Term 2, 2018
Time: Friday 3:15 pm
Location:
213 Peter Hall
Organizers: Jesse GellRedman and Ting Xue
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Date (DD/MM)

Speaker

Title

Abstract

07/12

Masoud Kamgarpour
(UQ)



23/11

Greg Martin
(UBC)

Prime number races

This talk is a survey of ``prime number races". Around 1850, Chebyshev noticed that for any given value of $x$, there always seem to be more primes of the form $4n+3$ less than $x$ than there are of the form $4n+1$. Similar observations have been made with primes of the form $3n+2$ and $3n+1$, primes of the form $10n+3,10n+7$ and $10n+1,10n+9$, and many others besides. More generally, one can consider primes of the form $qn+a,qn+b,qn+c, \dots$ for our favorite constants $q,a,b,c,\dots$ and try to figure out which forms are ``preferred" over the othersnot to mention figuring out what, precisely, being ``preferred" means. All of these ``races'' are related to the function $\pi(x)$ that counts the number of primes up to $x$, which has both an asymptotic formula with a wonderful proof and an associated ``race'' of its own; and the attempts to analyze these races are closely related to the Riemann hypothesisthe most famous open problem in mathematics. 
16/11




9/11

Gregoire Leoper
(Monash)

Reconstruction of missing data by optimal transport: applications in cosmology and finance 
Optimal Transport is an old optimisation problem that goes back to Gaspard Monge in 1781. I will give some historical perspective of the problem and its solutions, and then present some
recent results where techniques from Optimal Transport can be used, going from the problem of reconstruction of the early universe, to a problem of model calibration in finance. 
2/11

James Borger
(ANU)

Lambdarings and Hilbert's 12th Problem 
Hilbert's 12th Problem asks whether it's possible to generate in an explicit way all the extensions L of a number field K which are Galois with abelian Galois group. It had been known since Kronecker that one could do this using special values of the exponential function (roots of unity) when K is the field of rational numbers, or special values of elliptic and modular functions when K is an imaginary quadratic field. Since the mid20th century, this has typically been expressed in the language of commutative algebraic groups with large endomorphism rings, and partial results were established in some new cases by Shimura and others. In this talk, I'll review some of this history and then explain a new framework for Hilbert's 12th Problem based on a generalization of the notion of lambdaring in algebraic Ktheory. This talk reports on joint work with Bart de Smit. 
26/10

Anna Beliakova
(UZH)

Quantum annular Khovanov homology 
In 1999 Khovanov assigned to a knot (or link) a complex
whose homotopy type is a link invariant and whose Euler characteristic
is the Jones polynomial.
This construction is known as a categorification of the
Jones polynomial.
In the talk I will explain Khovanov’s approach and show that it can be
extended to annular links in a way sensible to annular link cobordisms.
This is a joint work with K. Putyra and S. Wehrli. 
19/10

Amnon Neeman
(ANU)

Approximable triangulated categories 
We will begin the talk with a theorem in algebraic geometry, about the relation between the derived categories $D^b_{coh}(X)$ and $D^{perf}(X)$ where $X$ is a noetherian scheme. The theorem represents a major improvement over what was known.
It turns out that the result is a relatively straightforward corollary of the (somewhat technical) theorem that the category $D_{qc}(X)$ is approximable. Approximability is a new notion, a recently introduced useful tool we will explain.
To illustrate the power of the new technique we will end the talk with several more applications, which we will then compare with what was known. 
12/10

Alex Ghitza
(U. Melbourne)

Differential operators on modular forms, and Galois representations 
Since a modular form is a holomorphic function, it is tempting to take
its derivative. However, this destroys the modularity property.
Several approaches exist for "fixing" this problem, and the resulting
objects have many arithmetic applications.
I will discuss such differential operators on various types of modular
forms (mod p), indicate a few ways of constructing them, and describe
the effect of these operators on the Galois representations attached
to Hecke eigenforms.
(This is an amalgamation of various projects joint with Owen Colman,
Ellen Eischen, Max Flander, Elena Mantovan, Angus McAndrew, and Takuya
Yamauchi.) 
27/09

Alexander Stoimenov
(Gwankju Institute of Science and Technology, Korea)

Exchange moves and nonconjugate braid representatives of links 
We prove that under fairly general conditions an iterated exchange
move gives infinitely many nonconjugate 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 BakerSchmidt Problem (1970) 
The Generalised BakerSchmidt 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 twentyfirst problem for padic varieties 
Hilbert’s twentyfirst 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 RiemannHilbert 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 padic 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)

SchurWeyl duality and Lie superalgebras 
In the classical setting, SchurWeyl 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 SchurWeyl 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
4:155:15pm,
PH 213

Graeme Wilkin
(National U. Singapore)

The topology and geometry of spaces of YangMillsHiggs 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 YangMillsHiggs 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 YangMillsHiggs flow in terms of their algebraic structure, which leads to an algebrogeometric classification of YangMillsHiggs 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 algebrogeometric compactification by iterated blowups of secant varieties. 
03/08
3:154:15pm,
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 Lfunctions 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 prosystems. 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 prosystems 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
34pm,
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 finitedimensional Lie groups. Prominent examples of such structures are Riemannian manifolds, conformal manifolds, projective structures and in general all geometric structures admitting equivalent descriptions as socalled 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 algebrogeometric 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 finitearea 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 nonselfintersecting 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 infinitesum 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 nondegenerate 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
nondegenerate 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 3sphere 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 3dimensional contact geometry. I'll survey this beautiful theory, and if time permits, describe work that extends it to other 3manifolds. 
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 3manifolds given from nontrivial 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 ZinnJustin
(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) 4step flag varieties. If time permits I will say a word about extensions to Ktheory and elliptic cohomology. 
04/05

Jessica Purcell
(Monash)

Cusp shape and tunnel number

Associated to a cusped hyperbolic 3manifold 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 3manifolds, which form a countable set, are dense in Teichmuller space. However, the 3manifolds constructed in that theorem are very complicated topologically. A natural question to ask is which cusp shapes arise for simpler manifolds. For example, every 3manifold 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 3manifolds 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 EhrhardRegnier 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 4manifolds from families SeibergWitten theory. 
Let X be a smooth, compact, oriented 4manifold 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 SeibergWitten theory for smooth families of 4manifolds and obtain obstructions to the existence of such lifts. For example, we construct compact simplyconnected 4manifolds 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 infinitedimensional 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 algebrogeometric structure is crucial to the construction. This is joint work with Olivier Biquard. 
08/06

Henry Segerman
(Oklahoma State)

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