
Almost twenty years after the book publication, both theories have advanced significantly. The Hecke algebra approach has been extended to affine type A, to super type A, and to finite types B/C/D, while the Hall algebra approach has been extended to the entire quantum groups, to i-quantum groups, and to some affine cases.
In this talk, I will focus on the Hecke algebra approach through their associated q-Schur algebras of classical types and show how to use certain multiplication formulas in q-Schur algebras to construct quantum/i-quantum groups and their canonical basis theory.
Research talk: Constructing quantum queer supergroups using Hecke-Clifford superalgebras
Abstract: Using a geometric setting of q-Schur algebras, Beilinson-Lusztig-MacPherson discovered a new basis for quantum gl_n (i.e., the quantum enveloping algebra Uq(gl_n) of the Lie algebra gl_n) and its associated matrix representation of the regular module of Uq(gl_n). This beautiful work shows that the structure of the quantum linear group is hidden in the structure of Hecke algebras. The work has been generalized (either geometrically or algebraically) to quantum affine gl_n, quantum super gl_{m|n}, and recently, to some i-quantum groups of type AIII. In this talk, I will report on a completion of the work for a new construction of the quantum queer supergroup using Hecke-Clifford superalgebras and their associated queer q-Schur superalgberas.
Wednesday, November 5, 1-3pm:
Speaker: Alexander Sherman (UNSW)
Title: Rep(G), tensor categories, and semisimplification
Abstract: Representation theory is, at its most basic level, the study of the category of representations of a group G, written Rep(G). Tensor categories are a natural generalisation of Rep(G), and provide a home for powerful constructions which can in turn tell us something new about Rep(G). In the first talk, I will give a high level discussion of tensor categories with many examples, and discuss the idea of Tannakian formalism and Deligne's theorem on super-Tannakian categories over C.
In the second talk, I will focus on semisimplification, which is a construction that takes in a tensor category and outputs a semisimple quotient tensor category. This construction comes with a functor, which has a tendency to produce interesting representation-theoretic constructions in a bevy of cases. We will look at several examples of semisimplification, and if there's time I'll discuss recent work on the study of such functors for modular representations of the symmetric group and/or algebraic groups.
Term 2
Wednesday June 18, 1-3pm:
Location: Red Centre 3085
Speaker: Joe Baine
Expository talk: Representations of SL2(C)
Abstract: The first hour will consist of a crash-course in the representation theory of SL2(C). The previously uninitiated attendee should walk away knowing the classification of finite dimensional SL2(C) representations, how to explicitly construct these representations, and the rules governing the decomposition of tensor products of these representations.
Research talk: On the coefficients of the Jones-Wenzl idempotent
Abstract: The Jones-Wenzl idempotent is an element of the Temperley-Lieb algebra which plays a crucial role in numerous fields: the representation theory of SL2, knot theory, Soergel bimodules, and more. A long-standing problem has been to express this idempotent in terms of the diagrammatic basis of the Temperley-Lieb algebra. Various recursive and algorithmic approaches to this problem have appeared over the years. In this talk I will explain how the coefficient that appear have deep representation-theoretic and geometric significance. This follows from categorifying a lift of the Jones-Wenzl idempotent to the Hecke algebra. I will then explain how certain structural properties of these coefficients are expected to hold more generally.
Wednesday July 2, 1-3pm:
Location: Red Centre 3085
Speaker: Chris Hone
Expository talk: What is a sheaf?
Abstract: In this talk we'll define what a sheaf is, and spend the rest of the time giving examples to illustrate the concept, with a view towards motivating sheaf cohomology.
Research talk: Geometric extensions
Abstract: The complex points of an algebraic variety has a rich topological structure, and many results in the smooth situation may be extended to the singular case by working with intersection cohomology in the constructible derived category of sheaves. In this talk I'll define the geometric extension E(X) of a singular algebraic variety X, a formally defined object that recovers intersection cohomology over the rationals and parity sheaves in the mod p schubert variety setting. This construction also gives a definition of (p completed) intersection K theory, and a real mod two version of intersection cohomology, answering an old question of Goresky-Macpherson.
Wednesday July 16, 1-3pm:
Location: Red Centre 3085
Speaker: Thomas Le Fils
Title: Teichmüller space, mapping class groups, and translation surfaces
Abstract: Topological surfaces are classified by their number of holes, but a single surface can support many distinct geometric structures. Among these, hyperbolic structures play a central role. The space that parameterises all such hyperbolic structures is called the Teichmüller space. In the first hour, I will give an introduction to these spaces and to mapping class groups, which capture the symmetries of a surface.
In the second hour, I will introduce translation surfaces and explain their connections to Teichmüller theory. I will then present an important system of coordinates on the space parameterising translation surfaces, the period coordinates, and discuss results characterising the periods that can arise.
Wednesday July 30, 1-3pm:
Location: Red Centre 3085
Speaker: Edmund Heng
Title: From surfaces to triangulated categories: an approach to Artin—Tits groups
Abstract: Artin—Tits groups are fundamental groups of complexified hyperplane complements. Examples of these groups include the braid groups, which can be viewed as mapping class groups of punctured disks. In these cases, Teichmuller theory offers extremely powerful tools, making braid groups one of the most well-understood family among the Artin—Tits groups. While one might hope to study Artin—Tits groups via mapping class groups in a similar fashion, Wajynrib has proven a technical result that makes it hard for Artin—Tits groups (in particular, type E) to act faithfully on surfaces.
The aim of this talk is to convince you of an a priori different approach that is nevertheless highly motivated by the theory of surfaces — we study Artin—Tits groups via actions on triangulated categories. More precisely, we will build on an analogy between surfaces and triangulated category initiated by Dmitrov, Haiden, Katzarkov and Kontsevich, where the theory of Bridgeland stability conditions plays the role of Teichmuller theory.
In the first talk, I will introduce the simplest example of a triangulated category: the homotopy category of complexes over nice additive categories. In fact, all triangulated categories of interests in both talks are of this form. I will introduce an important tool called “Gaussian elimination” (coined by Bar-Natan), which allows one to simplify any complex into a minimal form up to homotopy. In our setting, this simple tool already offers a solution to the word problem for spherical type Artin—Tits groups, which moreover conjecturally works for all types.
In the second talk, I will discuss a recent progress on the categorical dynamics of spherical Artin—Tits groups. In particular, I will speak about how periodic elements have fixed points on the space of Bridgeland stability conditions — similar to how periodic elements have fixed points on the Teichmuller space, in the sense of the dynamical classification of Nielsen and Thurston. This is joint work with Oded Yacobi and Tony Licata.
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