Past

We will discuss the Chapter "Cohomology" from the book "Elementary applied topology" by Robert Ghrist (available at https://www.math.upenn.edu/~ghrist/notes.html).

This is a lunch seminar, so feel free to bring your lunch along!

This talk is about a 4d "(ambi-)twistor string" formula for sEYM tree-level scattering amplitudes and is work in progress. The formula sheds some new light on the translation between CHY formulae and (ambi-)twistor formulae in general, potentially even at loop level.

In the mathematical theory of liquid crystals, a hedgehog is a universal equilibrium solution for Frank's elastic free-energy functional. It is characterized by a radial defect for the nematic director, reminiscent of the way spines are arranged in the spiny mammal. For certain choices of Frank's elastic constants, the free energy stored in a ball subject to radial boundary conditions for the director is minimized by a hedgehog with its defect in the centre of the ball. For other choices of Frank's constants, it is known that a radial hedgehog cannot be a minimizer for this variational problem. We shall gather evidence supporting the conjecture that a "twisted" hedgehog takes the place of a radial hedgehog as an energy minimizer (and we shall not fail to say in which sense it is "twisted"). We shall also show that a twisted hedgehog often accompanies, unseen, a radial hedgehog, as its virtual double, ready to beat its energy as a certain elastic anisotropy is reached.

I will described how ideas from constructive quantum field theory can be adapted to produce a systematic approach for analytic renormalization in the theory of regularity structures.

Knots and links naturally appear in the neighbourhood of the singularity of a complex curve; this creates a bridge between algebraic geometry and differential topology. I will discuss a topological approach to the study of 1-parameter families of singular curves, using correction terms in Heegaard Floer homology. This is joint work with József Bodnár and Daniele Celoria.

We will give an overview of the Soliton Resolution Conjecture, focusing mainly on the Wave Maps Equation. This is a program about understanding the formation of singularities for a variety of critical hyperbolic/dispersive equations, and stands as a remarkable topic of research in modern PDE theory and Mathematical Physics. We will be presenting our contributions to this field, elaborating on the required background, as well as discussing some of the latest results by various authors.

The Hastings-Levitov models describe the growth of random sets (or clusters) in the complex plane as the result of iterated composition of random conformal maps. The correlations between these maps are determined by the harmonic measure density profile on the boundary of the clusters. In this talk I will focus on the simplest case, that of i.i.d. conformal maps, and obtain a description of the local fluctuations of the harmonic measure density around its deterministic limit, showing that these are Gaussian. This is joint work with James Norris.

Calabi-Yau 3-folds play a large role in string theory.  Cohomology of sheaves on such varieties has many uses in string theory, including counting the number of particles or fields in a theory, as well as to help identify terms in the superpotential that determines the equations of motion of the corresponding string theory, and many other uses as well.  As a computational algebraic geometer, string theory provides a rich source of new computational problems to solve.

In this talk, we focus on the search for rigid divisors on these Calabi-Yau hypersurfaces of toric varieties.  We have had methods to compute sheaf cohomology on these varieties for many years now (Eisenbud-Mustata-Stillman, around 2000), but these methods fail for many of the examples of interest, in that they take a very long time, or the software (wisely) refuses to try!

We provide techniques and formulas for the sheaf cohomology of certain divisors of interest in string theory, that other current methods cannot handle.  Along the way, we describe a Macaulay2 package for computing with these objects, and show its use on examples.

This is joint work with Andreas Braun, Cody Long, Liam McAllister, and Benjamin Sung.

Research takes a long time while the attention span of the world is apparently decreasing, so today's researchers need to be able to get their message across quickly and succinctly. In this session we'll share some tips on how to communicate the key messages of your work in just a few minutes, and give you a chance to have a go yourself.  This will be helpful for job and funding applications and interviews, and also for public engagement. In September there will be an opportunity to do it for real, for our alumni, when we'll showcase Oxford Mathematics at the Alumni Weekend.

We study vortex sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. The problem is a nonlinear hyperbolic problem with a characteristic free boundary. The so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. A necessary condition for the weak stability is obtained by analyzing roots of the Lopatinskii determinant associated to the linearized problem. Under such stability condition,  we prove short-time existence and nonlinear stability of relativistic vortex sheets by the Nash-Moser iterative scheme.

