Apr 28 (Tue)
13:00 MCS2068 APDEAlexandre Girouard (Université Laval): Isoperimetric-type upper bounds for Steklov eigenvalues of warped products
We obtain isoperimetric-type upper bounds for the Steklov eigenvalues of Riemannian warped products \(D\times_h\Sigma\), where \(D\) is compact with boundary and \(\Sigma\) is closed. The bounds depend on volume, dimension and \(L^p\) norms of the warping function \(h\). The bounds are sharp and in some cases we obtain quantitative stability improvement. Homogenization theory will play a role in saturating some bounds. I will compare this use of homogenization theory to some of its other uses in spectral geometry. This is based on joint work with Jade Brisson, Bruno Colbois, Katie Gittins and Jean Lagacé.
Venue: MCS2068
15:00 MCS0001 PureSergey Fomin (U Michigan): Incidence geometry and tiled surfaces
We show that various classical theorems of linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a general result that involves a tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing and interpreting the known ones.
This is joint work<https://arxiv.org/abs/2305.07728> with Pavlo Pylyavskyy<https://cse.umn.edu/math/pavlo-pylyavskyy>.
Venue: MCS0001
Apr 29 (Wed)
14:00 MCS2068 S&MYulin Gong (University of Bristol): Observability and Semiclassical Control for Schrödinger Equations on Non-compact Hyperbolic Surfaces
In this talk, we study the observability of the Schrödinger equation on $X$, a noncompact covering space of a compact hyperbolic surface $M$. Using a generalized Bloch theory, wave functions on $X$ are identified as sections of a unitary flat Hilbert bundle over $M$. We extend the semiclassical analysis to unitary flat Hilbert bundles and generalize Dyatlov and Jin's semiclassical control to all flat unitary Hilbert bundles over $M$, with uniform constants independent of the choice of bundle. Furthermore, if the Riemannian cover $X \rightarrow M$ is a normal cover with a virtually Abelian deck transform group $\Gamma$, we apply the generalized Bloch theorem to derive the observability from all $\Gamma$-periodic open subsets of $X$. We will also discuss the application of uniform semiclassical control in spectral geometry. This is joint work with Xin Fu (Westlake University) and Yunlei Wang (Louisiana State University).
Venue: MCS2068
16:00 zoom A&CYao Qi Zhang (University of Hertfordshire): To square or not to square: from \(A\) to \(\abs{A}^2\) and back in ABJM theory
We discuss recent progress on squared amplitudes in Aharony-Bergman-Jafferis-Maldacena (ABJM) theory, where amplitudes with different multiplicities and loop orders can be unified into a single generating function with a hidden $S_{n+\ell}$ permutation symmetry. This object admits graphical representations in terms of weight-3 $f$-graphs. Building on this framework, we outline how planar integrands of amplitudes, as well as bipartite integrands for the logarithm of amplitudes, can be systematically extracted. This provides evidence that, despite their highly unified structure, squared amplitudes contain sufficient information to reconstruct amplitudes. We conclude by commenting on possible connections to squared amplituhedron and correlahedron in three-dimension.
Venue: zoom
Online: https://teams.microsoft.com/meet/32190483378522?p=bLb3UsnZxoD344qQ73
Apr 30 (Thu)
13:00 MCS2068 G&TAnthea Monod (Imperial): Topological Graph Kernels from Tropical Geometry
We introduce a new class of graph kernels for machine
learning with metric graphs based on tropical geometry and the graph
topologies. Unlike traditional graph kernels that are defined by graph
combinatorics (nodes, edges, subgraphs), our approach considers only the
geometry and topology of the underlying metric space. A key property of
our construction is its invariance under edge subdivision, making the
kernels intrinsically well-suited for comparing graphs that represent
different underlying spaces. Our kernels are efficient to compute and
depend only on the graph genus rather than the size. In label-free
settings, our kernels outperform existing methods, which we showcase on
synthetic, benchmarking, and real-world road network data. Joint work
with Yueqi Cao (KTH Sweden).
Venue: MCS2068
14:00 MCS2068 ProbZhen-Qing Chen (University of Washington): Quantitative homogenization for long range random walks in dynamic random environments
In this talk, I will present quantitative homogenization results for stable-like long range random walks in time-dependent random conductance models, where the conductance is bounded above but can be degenerate. Based on joint work with X. Chen, T. Kumagai and J. Wang.
