Pure Mathematics Seminar 2018 Semester 1

The Pure Mathematics Seminar meets Fridays 3:15pm-4:15pm in Peter Hall Room 213.

Organisers: Jesse Gell-Redman, Ting Xue

Schedule

June 8 Henry Segerman (Oklahoma State)

Title: From veering triangulations to pseudo-Anosov flows

Abstract: Agol introduced veering triangulations of mapping tori, whose combinatorics are canonically associated to the pseudo-Anosov monodromy. Guéritaud and Agol generalised an alternative construction to any closed manifold equipped with a pseudo-Anosov flow without perfect fits. Using Mosher’s dynamic pairs, we prove the converse, showing that veering triangulations are a perfect combinatorialisation of such flows.

May 25 Asilata Bapat (ANU)

Title: Perverse sheaves on hyperplane arrangements and gluing

Abstract: A hyperplane arrangement cuts up a vector space into several pieces. The combinatorics and topology of this subdivision is encoded in the associated abelian category of perverse sheaves. This category has an alternate algebraic description due to Kapranov and Schechtman, in terms of representations of a quiver with relations. I will first explain this description and some further simplifications. I will then focus on gluing, or “recollement”, which is a recipe to reconstruct the category of perverse sheaves on a space from an open subset and its complement. I will describe how recollement on the above category of perverse sheaves translates to the category of quiver representations.

May 18 Stephan Tillmann (Sydney)

Title: Three angles on tropical geometry

Abstract: The field of tropical geometry is relatively young and developed in several areas of mathematics and computer science independently. I begin with a classical source of tropical geometry: Newton’s work on “sketching” polynomials. This motivates two different, but related viewpoints that are useful in describing limits of geometric objects, as well as the computation and enumeration of algebraic sets. As an application of tropical convexity, I will describe work in progress with Dominic Tate on describing limits of real projective structures of surfaces as singular Euclidean structures modelled on buildings.

May 23 1:15-2:15pm Cheng-Chiang Tsai (Stanford)

Special location: Peter Hall 107

Title: Springer theory and characters

Abstract: A finite group of Lie type is the group of elements of a reductive algebraic groups defined over a finite field. For example, all finite simple groups are finite groups of Lie type except for a finite list of exceptions. We will begin by describing the basic elements of the theory of characters of finite groups of Lie type, and how Springer theory plays a role. If time permits, we will proceed to discuss their possible affine generalization.

May 11 Daniel Murfet (Melbourne)

Title: Derivatives of Turing machines in linear logic

Abstract: Small changes in a program typically result in large changes in its behaviour, so it is not obvious that there should be any reasonable general notion of a derivative for programs. For similar reasons, one cannot search the “space” of programs by typical optimisation methods like gradient descent. It was therefore surprising when Ehrhard and Regnier discovered in 2003 a general syntactic derivative for programs in the setting of lambda calculus, and for programs in the language of linear logic. I will present recent joint work with James Clift which uses the Ehrhard-Regnier derivative, together with encodings of Turing machines into linear logic, to study derivatives of Turing machines. One potential application of these ideas is that they give a natural way to make sense of gradient descent as a method for constructing programs. Finally, I will explain how the mathematics lying behind all of this is the theory of coalgebras.

May 4 Jessica Purcell (Monash)

Title: Cusp shape and tunnel number

Abstract: Associated to a cusped hyperbolic 3-manifold is a cusp shape, which is a point in the Teichmuller space of the torus. It is natural to ask which points in Teichmuller space arise. In the 1990s, Nimmersheim showed that the cusp shapes of finite volume hyperbolic 3-manifolds, which form a countable set, are dense in Teichmuller space. However, the 3-manifolds constructed in that theorem are very complicated topologically. A natural question to ask is which cusp shapes arise for simpler manifolds. For example, every 3-manifold has a Heegaard splitting. If we restrict to simple Heegaard splittings, of bounded genus g, which cusp shapes arise? In this talk, I will show that for fixed genus g, cusp shapes of finite volume 3-manifolds of genus g are still dense in Teichmuller space. This is joint with Vinh Dang.