We consider the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. 
In this talk we present our results about the well-posedness of the problem in the sense of Hadamard, under a suitable stability condition, that is the 
local-in-time existence in Sobolev spaces and uniqueness of smooth solutions to the Cauchy problem, and the strong continuous dependence on the data in the same topology.
Joint works with: Alessandro Morando and Paola Trebeschi.
 

 I will start with a motivation of what algebraic and model-theoretic properties an algebraically closed field of characteristic 1 is expected to have. Then I will explain how these properties forces one to follow the route of Hrushovski's construction leading to a a 'pseudo-analytic' structure which we identify as an algebraically closed field of characteristic 1 . Then I am able to formulate very precise axioms that such a field must satisfy.  The main theorem then states that under the axioms the structure has the desired algebraic and analytic properties. The axioms have a form of statements about existence of solutions to systems of equations in terms of a 'multi-dimensional' valuation theory and the validity of these statements is an open problem to be discussed. 
This is a joint work with Alex Cruz Morales.
 

1.Kremnitzer. I will explain an approach to constructing geometries relative to a symmetric monoidal 
category. I will then introduce the category of normed sets as a possible analytic geometry over 
the field with one element. I will show that the Fargues-Fontaine curve from p-adic Hodge theory and 
the Connes-Bost system are naturally interpreted in this geometry. This is joint work with Federico Bambozzi and 
Oren Ben-Bassat.
 

In this talk, I discuss the positive geometry of the Wilson Loop Diagrams appearing in SYM N-4 theory. In particular, I define an algorithm for associating Wilson Loop diagrams to convex cells of the positive Grassmannians. Using combinatorics of these cells, I then consider the geometry of N^2MHV diagrams on 6 points.

The "Weyl fermion" was discovered in a topological semimetal in
2015. Its mathematical characterisation turns out to involve deep and subtle
results in differential topology. I will outline this theory, and explain
some connections to Euler structures, torsion of manifolds,
and Seiberg-Witten invariants. I also propose interesting generalisations
with torsion topological charges arising from Kervaire semicharacteristics
and ``Quaternionic'' characteristic classes.

The technique of scanning, or the parameterised Pontrjagin--Thom construction, has been extraordinarily successful in calculating the cohomology of configuration spaces (McDuff), moduli spaces of Riemann surfaces (Madsen, Tillmann, Weiss), moduli spaces of graphs (Galatius), and moduli spaces of manifolds of higher dimension (Galatius, R-W, Botvinnik, Perlmutter), with constant coefficients. In each case the method also works to study the cohomology of moduli spaces of objects equipped with a "tangential structure". I will explain how choosing an auxiliary highly-symmetric tangential structure often lets one calculate the cohomology of these moduli spaces with large families of twisted coefficients, by exploiting the symmetries of the tangential structure and using a little representation theory.

 

 Markus Szymik and I computed the homology of the Higman-Thompson groups by first showing that they stabilize (with slope 0), and then computing the stable homology. I will in this talk give a new point of view on the computation of the stable homology using Thumann's "operad groups". I will also give an idea of how scanning methods can enter the picture. (This is partially joint work with Søren Galatius.) 

It is hard to detect the exotic nature of an exotic n-sphere M 
in homotopical features of the diffeomorphism group Diff(M). The well 
known reason is that Diff(M) contains a big topological subgroup H which 
is identified with the group of diffeomorphisms rel boundary of the 
n-disk, with a small coset space Diff(M)/H which is invariably homotopy 
equivalent to O(n+1). Therefore it seems that our only chance to detect 
the exotic nature of M in homotopical features of Diff(M) is to see 
something in this extension.  (To make sense of "homotopical features of 
Diff(M)" one should think of Diff(M) as a space with a multiplication 
acting on an n-sphere.) I am planning to report on PhD work of O Sommer 
and calculations due to myself and Sommer which, if all goes well, would 
show that Diff(M) has some exotic homotopical properties in the case 
where M is the 7-dimensional exotic sphere of Kervaire-Milnor fame which 
bounds a compact smooth framed 8-manifold of signature 8. The 
theoretical work is based on classical smoothing theory and the 
calculations would be based on ever-ongoing (>30 years) joint work 
Weiss-Williams, and might give me and Williams another valuable 
incentive to finish it.

We construct the diffeomorphism-equivariant “scanning map” associated to the configuration spaces of manifolds with twisted summable labels. The scanning map is also functorial with respect to embeddings of manifolds. To adapt P. Salvatore's idea of non-commutative summation into twisted setting, we define a bundle of Fulton-MacPherson operads over a manifold M whose fibres are built within tangent spaces of M.

