Seminars in Mathematical Sciences

Seminars in the next week
May 29 (Wed)

11:00 zoom A&CGabriele Dian (DESY): The Weighted Cosmological Polytope

In a seminal paper, Akrani-Hamed, Benicasa and Postikov showed how any graph contributing to the wavefunction of the universe in a class of toy models of conformally coupled scalars (with non-conformal interactions) in FRW cosmologies can be computed as the canonical form of polytope named the cosmological polytope. In this talk, I will show how to extend this geometrical approach directly to cosmological correlators introducing novel geometries we name weighted cosmological polytopes. In this picture, all the possible ways of organising, and computing, cosmological correlators correspond to triangulations and subdivisions of the geometry, containing the one in terms of wavefunction coefficients and many others. (based on arXiv:2401.05207)

Venue: zoom

Zoom: https://durhamuniversity.zoom.us/j/95077429183?pwd=Y2JlcWE0QmVLRFhwOVhXU3huQmp0UT09


Click on title to see abstract.

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Current and Upcoming Events

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

Jun 13 [MCS2068/0001] Spectral Geometry Day

Speakers: G Berkolaiko, TV Chakradhar, G Habib, A Hassannezhad. See G&T seminar list for titles & abstracts.

Venue: MCS2068/0001

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

Upcoming Seminars by Series

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• Amplitudes and Correlators

Usual Venue: zoom

Contact: arthur.lipstein@durham.ac.uk

May 29 11:00 Gabriele Dian (DESY): The Weighted Cosmological Polytope

In a seminal paper, Akrani-Hamed, Benicasa and Postikov showed how any graph contributing to the wavefunction of the universe in a class of toy models of conformally coupled scalars (with non-conformal interactions) in FRW cosmologies can be computed as the canonical form of polytope named the cosmological polytope. In this talk, I will show how to extend this geometrical approach directly to cosmological correlators introducing novel geometries we name weighted cosmological polytopes. In this picture, all the possible ways of organising, and computing, cosmological correlators correspond to triangulations and subdivisions of the geometry, containing the one in terms of wavefunction coefficients and many others. (based on arXiv:2401.05207)

Venue: zoom

• Analysis and PDE

Usual Venue: MCS3070

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

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

• Applied Mathematics

Usual Venue: MCS2052

Contact: andrew.krause@durham.ac.uk

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

• Arithmetic Study Group

Usual Venue: MCS2068

Contact: alexander.mangerel@durham.ac.uk

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

• CPT Colloquium

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).

• Department Research Colloquium

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).

• Distinguished and Public Lectures

Usual Venue: MCS0001

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

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

• Education and Pedagogy

Usual Venue: MCS3052

Contact: andrew.krause@durham.ac.uk

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

• Ergodic Theory and Dynamics

Usual Venue: MCS3070

Contact: gabriel.fuhrmann@durham.ac.uk

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

• Geometry and Topology

Usual Venue: MCS2068

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

Recordings of past seminars can be found HERE.


Jun 13 09:00 Tirumala Venkata Chakradhar (Bristol): Eigenvalue bounds for the Steklov problem on differential forms in warped product manifolds

The Steklov eigenvalue problem is known to have generalisations to the framework of differential forms. We consider a version that is of interest from geometric perspective and present eigenvalue bounds in the case of warped product manifolds, in various settings such as manifolds with non-negative Ricci curvature and convex boundary, hypersurfaces of revolution, etc. We compare and contrast the behaviour with known results in the case of functions (i.e., 0-forms), highlighting the influence of the underlying topology on the spectrum for p-forms in general.

Venue: MCS2068

Jun 13 10:00 Georges Habib (Lebanese University/IECL Lorraine): A Poincaré formula for differential forms and applications

We prove a new general Poincaré type inequality for differential forms on compact Riemannian manifolds with nonempty boundary.

When the boundary is isometrically immersed in Euclidean space, we derive a new inequality involving mean and scalar curvatures of the boundary only and characterize the limiting case. Also a new Ros-type inequality for differential forms is derived assuming the existence of a nonzero parallel form on the manifold. This work is joint with Nicolas Ginoux and Simon Raulot.

Venue: MCS2068

Jun 13 12:00 Asma Hassannezhad (Bristol): Steklov eigenvalues of hyperbolic manifolds with totally geodesic boundary

The geometry and topology of closed negatively curved manifolds are subtly reflected in a geometric bound for the Laplace eigenvalues. In 1980, Schoen, Wolpert, and Yau showed that the small Laplace eigenvalues can be bounded from below and above by the length of a collection of closed simple geodesics cutting the surface into disjoint connected components. Schoen later obtained a spectral gap on negatively curved manifolds in higher dimensions which is in contrast with the result for hyperbolic surfaces.  In this talk, we discuss how these results can be extended to the setting of the Steklov eigenvalue problem.

Venue: MCS0001

Jun 13 16:00 Gregory Berkolaiko (Texas A&M University): Duistermaat index and eigenvalue interlacing for perturbations in boundary conditions.

Eigenvalue interlacing is a tremendously useful tool in linear algebra and spectral analysis.  In its simplest form, the interlacing inequality states that a rank-one positive perturbation shifts the eigenvalue up, but not further than the next unperturbed eigenvalue.  For different types of perturbations, this idea is known as the "Weyl interlacing" (additive perturbations), "Cauchy interlacing" (for principal submatrices of Hermitian matrices), "Dirichlet-Neumann bracketing" and so on.

We discuss the extension of this idea to general "perturbations in boundary conditions", encoded as interlacing between eigenvalues of two self-adjoint extensions of a fixed symmetric operator with finite (and equal) defect numbers.  In this context, even the terms such as "signature of the perturbation" are not immediately clear, since one cannot take the difference of two operators with different domains.  However, it turns out that definitive answers can be obtained, and they are expressed most concisely in terms of the Duistermaat index, an integer-valued topological invariant describing the relative position of three Lagrangian planes in a symplectic space.  Two of the planes describe the two self-adjoint extensions being compared, while the third one corresponds to the distinguished Friedrichs extension.

We will illustrate our general results with simple examples, avoiding technicalities as much as possible and giving intuitive explanations of the Duistermaat index, the rank and signature of the perturbation in the self-adjoint extension, and the curious role of the third extension (Friedrichs) appearing in the answers.

Based on a work in progress with Graham Cox, Yuri Latushkin and Selim Sukhtaiev.

Venue: MCS2068

• HEP Journal Club

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).

• HEP Lunchtime

Usual Venue: MCS0001

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

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

• Probability

Usual Venue: MCS2068

Contact: kohei.suzuki@durham.ac.uk

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

• Pure Maths Colloquium

Usual Venue: MCS0001

Contact: raphael.zentner@durham.ac.uk

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

• Spectra and Moduli

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).

• Statistics

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).

• Stats4Grads

Contact: adam.stone2@durham.ac.uk

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