Seminars in the next week

Apr 22 (Mon)

13:00 MCS0001 PureAbigail Ward (Cambridge): *Defining and computing algebraic invariants of symplectic manifolds*

Symplectic topology has been studied since Hamilton wrote
down his equations describing classical mechanics, but it was not until
work of Gromov and Floer in the 1980s that the field came into its
modern form. We will first discuss how Gromov's work on
pseudo-holomorphic curves has been used to produce algebraic,
Floer-theoretic invariants of symplectic manifolds such as the Fukaya
category, and give some example applications. We will then discuss the
arborealization program of Nadler which aims to systematically reduce
the calculation of these invariants to a combinatorial problem, in
analogy to how one might compute the homology of a space via a cellular
decomposition, and exhibit topological obstructions to this approach
found in joint work with Daniel Alvarez-Gavela and Tim Large.

**Venue:** MCS0001

Apr 23 (Tue)

15:00 MCS3070 APDEErik Duse (KTH Stockholm): *Morrey inequalities and subelliptic estimates via Weitzenböck identities*

In joint work with Andreas Rosén we prove a general Weitzenböck identity for arbitrary pairs of constant coefficients homogeneous first order PDE operators on domains for fields that satisfy natural boundary conditions. This identity gives rise to a generalization of the Levi form for the classical d-bar complex in several complex variables. Under the assumption that one of the operators is cocancelling, a concept introduced by J. Van Schaftingen in his work on endpoint Sobolev estimates, and an additional algebraic condition we prove a generalized Morrey inequality. We derive from this a weighted Sobolev inequality as well as giving new proof of the equivalence of Morrey inequalities and subelliptic estimates. Using the theory of J. Kohn and L. Nirenberg this in particular implies solvability for the generalized Neumann-dbar problem on generalized strongly pseudoconvex domains for constant rank operators satisfying our conditions.

**Venue:** MCS3070

Apr 24 (Wed)

11:00 zoom A&CGiulio Salvatori (Max Planck Institute): *Positive Geometries and Scattering Amplitudes*

I will present a formulation of scattering amplitudes in the simplest colored, cubic, scalar theory - Tr \phi^3 - as an integral over the
space of curves on Riemann surfaces, valid at all loop orders and at all
orders in the topological 't Hooft expansion. This so-called "curve integral" has the main advantage of describing amplitudes as a unique object, rather than as a sum over Feynman diagrams, allowing to study phenomena which are hidden graph-by-graph and suggesting powerful techniques for the numerical evaluation of amplitudes. Furthermore, the singularity structure of the propagators of Tr \phi^3 theory is shared by any colored theory, thus suggesting the generalization of the formalism to more realistic theories by insertion of appropriate numerators in the curve integral. At the heart of the formalism is a simple counting problem associated to curves on surfaces, which surprisingly provides a combinatorial origin for the physics of scattering amplitudes. The talk is based on 2309.15913, 2311.09284 and 2402.06719.

**Venue:** zoom

Zoom: https://durhamuniversity.zoom.us/j/91888793409?pwd=eUc2RW5rY3BCWU90dEx5QnBYZ1RDZz09

14:00 MCS2068 HEPMValentina Forini (Humboldt University): *Conformal field theories from line defects, holography and the analytic bootstrap*

Wilson lines are a prototypical example of defect in quantum field theory. After reviewing the superconformal case - in which the one-dimensional defect CFT that they define is particularly interesting - I will discuss some analytic tools that may prove useful in this context, but are developed for generic 1d CFTs. Among them, a representation of the four-point correlator as a Mellin amplitude and via a recently derived dispersion relation.

**Venue:** MCS2068

Apr 25 (Thu)

13:00 MCS2068 G&TLuc Vrancken (KU Leuven/Université Polytechnique Hauts-de-France): *Homogeneous 6 dimensional nearly Kaehler manifolds and their
submanifolds*

We present a survey of how the curvature tensor of all known
homogeneous 6 dimensional nearly Kähler spaces (both in the definite and
in the pseudo Riemannian case) can be expressed in an invariant way
using the induced geometric structures on the 6 dimensional nearly
Kähler space.

As an application we show how this can be used to study special classes of submanifolds in these spaces. In the latter case we will in particular focus on totally geodesic Lagrangian submanifolds and equivariant Lagrangian immersions.