In 1956 Milnor published a paper proving that there are manifolds homeomorphic to the 7-sphere but not diffeomorphic to it. Seeking to generalise this example, he was led in around 1960 to introduce a construction for  killing homotopy groups of manifolds. When this was generalised to killing relative homotopy groups it became a general and powerful method of construction. An obstruction arises to killing the last group, and the analysis of this obstruction in general leads to a new theory.

Categorification of knot polynomials -- Daniele Celoria

Classically, the most powerful and versatile knot invariants take the form of polynomials. These can usually be defined by simple recursive equations, known as skein relations; after giving the main examples of polynomial knot invariants (Alexander and Jones polynomials), we are going to informally introduce categorifications. Finally we are going to present the Knot Floer and the Khovanov homologies, and show that they provide a categorification of the aforementioned polynomial knot invariants.

Network science for online social media: an x-ray or a stethoscope for society -- Mariano Beguerisse

The abundance of data from social media outlets such as Twitter provides the opportunity to perform research at a societal level at a scale unforeseen. This has spurred the development of mathematical and computational methods such as network science, which uses the formalism and language of graph theory to study large systems of interacting agents. In this talk, I will provide a sketch of network science and its application to study online social media. A number of different networks can be constructed from Twitter data, which can be used to ask questions about users, ranging from the structural (an 'x-ray' to see how societies are connected online) to the topical ('stethoscope' to feel how users interact in the context of specific event). I will provide concrete examples from the UK riots of 2011, applications to medical anthropology, and political referenda, and will also highlight distinct challenges such as the directionality of connections, the size of the network, the use of temporal information and text, all of which are active areas of research.

The efficient pricing and hedging of vanilla and exotic foreign exchange options requires an adequate model that takes into account both the local and the stochastic features of the volatility dynamics. In this joint work, we put forward a four-factor hybrid local-stochastic volatility (LSV) model that combines state-of-the-art dynamics for the exchange rate with stochastic dynamics for the domestic and foreign short rates, and provide a consistent and self-contained calibration and pricing framework.
For the calibration, we propose a novel and generic algorithm that builds on the particle method of Guyon and Labordere. We combine it with new variance reduction techniques to accelerate convergence and use control variates derived from a pure local volatility model, the stochastic short rates and the two-factor Heston-type LSV model. Our numerical experiments show a dramatic variance reduction that allows us to calibrate the four-factor model at almost no extra computational cost. The method can be applied to a large class of hybrid LSV models and is not restricted to our particular choice of the diffusion.
For the pricing, we propose a Monte Carlo simulation scheme that combines the full truncation Euler (FTE) scheme for the stochastic volatility and the stochastic short rates with the log-Euler scheme for the exchange rate. We find a lower bound on the explosion time of exponential moments of FTE approximations, and prove the strong convergence of the exchange rate approximations and the convergence of Monte Carlo estimators for a number of vanilla and exotic options. We then carry out numerical experiments to justify our choice of model and demonstrate convergence.
 

We discuss how one can define transseries in several variables. The idea is
to combine the construction of the univariate transseries with a blow up procedure. The
latter allows to normalize transseries in an arbitrary number of variables which makes
them manageable as usual transseries.
 

Given a Calabi-Yau threefold X, one can count curves on X using various approaches, for example using stable maps or ideal sheaves; for any curve class on X, this produces an infinite sequence of invariants, indexed by extra discrete data (e.g. by the domain genus of a stable map).  Conjecturally, however, this sequence is determined by only a finite number of integer invariants, known as Gopakumar-Vafa invariants.  In this talk, I will propose a direct definition of these invariants via sheaves of vanishing cycles, building on earlier approaches of Kiem-Li and Hosono-Saito-Takahashi.  Conjecturally, these should agree with the invariants as defined by stable maps.  I will also explain how to prove the conjectural correspondence for irreducible curves on local surfaces.  This is joint work with Yukinobu Toda.

I will discuss the definitions and the basic properties of some infinite dimensional generalizations of Euclidean lattices and of their invariants defined in terms of theta series. Then I will present some of their applications to transcendence theory and Diophantine geometry.

Cover’s celebrated theorem states that the long run yield of a properly chosen “universal” portfolio is as good as the long run yield of the best retrospectively chosen constant rebalanced portfolio. We formulate an abstract principle behind such a universality phenomenon valid for general optimization problems in the long run. This allows to obtain new results on modelfree portfolio optimization, in particular in continuous time, involving larger classes of investment strategies. These modelfree results are complemented by a comparison with the log-optimal numeraire portfolio when fixing a stochastic model for the asset prices. The talk is based on joint work with Walter Schachermayer and Leonard Wong.