Apr 27 Paul Zinn-Justin (Melbourne)

Title: Stable classes and Schubert calculus

Abstract: About 10 years ago, I noticed that a classical problem of 19th century geometry, now known as Schubert calculus, could be solved by making use of the methods of quantum integrable systems. In an unrelated development, culminating with the work of Okounkov and collaborators in the early 2010s, a connection was established between quantum integrable systems and the equivariant cohomology of certain symplectic algebraic varieties. In this talk I will try to explain the interrelations between all these ideas, and how this led A. Knutson and me to our recent solution of the Schubert calculus problem for 2-, 3- and even (in a more limited sense) 4-step flag varieties. If time permits I will say a word about extensions to K-theory and elliptic cohomology.

Apr 13 Yaping Yang (U Melbourne)

Title: Double current algebras and applications

Abstract: The deformed double current algebra associated to a complex simple Lie algebra 𝖌 is defined by Guay recently as a rational degeneration of the quantum toroidal algebra of 𝖌. It deforms the universal central extension of the double current algebra 𝖌[u, v]. In my talk, I will introduce the deformed double current algebra and give two applications. The first is the elliptic Casimir connection constructed by Toledano Laredo and myself. It is a flat connection with logarithmic singularities on the elliptic configuration space. The second is the work of Kevin Costello on the AdS/CFT correspondence in the case of M2 branes in an Ω-background.

Mar 23 Jonathan Bowden (Monash)

Title: From Foliations to Contact Structures.

March 16 Joan Licata (ANU)

Title: Grid Diagrams

Abstract: Grid diagrams are a classical approach to encoding links in the 3-sphere combinatorially, and the grid diagrams associated to topologically equivalent links are related by sequences of simple moves. Restricting the allowed moves captures stronger notions of equivalence appearing in braid theory and 3-dimensional contact geometry. I’ll survey this beautiful theory, and if time permits, describe work that extends it to other 3-manifolds.

Mar 2 Frank Calegari  (U Chicago)

Title: Point counting on curves and random matrices.

Feb 19, 3:15pm, Yi Huang (Tsinghua U)

Title: McShane identities for finite-area convex real projective surfaces

Abstract: Although Teichmueller theory began as the study of Riemann surface structures, one popular modern approach is via hyperbolic surfaces. Every point in the Teichmueller space T(S) describes a different possible hyperbolic structure on the topological surface S. Hyperbolic geometry allows us to define geometrically meaningful coordinates for T(S), such as length and twist coordinates, which explicitly describe the underlying hyperbolic structures on S. One major success story in this direction, is that of McShane discovering geometric identities which are valid for all cusped hyperbolic surfaces and Mirzakhani’s later generalization and application of these identities to prove Witten’s conjecture and to study the growth rates of the number of non-self-intersecting closed geodesics on hyperbolic surfaces.

Another popular approach to Teichmueller theory is more algebraic: the hyperbolic structure on a surface S may be encoded as a SL(2,R) representation of the fundamental group of S. This approach lends itself to natural generalizations of Teichmueller theory where we increase the rank of SL(2,R) to SL(n,R). For n=3, there is a geometric interpretation of this higher (rank) Teichmueller theory as the theory of strictly convex real projective structures on S. We show that there is a generalization of McShane’s identity to this context: a type of infinite-sum trigonometric identity which holds for all cusped convex real projective surfaces. This is work in collaboration with Zhe Sun (YMSC).

Feb 16 Oded Yacobi (Sydney)

Title: The category O of slices in the affine Grassmannian

Abstract: The affine Grassmannian $Gr_G$ is an important algebraic (ind-)variety in geometric representation theory associated to a reductive group $G$. The slices in $Gr_G$ are naturally occurring subvarieties which, by the geometric Satake correspondence of Mirkovic and Vilonen, geometrise weight spaces of irreducible representations of $G^L$, the Langlands dual group. They carry a natural Poisson structure, and under symplectic duality (due to Braden, Licata, Proudfoot, and Webster) they are dual to another class of important varieties called Nakajima quiver varieties. The essential feature of this duality is formulated as a Koszul duality between categories associated to these varieties called categories $O$ (these categories generalise the usual BGG category $O$ of $\mathfrak{g}=Lie(G)$-modules).

I will explain these ideas in a basic example, and use this to motivate the study of the category O associated to the slices in the affine Grassmannian. The main result I want to explain is a combinatorial description of the set of simple objects in this category, which turns out is governed by a finite dimensional representation of g^L. We conjectured this description in 2014, and recently proved it by relating the category to Webster’s tensor product algebras. I will try to explain the basic ideas of this proof.

This work is joint with various subsets of {J. Kamnitzer, P. Tingley, B. Webster, A. Weekes}.