**Venue:** MCS2068

Apr 26 (Fri)

13:00 online ( ProbSimon Wittmann (Hong Kong Polytechnic University): *Construction of a Diffusion on the Wasserstein Space*

For stochastic analysis on the Wasserstein space, it is crucial to construct a diffusion process which plays a role of Brownian motion in finite-dimensions, or the Ornstein-Uhlenbeck process on a separable Hilbert space. This has been a long standing open problem due to the lack of a volume or Gaussian measure on the Wasserstein space, which could serve as an invariant measure.
To study diffusion processes on the $p$-Wasserstein space $\mathcal P_p $ for $p\in [1,\infty)$ over a separable, reflexive Banach space $X$, we present a criterion on the quasi-regularity of Dirichlet forms in $L^2(\mathcal P_p,\Lambda)$ for a reference probability $\Lambda$ on $\mathcal P_p$ by using an upper bound condition with the uniform norm of the intrinsic derivative. The condition is easy to check in applications. As a consequence, a class of quasi-regular local Dirichlet forms are constructed on $\mathcal P_p$ by using image of Dirichlet forms on the tangent space $L^p(X\to X,\mu_0)$ at a reference point $\mu_0\in \mathcal P_p$. In particular, the quasi-regularity is confirmed for Ornstein-Uhlenbeck type Dirichlet forms, and an explicit heat kernel estimate is derived based on the eigenvalues of the covariance operator of the underlying Gaussian measure.

**Venue:** online (streamed into MCS2068)

Click on title to see abstract.

Back to Homepage

Current and Upcoming Events

These events are hosted in and/or organised by members of the Department (follow links for details):

Aug 27--30 [MCS-tba] Durham Symposium: Large-scale behaviour of critical and near critical statistical physics models

Upcoming Seminars by Series

Click on series to expand.

Back to Homepage

Usual Venue: zoom

Contact: arthur.lipstein@durham.ac.uk

Apr 24 11:00 Giulio Salvatori (Max Planck Institute): *Positive Geometries and Scattering Amplitudes*

I will present a formulation of scattering amplitudes in the simplest colored, cubic, scalar theory - Tr \phi^3 - as an integral over the
space of curves on Riemann surfaces, valid at all loop orders and at all
orders in the topological 't Hooft expansion. This so-called "curve integral" has the main advantage of describing amplitudes as a unique object, rather than as a sum over Feynman diagrams, allowing to study phenomena which are hidden graph-by-graph and suggesting powerful techniques for the numerical evaluation of amplitudes. Furthermore, the singularity structure of the propagators of Tr \phi^3 theory is shared by any colored theory, thus suggesting the generalization of the formalism to more realistic theories by insertion of appropriate numerators in the curve integral. At the heart of the formalism is a simple counting problem associated to curves on surfaces, which surprisingly provides a combinatorial origin for the physics of scattering amplitudes. The talk is based on 2309.15913, 2311.09284 and 2402.06719.

**Venue:** zoom

Usual Venue: MCS3070

Contact: alpar.r.meszaros@durham.ac.uk

Apr 23 15:00 Erik Duse (KTH Stockholm): *Morrey inequalities and subelliptic estimates via Weitzenböck identities*

In joint work with Andreas Rosén we prove a general Weitzenböck identity for arbitrary pairs of constant coefficients homogeneous first order PDE operators on domains for fields that satisfy natural boundary conditions. This identity gives rise to a generalization of the Levi form for the classical d-bar complex in several complex variables. Under the assumption that one of the operators is cocancelling, a concept introduced by J. Van Schaftingen in his work on endpoint Sobolev estimates, and an additional algebraic condition we prove a generalized Morrey inequality. We derive from this a weighted Sobolev inequality as well as giving new proof of the equivalence of Morrey inequalities and subelliptic estimates. Using the theory of J. Kohn and L. Nirenberg this in particular implies solvability for the generalized Neumann-dbar problem on generalized strongly pseudoconvex domains for constant rank operators satisfying our conditions.

**Venue:** MCS3070

Apr 30 15:00 Eugene Shargorodsky (King's College London): *Variations on Liouville's theorem*

The talk discusses generalisations of Liouville's theorem to nonlocal translation-invariant operators. It is based on a joint work with D. Berger and R.L. Schilling, and a further joint work with the same co-authors and T. Sharia. We consider operators with continuous but not necessarily infinitely smooth symbols.