In Soft Matter, octupolar order is not just an exotic mathematical curio. Liquid crystals have already provided a noticeable case of soft ordered materials for which a (second-rank) quadrupolar order tensor may not suffice to capture the complexity of the condensed phases they can exhibit. This lecture will discuss the properties of a third-rank order tensor capable of describing these more complex phases. In particular, it will be shown that octupolar order tensors come in two separate, equally abundant variants. This fact, which will be given a simple geometric interpretation, anticipates the possible existence of two distinct octupolar sub-phases. 

With the purpose of examining some relevant geometric properties of the moduli space of sheaves over an algebraic surface, Le Potier conjectured some unexpected duality between the complete linear series of certain natural divisors, called Theta divisors, on the moduli space. Such conjecture is widely known as Strange Duality conjecture. After having motivated the problem by looking at certain instances of quantization in physics, we will work in the setting of surfaces. We will then sketch the proof in the case of abelian surfaces, giving an idea of the techniques that are used. In particular, we will show how the theory of discrete Heisenberg groups and fiber wise Fourier-Mukai transforms, which might be applied to other cases of interest, enter the picture. This is joint work with Alina Marian, Dragos Opera and Kota Yoshioka.

During the last decade, the progress in the computational performance of commercial mixed-integer programming solvers have been significant. Part of this success is due to faster computers and better software engineering but a more significant part of it is due to the power of the cutting planes used in these solvers.
In the first part of this talk, we will discuss main components of a MIP solver and describe some classical families of valid inequalities (Gomory mixed integer cuts, mixed integer rounding cuts, split cuts, etc.) that are routinely used in these solvers. In the second part, we will discuss recent progress in cutting plane theory that has not yet made its way to commercial solvers. In particular, we will discuss cuts from lattice-free convex sets and answer a long standing question in the affirmative by deriving a finite cutting plane algorithm for mixed-integer programming.

Consider a family of uniformly bounded $W^{2,p}$ isometric immersions of an $n$-dimensional (semi-) Riemannian manifold into (resp., semi-) Euclidean spaces. Are the weak limits still isometric immersions?

We answer the question in the affirmative for $p>n$ in the Riemannian case, by exploiting the div-curl structure of the Gauss-Codazzi-Ricci equations, which describe the curvature flatness of the isometric immersions. Along the way a generalised div-curl lemma in Banach spaces is established. Moreover, the endpoint case $p=n=2$ is settled. 

In the semi-Riemannian case we reduce the problem to the weak continuity of H. Cartan's structural equations in $W^{1,p}_{\rm loc}$, which is proved by a generalised compensated compactness theorem relating the weak continuity of quadratic forms to the principal symbols of differential constraints. Again for $p>n$ we obtain the weak rigidity. The case of degenerate hypersurfaces are also discussed, as well as connections to PDEs in fluid dynamics.

Manifolds with corners are similar to manifolds, yet are locally modelled on subsets $[0,\infty)^k \times R^{n-k}$. I will discuss some of the theory of these objects, as well as introducing $C^\infty$-rings. This will explain the background to my current research in $C^\infty$-Algebraic Geometry. Time permitting, I will briefly discuss my current research on $C^\infty$-schemes with corners and motivation of this research.

The amount of digital data that requires long-term protection 
of integrity, authenticity, and confidentiality protection is steadily 
increasing. Examples are health records and genomic data which may have 
to be kept and protected for 100 years and more. However, current 
security technology does not provide such protection which I consider a 
major challenge. In this talk I report about a storage system that 
achieves the above protection goals in the long-term. It is based on 
information theoretic secure cryptography (both classical and quantum) 
as well as on chains of committments. I discuss its security and present 
a proof-of-concept implementation including an experimental analysis.

A variety of groups is an equationally defined class of groups, namely the class of groups in which each of a set of "laws" (or "identical relations") holds. Examples include the abelian groups (defined by the law $xy = yx$), the groups of exponent dividing $d$ (defined by the law $x^d$), the nilpotent groups of class at most some fixed integer, and the solvable groups of derived length at most some fixed integer. This talk will give an introduction to varieties of groups, and then conclude with recent work on determining for certain varieties whether, for fixed coprime $m$ and $n$, a group $G$ is in the variety if and only if the power subgroups $G^m$ and $G^n$ (generated by the $m$-th and $n$-th powers) are in the variety.

