### University of Melbourne Pure Mathematics Seminar

#### Term 2,   2018

Time: Friday 3:15 pm
Location:
213 Peter Hall
Organizers: Jesse Gell-Redman and Ting Xue

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 Date (DD/MM) Speaker Title Abstract 07/12 Masoud Kamgarpour (UQ) 23/11 Greg Martin (UBC) 16/11 Cheng-Chiang Tsai (Stanford) 9/11 Gregoire Leoper (Monash) 2/11 James Borger (ANU) 19/10 Amnon Neeman (ANU) 27/09 Alexander Stoimenov (Gwankju Institute of Science and Technology, Korea) Exchange moves and non-conjugate braid representatives of links We prove that under fairly general conditions an iterated exchange move gives infinitely many non-conjugate braid representatives of links. As a consequence, every knot has infinitely many conjugacy classes of $n$-braid representatives if and only if it has one admitting an exchange move. We discuss a project to give some fairly general conditions on the linking numbers of a link, so that it has infinitely many conjugacy classes of $n$-braid representatives if and only if it has one admitting an exchange move. 07/09 Mumtaz Hussain (La Trobe) The Generalised Baker--Schmidt Problem (1970) The Generalised Baker--Schmidt Problem, inspired by the pioneering work of Alan Baker and Wolfgang Schmidt (1970), is a central problem in metric Diophantine approximation on manifolds. It concerns the estimation of $f$-dimensional Hausdorff measure of the set of $\psi$-approximable points on a nondegenerate manifold. In this talk, I will explain resolution of this problem for a parabola, planar curves and hypersurfaces. These result are the first of their kind. This is a joint work with David Simmons (York) and Johannes Schleischitz (Ottawa). 31/08 Chenyan Wu (U Melbourne) Arthur parameters, Theta Correspondence and Period Integrals We give a brief overview of the theory of theta correspondence and show how it manifests in the Arthur parameter attached to an irreducible cuspidal representation of a symplectic group. We also propose a refinement in terms of period integrals. 24/08 Xinwen Zhu (Caltech) Hilbert’s twenty-first problem for p-adic varieties Hilbert’s twenty-first problem, formulated to generalize Riemann’s work on hypergeometric equations, concerns the existence of linear differential equations of Fuchsian type on the complex plane with specified singular points and monodromic group. Its modern solution, due to Deligne and known as the Riemann-Hilbert correspondence, establishes an equivalence between two different types of data on a complex algebraic manifold X: the representations of the fundamental group of X (topological data) and the linear systems of algebraic differential equations on X with regular singularieties (algebraic data). I’ll review this classical theory, and discuss some recent progress to solve similar problems for p-adic manifolds. 17/08 Tony Licata (ANU) Categorical taffy The Artin braid group appears prominently in several mathematical subjects, including both the geometry of surfaces and the representation theory of Lie algebras and quantum groups. The goal of this talk will be to motivate the study of higher, categorical representations of braid groups by illustrating how some of the structure of interest on the geometric side of braid theory (Teichmuller space, dynamics...) also emerges from higher representation theory. 10/08 Iva Halacheva (U Melbourne) Schur-Weyl duality and Lie superalgebras In the classical setting, Schur-Weyl duality describes an interaction between the symmetric group on d elements and the general linear Lie algebra gl(n), in terms of their action on d tensor copies of the vector representation of gl(n). This approach has been extended by Arakawa and Suzuki, and later Brundan and Kleshchev, to more general gl(n)-representations by upgrading the symmetric group to the degenerate affine Hecke algebra. A further generalization includes replacing gl(n) by sp(2n) or so(n), and the symmetric group by the Brauer algebra respectively. I will review some of these constructions and then discuss another instance of Schur-Weyl duality for the periplectic Lie superalgebra. One aspect which makes this case more unusual is the trivial action of the center of the universal enveloping algebra, and so a more elaborate construction than the standard Casimir element is required. 03/08 4:15-5:15pm, PH 213 Graeme Wilkin (National U. Singapore) The topology and geometry of spaces of Yang-Mills-Higgs flow lines Given a smooth complex Hermitian vector bundle over a compact Riemann surface, one can define the space of Higgs bundles and an energy functional on this space: the Yang-Mills-Higgs functional. The gradient flow of this functional resembles a nonlinear heat equation, and the limit of the flow detects information about the algebraic structure of the initial Higgs bundle (for example, whether or not it is semistable). In this talk I will explain my work to classify ancient solutions of the Yang-Mills-Higgs flow in terms of their algebraic structure, which leads to an algebro-geometric classification of Yang-Mills-Higgs flow lines. Critical points connected by flow lines can then be interpreted in terms of the Hecke correspondence. This classification also gives a geometric description of spaces of unbroken flow lines in terms of secant varieties of the underlying Riemann surface, and in the remaining time I will describe work in progress to relate the (analytic) Morse compactification of these spaces by broken flow lines to an algebro-geometric compactification by iterated blowups of secant varieties. 03/08 3:15-4:15pm, PH 213 Deepam Patel (Purdue) Hypergeometric Motives and periods in families. In this talk, we will first recall some conjectures/results on special values of L-functions of varieties over a number field and their relation to `regulators of extensions of motive'. In the literature, these extensions generally appear in families of pro-systems. We will discuss some recent joint work with M. Nori on the construction of a universal category of hypergeometric motives where one can uniformly recover the pro-systems of extensions of motives that appear in the literature. If time remains, we will discuss some applications to periods and cohomology jump loci. 02/08 3-4pm, Building 165 (Chem/Bio), G20 Jack Hall (Arizona) GAGA theorems Given an algebraic variety X over a topological field (e.g. R, C or Qp), one can often make some sort of analytic space Xan from X. The topology on Xan reflects the topology of the topological field and the functions on Xan should be appropriately holomorphic. The relationship between the vector bundles, subvarieties, cohomology, and coherent sheaves on X and Xan is typically referred to as a “GAGA theorem”. This name goes back to Serre’s paper Géométrie algébrique et géométrie analytique (1956), where the relationship was considered over C. Over the decades, corresponding to different types of analytifications of varieties and schemes, various GAGA theorems have been established (e.g., rigid, adic, and formal). I will discuss a new and unified GAGA theorem. This gives all existing results in the literature and also puts them into a broader context.