It follows from our results that if $\left\{\eta \in \mathbb{R}^n \mid m(\eta) = 0\right\} \subseteq \{0\}$, then, under suitable conditions, every polynomially bounded weak solution $f$ of the equation $m(D)f=0$ is in fact a polynomial, while sub-exponentially growing solutions admit analytic continuation to entire functions on $\mathbb{C}^n$.

**Venue:** MCS3070

Usual Venue: MCS2052

Contact: andrew.krause@durham.ac.uk

No upcoming seminars have been scheduled (not unusual outside term time).

Usual Venue: MCS2068

Contact: alexander.mangerel@durham.ac.uk

No upcoming seminars have been scheduled (not unusual outside term time).

Usual Venue: OC218

Contact: mohamed.anber@durham.ac.uk

For more information, see HERE.

No upcoming seminars have been scheduled (not unusual outside term time).

Usual Venue: MCS0001

Contact: inaki.garcia-etxebarria@durham.ac.uk,sunil.chhita@durham.ac.uk

No upcoming seminars have been scheduled (not unusual outside term time).

Usual Venue: MCS0001

Contact: sabine.boegli@durham.ac.uk,alpar.r.meszaros@durham.ac.uk

May 01 16:00 Endre Süli (Oxford): *Hilbert’s 19th problem and discrete De Giorgi–Nash–Moser theory: analysis and applications*

Mathematical models of non-Newtonian fluids play an important role in science and engineering, and their analysis has been an active field of research over the past decade. This talk is concerned with the mathematical analysis of numerical methods for the approximate solution of systems of nonlinear elliptic partial differential equations that arise in models of chemically reacting viscous incompressible non-Newtonian fluids, such as the synovial fluid found in the cavities of synovial joints. The synovial fluid consists of an ultra filtrate of blood plasma that contains hyaluronic acid, whose concentration influences the shear-thinning property and helps to maintain a high viscosity; its function is to reduce friction during movement. The shear-stress appearing in the model involves a power-law type nonlinearity, where, instead of being a fixed constant, the power law-exponent is a function of a spatially varying nonnegative concentration function, which, in turn, solves a nonlinear
convection-diffusion equation. In order to prove the convergence of the sequence of numerical approximations to a solution of this coupled system of nonlinear partial differential equations, a uniform Hölder norm bound needs to be derived for the sequence of numerical approximations to the concentration in a setting, where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely an L^∞ function. This necessitates the development of a discrete counterpart of the De Giorgi–Nash–Moser theory. Motivated by an early paper by Aguilera and Caffarelli (1986) in the simpler setting of Laplace’s equation, we derive such uniform Hölder norm bounds on the sequence of continuous piecewise linear finite element approximations to the concentration. We then use these to deduce the convergence of the sequence of approximations to a weak solution of the coupled system of nonlinear partial differential equations under consideration.

**Venue:** MCS0001

Usual Venue: MCS3052

Contact: andrew.krause@durham.ac.uk

No upcoming seminars have been scheduled (not unusual outside term time).

Usual Venue: MCS3070

Contact: gabriel.fuhrmann@durham.ac.uk

No upcoming seminars have been scheduled (not unusual outside term time).

Usual Venue: MCS2068

Contact: martin.p.kerin@durham.ac.uk

Recordings of past seminars can be found HERE.

Apr 25 13:00 Luc Vrancken (KU Leuven/Université Polytechnique Hauts-de-France): *Homogeneous 6 dimensional nearly Kaehler manifolds and their
submanifolds*

We present a survey of how the curvature tensor of all known
homogeneous 6 dimensional nearly Kähler spaces (both in the definite and
in the pseudo Riemannian case) can be expressed in an invariant way
using the induced geometric structures on the 6 dimensional nearly
Kähler space.

As an application we show how this can be used to study special classes of submanifolds in these spaces. In the latter case we will in particular focus on totally geodesic Lagrangian submanifolds and equivariant Lagrangian immersions.