Every finite regular CW complex is, ipso facto, a cohomologically stratified space when filtered by skeleta. We outline a method to recover the canonical (i.e., coarsest possible) stratification of such a complex that is compatible with its underlying cell structure. Our construction proceeds by first localizing and then resolving a complex of cosheaves which capture local cohomology at every cell. The result is a sequence of categories whose limit recovers the desired strata via its (isomorphism classes of) objects. As a bonus, we observe that the entire process is algorithmic and amenable to efficient computations!

Given two $k$-graphs $H$ and $F$, a perfect $F$-packing in $H$ is a collection of vertex-disjoint copies of $F$ in $H$ which together cover all the vertices in $H$. In the case when $F$ is a single edge, a perfect $F$-packing is simply a perfect matching. For a given fixed $F$, it is generally the case that the decision problem whether an $n$-vertex $k$-graph $H$ contains a perfect $F$-packing is NP-complete.

In this talk we describe a general tool which can be used to determine classes of (hyper)graphs for which the corresponding decision problem for perfect $F$-packings is polynomial time solvable. We then give applications of this tool. For example, we give a minimum $\ell$-degree condition for which it is polynomial time solvable to determine whether a $k$-graph satisfying this condition has a perfect matching (partially resolving a conjecture of Keevash, Knox and Mycroft). We also answer a question of Yuster concerning perfect $F$-packings in graphs.

This is joint work with Jie Han (Sao Paulo).
 

The Morita Frobenius number of an algebra is the number of Morita equivalence classes of its Frobenius twists. Morita Frobenius numbers were introduced by Kessar in 2004 in the context of Donovan’s Conjecture in block theory. I will present the latest results of a project in which we aim to calculate the Morita Frobenius numbers of the blocks of quasi-simple finite groups. I will also discuss the importance of a recent result of Bonnafe-Dat-Rouquier for our methods, and explain the relationship between Morita Frobenius numbers and Donovan’s Conjecture. 

In various research fields such as machine learning, compressed sensing and operations research, optimization problems which seek sparsity of solutions by the cardinality constraint or rank constraint are studied. We formulate such problems as DC (Difference of two Convex functions) optimization problems and apply DC Algorithm (DCA) to them. While a subproblem needs to be solved in each DCA iteration, its closed-form solution can be easily obtained by soft-thresholding operation. Numerical experiments demonstrate the efficiency of the proposed DCA in comparison with existing methods.
This is a joint work with J. Gotoh (Chuo Univ.) and K. Tono (U. Tokyo). 

Quantization is the study of the interface between commutative and
noncommutative geometry. There are myriad approaches to it, mostly presented
as ad hoc recipes. I shall discuss the motivating ideas, and the relations
between some of the methods, especially the relation between 'deformation'
and 'geometric' quantization.

What makes an intersection likely or unlikely? A simple dimension count shows that two varieties of dimension r and s are non "likely" to intersect if r < codim s, unless there are some special geometrical relations among them. A series of conjectures due to Bombieri-Masser-Zannier, Zilber and Pink rely on this philosophy. I will speak about a joint work with F. Barroero (Basel) in this framework in the special case of a curve in a family of elliptic curves. The proof is based on Pila-Zannier method, combining diophantine ingredients with a refinement of a theorem of Pila and Wilkie about counting rational points in sets definable in o-minimal structures.
   Everyone welcome!
 

A familiar technique in PDE theory is to use mollification to adjust a function controlled in some weak norm into a smooth function with corresponding control on its $C^k$ norm. It would be extremely useful to be able to do the same sort of regularisation for Riemannian metrics, and one might hope to use Ricci flow to do this. However, attempting to do so throws up some fundamental problems concerning the well-posedness of Ricci flow. I will explain some recent developments that allow us to use Ricci flow in this way in certain important cases. In particular, the Ricci flow will now allow us to adjust a `noncollapsed’ 3-manifold with a lower bound on its Ricci curvature through a family of such manifolds, without disturbing the Riemannian distance function too much, and so that we instantly obtain uniform bounds on the full curvature tensor and all its derivatives. These ideas lead to the resolution of some long-standing open problems in geometry.

No previous knowledge of Ricci flow will be assumed, and differential geometry prerequisites will be kept to a minimum.

Joint work with Miles Simon.