Venue: MCS2068
May 05 (Tue)
13:00 MCS2068 APDEYufei Zhang (Imperial College London): Continuous-time mean field games: a primal-dual characterization
This paper establishes a primal-dual formulation for continuous-time mean field games (MFGs) and provides a complete analytical characterization for the set of all Nash equilibria (NEs). We first show that for any given mean field flow, the representative player’s control problem with measurable coefficients is equivalent to a linear program over the space of occupation measures. We then establish the dual formulation of this linear program as a maximization problem over smooth subsolutions of the associated Hamilton-Jacobi-Bellman (HJB) equation. Finally, a complete characterization of all NEs for MFGs is established by the strong duality between the linear program and its dual problem. This strong duality is obtained by the solvability of the dual problem, and in particular through the regularity of the associated HJB equation.
A key new insight of our analysis for MFGs is that the dual variable within the primal-dual formulation is only required to coincide with the solution of the HJB equation on the support of the mean field flow, reminiscent of the adjoint process in the stochastic maximum principle which only needs to be defined along the optimal state trajectory.
Compared with existing approaches for MFGs, the primal-dual formulation and its NE characterization require neither the convexity of the associated Hamiltonian nor the uniqueness of its optimizer, and remain applicable when the HJB equation lacks classical or even continuous solutions.
Venue: MCS2068
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Contact: arthur.lipstein@durham.ac.uk
Apr 29 16:00 Yao Qi Zhang (University of Hertfordshire): To square or not to square: from \(A\) to \(\abs{A}^2\) and back in ABJM theory
We discuss recent progress on squared amplitudes in Aharony-Bergman-Jafferis-Maldacena (ABJM) theory, where amplitudes with different multiplicities and loop orders can be unified into a single generating function with a hidden $S_{n+\ell}$ permutation symmetry. This object admits graphical representations in terms of weight-3 $f$-graphs. Building on this framework, we outline how planar integrands of amplitudes, as well as bipartite integrands for the logarithm of amplitudes, can be systematically extracted. This provides evidence that, despite their highly unified structure, squared amplitudes contain sufficient information to reconstruct amplitudes. We conclude by commenting on possible connections to squared amplituhedron and correlahedron in three-dimension.
Venue: zoom
Usual Venue: MCS2068
Contact: yohance.a.osborne@durham.ac.uk
Apr 28 13:00 Alexandre Girouard (Université Laval): Isoperimetric-type upper bounds for Steklov eigenvalues of warped products
We obtain isoperimetric-type upper bounds for the Steklov eigenvalues of Riemannian warped products \(D\times_h\Sigma\), where \(D\) is compact with boundary and \(\Sigma\) is closed. The bounds depend on volume, dimension and \(L^p\) norms of the warping function \(h\). The bounds are sharp and in some cases we obtain quantitative stability improvement. Homogenization theory will play a role in saturating some bounds. I will compare this use of homogenization theory to some of its other uses in spectral geometry. This is based on joint work with Jade Brisson, Bruno Colbois, Katie Gittins and Jean Lagacé.
Venue: MCS2068
May 05 13:00 Yufei Zhang (Imperial College London): Continuous-time mean field games: a primal-dual characterization
This paper establishes a primal-dual formulation for continuous-time mean field games (MFGs) and provides a complete analytical characterization for the set of all Nash equilibria (NEs). We first show that for any given mean field flow, the representative player’s control problem with measurable coefficients is equivalent to a linear program over the space of occupation measures. We then establish the dual formulation of this linear program as a maximization problem over smooth subsolutions of the associated Hamilton-Jacobi-Bellman (HJB) equation. Finally, a complete characterization of all NEs for MFGs is established by the strong duality between the linear program and its dual problem. This strong duality is obtained by the solvability of the dual problem, and in particular through the regularity of the associated HJB equation.
A key new insight of our analysis for MFGs is that the dual variable within the primal-dual formulation is only required to coincide with the solution of the HJB equation on the support of the mean field flow, reminiscent of the adjoint process in the stochastic maximum principle which only needs to be defined along the optimal state trajectory.
Compared with existing approaches for MFGs, the primal-dual formulation and its NE characterization require neither the convexity of the associated Hamiltonian nor the uniqueness of its optimizer, and remain applicable when the HJB equation lacks classical or even continuous solutions.
Venue: MCS2068
Usual Venue: MCS3070
Contact: andrew.krause@durham.ac.uk
No upcoming seminars have been scheduled (not unusual outside term time).