**Venue:** MCS2068

May 02 13:00 Hendrik Süß (INI/Jena): *TBA*

**Venue:** MCS2068

May 09 13:00 Andrey Lazarev (Lancaster): *TBA*

**Venue:** MCS2068

Jun 13 10:00 Tirumala Venkata Chakradhar (Bristol): *TBA*

**Venue:** MCS2068

Jun 13 13:00 Asma Hassannezhad (Bristol): *TBA*

**Venue:** MCS2068

Jun 13 15:00 Georges Habib (Lebanese University/IECL Lorraine): *TBA*

**Venue:** MCS2068

Usual Venue: MCS3070

Contact: andrea.grigoletto@durham.ac.uk,nakarin.lohitsiri@durham.ac.uk

No upcoming seminars have been scheduled (not unusual outside term time).

Usual Venue: MCS0001

Contact: silvia.nagy@durham.ac.uk,ana.retore@durham.ac.uk

Apr 24 14:00 Valentina Forini (Humboldt University): *Conformal field theories from line defects, holography and the analytic bootstrap*

Wilson lines are a prototypical example of defect in quantum field theory. After reviewing the superconformal case - in which the one-dimensional defect CFT that they define is particularly interesting - I will discuss some analytic tools that may prove useful in this context, but are developed for generic 1d CFTs. Among them, a representation of the four-point correlator as a Mellin amplitude and via a recently derived dispersion relation.

**Venue:** MCS2068

Usual Venue: MCS2068

Contact: kohei.suzuki@durham.ac.uk

Apr 26 13:00 Simon Wittmann (Hong Kong Polytechnic University): *Construction of a Diffusion on the Wasserstein Space*

For stochastic analysis on the Wasserstein space, it is crucial to construct a diffusion process which plays a role of Brownian motion in finite-dimensions, or the Ornstein-Uhlenbeck process on a separable Hilbert space. This has been a long standing open problem due to the lack of a volume or Gaussian measure on the Wasserstein space, which could serve as an invariant measure.
To study diffusion processes on the $p$-Wasserstein space $\mathcal P_p $ for $p\in [1,\infty)$ over a separable, reflexive Banach space $X$, we present a criterion on the quasi-regularity of Dirichlet forms in $L^2(\mathcal P_p,\Lambda)$ for a reference probability $\Lambda$ on $\mathcal P_p$ by using an upper bound condition with the uniform norm of the intrinsic derivative. The condition is easy to check in applications. As a consequence, a class of quasi-regular local Dirichlet forms are constructed on $\mathcal P_p$ by using image of Dirichlet forms on the tangent space $L^p(X\to X,\mu_0)$ at a reference point $\mu_0\in \mathcal P_p$. In particular, the quasi-regularity is confirmed for Ornstein-Uhlenbeck type Dirichlet forms, and an explicit heat kernel estimate is derived based on the eigenvalues of the covariance operator of the underlying Gaussian measure.

**Venue:** online (streamed into MCS2068)

Usual Venue: MCS0001

Contact: raphael.zentner@durham.ac.uk

Apr 22 13:00 Abigail Ward (Cambridge): *Defining and computing algebraic invariants of symplectic manifolds*

Symplectic topology has been studied since Hamilton wrote
down his equations describing classical mechanics, but it was not until
work of Gromov and Floer in the 1980s that the field came into its
modern form. We will first discuss how Gromov's work on
pseudo-holomorphic curves has been used to produce algebraic,
Floer-theoretic invariants of symplectic manifolds such as the Fukaya
category, and give some example applications. We will then discuss the
arborealization program of Nadler which aims to systematically reduce
the calculation of these invariants to a combinatorial problem, in
analogy to how one might compute the homology of a space via a cellular
decomposition, and exhibit topological obstructions to this approach
found in joint work with Daniel Alvarez-Gavela and Tim Large.

**Venue:** MCS0001

Usual Venue: MCS3070

Contact: irving.d.calderon-camacho@durham.ac.uk,joe.thomas@durham.ac.uk

No upcoming seminars have been scheduled (not unusual outside term time).

Usual Venue: MCS2068

Contact: hyeyoung.maeng@durham.ac.uk,andrew.iskauskas@durham.ac.uk

No upcoming seminars have been scheduled (not unusual outside term time).

Contact: adam.stone2@durham.ac.uk

No upcoming seminars have been scheduled (not unusual outside term time).