PAST SEMINARS, TERM 1 2018:

 Date (MM/DD) Speaker Title Abstract 02/02 Katharina Neusser  (Charles University) Symmetry and Geometric Rigidity In differential geometry many important geometric structures are geometrically rigid in the sense that their automorphism groups in some natural topology are finite-dimensional Lie groups. Prominent examples of such structures are Riemannian manifolds, conformal manifolds, projective structures and in general all geometric structures admitting equivalent descriptions as so-called Cartan geometries, which comprise a huge variety of geometric structures. Generically these geometric structures have trivial automorphism groups and so the ones among them with large automorphism groups or special types of automorphisms are typically geometrically and topologically very constrained and hence can often be classified. Recall for instance that a Riemannian manifold with an isometry group of largest possible dimension is isometric to a space of constant curvature. In this talk I will present several new and also discuss some classical results along these lines, concerned with (local) automorphism groups of various geometric structures and local and global questions of geometric rigidity.  This talk is intended for a general audience and will not require any special knowledge about differential geometry and Lie groups. 14/02 Qizheng Yin (Peking U) Curves, sheaves, and cycles on K3 surfaces The study of K3 surfaces is a classical subject in algebraic geometry. In my talk I will build connections between various algebro-geometric objects centering around K3 surfaces: algebraic curves, coherent sheaves, algebraic cycles, derived categories, and moduli spaces. Joint work with Junliang Shen and Xiaolei Zhao. 16/02 Oded Yacobi (U Sydney) The category O of slices in the affine Grassmannian 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 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}. 19/02 Yi Huang (Tsinghua) McShane identities for finite-area convex real projective surfaces. Although Teichm\"{u}ller theory began as the study of Riemann surface structures, one popular modern approach is via hyperbolic surfaces. Every point in the Teichm\"{u}ller space $\mathcal{T}(S)$ describes a different possible hyperbolic structure on $S$. Hyperbolic geometry allows us to define geometrically meaning coordinates, such as length and twisting 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 Teichm\"{u}ller theory is more algebraic: the hyperbolic structure on a surface $S$ may be encoded as a $\mathrm{SL}(2,\mathbb{R})$ representation of the fundamental group $\pi_1(S)$ of $S$. This approach lends itself to natural generalizations of Teichm\"{u}ller theory where we increase the rank of $\mathrm{SL}(2,\mathbb{R})$ to $\mathrm{SL}(n,\mathbb{R})$ (i.e.: a \emph{higher (rank) Teichm\"{u}ller theory}). For $n=3$, there is a geometric interpretation of higher Teichm\"uller 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). 23/02 Tarig Abdelgadir (UNSW) Moduli stacks of tensor stable points This is based on a joint work in progress with Daniel Chan. The idea of moduli spaces features heavily in modern algebraic geometry. They classically stem from a need to classify geometric objects and have since developed into essential tools in studying interactions between geometry, representation theory and string theory to name but a few. Here we will discuss the moduli spaces of points on a stack and how one may recover the given stack from them. The examples we will use will be derived equivalent to algebras and hence provide a connection to representation theory. Time permitting, we will them discuss an application to the McKay correspondence in its derived categories formulation. 02/03 Frank Calegari (U Chicago) Point counting on curves and random matrices 09/03 Andrew Schopieray (U. Oregon) Toward Categorical Witt Group Generators Joyal & Street provided a correspondence between metric groups and non-degenerate braided fusion categories about 25 years ago. Metric groups can be organized by Witt equivalence classes which form an abelian group. We will discuss a categorical Witt group for non-degenerate braided fusion categories (due to Davydov, Muger, Nikshych & Ostrik) and demonstrate how finding relations in this group amounts to studying the representation theory of fusion categories. Lastly we will discuss recent results in this direction related to extending a "quantum" McKay correspondence to arbitrary Lie algebras (based on the work of Ocneanu & Ostrik). 16/03 Joan Licata (ANU) Grid Diagrams 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. 23/03 Jonathan Bowden (Monash) From Foliations to Contact Structures In the mid 90’s Eliashberg and Thurston established a fundamental link between the more classical theory of (smooth!) foliations and that of contact topology in dimension 3, which, amongst other things, played an important role in Mrowka and Kronheimer’s proof of Property P Conjecture. Their theory gains its potency from the fact that Gabai gave a very general method for constructing (smooth) taut foliations on 3-manifolds given from non-trivial homology classes.  On the other hand most foliations that occur in nature via (pseudo)-Anosov flows, surgery, gluing, blows ups... are not smooth in general. This naturally motivates the need to apply Eliashberg and Thurston’s theory to foliations of lower regularity. In this talk I will report on how their theory generalises. Time permitting I will discuss some applications and related questions. 30/03 no seminar (easter holiday) 06/04 no seminar (easter holiday) 13/04 Yaping Yang (Melbourne) Double current algebras and applications The deformed double current algebra associated to a complex simple Lie algebra $\mathfrak{g}$ is defined by Guay recently as a rational degeneration of the quantum toroidal algebra of $\mathfrak{g}$. It deforms the universal central extension of the double current algebra $\mathfrak{g}[u, v]$. 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 $\Omega$-background. 20/04 Yann Bernard (Monash) On the Willmore energy The Willmore energy of a surface captures the way it bends. Originally discovered 200 years ago by Sophie Germain in the context of elasticity theory, it has since then been rediscovered numerous times in several areas of science: general relativity, optics, string theory, conformal geometry, and cell biology. For example, our red blood cells assume a peculiar shape that minimises (a close relative of) the Willmore energy.  In this talk, I will present the history of the Willmore energy, its applications, and its main properties. I will also show some recent advances in the study of the Willmore energy and related problems. The presentation will be accessible to all mathematicians as well as to advanced undergraduate students. 27/04 Paul Zinn-Justin (Melbourne) Stable classes and Schubert calculus 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. 04/05 Jessica Purcell (Monash) Cusp shape and tunnel number 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. 11/05 Daniel Murfet (Melbourne) Derivatives of Turing machines in linear logic 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. 18/05 Stephan Tillmann (U. Sydney) Three angles on tropical geometry 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. 25/05 Asilata Bapat (ANU) Perverse sheaves on hyperplane arrangements and gluing 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. 01/06 David Baraglia (Adelaide) 1pm  Peter Hall 213 Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory. Let X be a smooth, compact, oriented 4-manifold and consider the following problem. Let G be a group which acts on the second cohomology of X preserving the intersection form. Can this action of G on H^2(X) be lifted to an action of G on X by diffeomorphisms? We study a parametrised version of Seiberg-Witten theory for smooth families of 4-manifolds and obtain obstructions to the existence of such lifts. For example, we construct compact simply-connected 4-manifolds X and involutions on H^2(X) that can be realised by a continuous involution on X, or by a diffeomorphism, but not by an involutive diffeomorphism for any smooth structure on X. 01/06 Heather Macbeth (MIT) 3:15pm Peter Hall 213 Ricci solitons on complex manifolds When a dynamical system possesses a symmetry, the trajectories of this system which evolve by that symmetry are of special importance in understanding the qualitative dynamics of the system. Such trajectories are called solitons. I will give an introduction to the Ricci flow, and to steady Ricci solitons, which are the solitons of the Ricci flow when considered as an infinite-dimensional dynamical system. It turns out that steady Ricci solitons can be characterized as solutions to a certain nonlinear elliptic partial differential equation. I will then describe the construction of a large new family of steady Ricci solitons, the first known examples which are not explicit (but, rather, constructed by solving this PDE in some generality). The underlying manifolds of these solitons are crepant resolutions of finite group quotients of C^n, and this algebro-geometric structure is crucial to the construction. This is joint work with Olivier Biquard. 08/06 Henry Segerman (Oklahoma State) From veering triangulations to pseudo-Anosov flows 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.