Usual Venue: MCS2068
Contact: herbert.gangl@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: alpar.r.meszaros@durham.ac.uk
May 06 15:00 Giuseppe Savaré (Bocconi University, Milan, Italy.): Diffusion, Optimal Transport and Ricci Curvature
The interplay between diffusion, optimal transport and Ricci curvature has been an active theme at the interface of analysis, geometry and probability over the last two decades.
On the analytic side, Bakry—Émery's \(Gamma\)-calculus characterises lower Ricci curvature bounds through gradient estimates for the heat semigroup, within the framework of Dirichlet forms.
On the geometric side, the work of Lott, Sturm and Villani shows that the same curvature information is encoded in the displacement convexity of the relative entropy along Wasserstein geodesics, leading to a synthetic notion of Ricci curvature that makes sense on general metric measure spaces.
The two viewpoints meet on the class of RCD spaces, developed in a series of papers with L. Ambrosio and N. Gigli. On such spaces the heat flow is linear --- equivalently, the Cheeger energy is quadratic --- and admits a double description as the \(L^2\)-gradient flow of the Dirichlet energy and as the Wasserstein EVI flow of the entropy. Together with stability under measured Gromov--Hausdorff convergence, this allows RCD spaces to inherit many of the structural properties and calculus tools available on smooth Riemannian manifolds.
The lecture will present the main ideas behind this circle of results and will close with some recent developments along related directions: gradient flows for convex functionals on RCD spaces, Hellinger--Kantorovich contractions, and metric Sobolev structures on spaces of random measures.
Venue: MCS0001
Usual Venue: MCS3052
Contact: andrew.krause@durham.ac.uk
No upcoming seminars have been scheduled (not unusual outside term time).
Usual Venue: MCS2068
Contact: martin.p.kerin@durham.ac.uk
Apr 30 13:00 Anthea Monod (Imperial): Topological Graph Kernels from Tropical Geometry
We introduce a new class of graph kernels for machine
learning with metric graphs based on tropical geometry and the graph
topologies. Unlike traditional graph kernels that are defined by graph
combinatorics (nodes, edges, subgraphs), our approach considers only the
geometry and topology of the underlying metric space. A key property of
our construction is its invariance under edge subdivision, making the
kernels intrinsically well-suited for comparing graphs that represent
different underlying spaces. Our kernels are efficient to compute and
depend only on the graph genus rather than the size. In label-free
settings, our kernels outperform existing methods, which we showcase on
synthetic, benchmarking, and real-world road network data. Joint work
with Yueqi Cao (KTH Sweden).
Venue: MCS2068
May 07 13:00 Wilhelm Klingenberg (Durham): Smirnov Decomposition of horizontal vector currents in
Heisenberg space
Joint work with Zhengyao Huang. Note that a divergence free
horizontal vector current in Heisenberg space may be viewed as an
element of the dual space of test vector horizontal fields. Using a
horizontal Liouville theorem in Heisenberg space, the resulting flow
lines of the divergence free vector field give rise to a family of
horizontal curves and a measure on the collection of such. This proves a
Smirnov-type decomposition as a current. As application we prove a
result on horizontal free approximation in C^1 on compact subsets of
Heisenberg space for which all rectifiable curves are constant.
Venue: MCS2068
May 14 13:00 Victoria Pelayo Alvaredo (Universidad Autónoma de Madrid/ICMAT): Beyond Dvoretzky's Theorem
Dvoretzky's theorem states that in real normed spaces there
exist subspaces of sufficiently high dimension that are nearly
Euclidean. Equivalently, every high-dimensional symmetric convex body
has a section of dimension on the order of log(n) that is close to a
Euclidean ball.
We study a functional version of this phenomenon in terms of radial
functions on the sphere. We also give versions of Dvoretzky's Theorem in
complex and quaternionic normed spaces. Finally, using the Hopf
fibration, which lifts positive functions on complex or quaternionic
projective spaces to radial functions on the sphere, we obtain
functional versions of these extensions.
Venue: MCS2068
Usual Venue: MCS2068
Contact: michael.r.magee@durham.ac.uk
Apr 28 15:00 Sergey Fomin (U Michigan): Incidence geometry and tiled surfaces
We show that various classical theorems of linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a general result that involves a tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing and interpreting the known ones.
This is joint work<https://arxiv.org/abs/2305.07728> with Pavlo Pylyavskyy<https://cse.umn.edu/math/pavlo-pylyavskyy>.
Venue: MCS0001
Usual Venue: MCS3070
Contact: joe.thomas@durham.ac.uk
Apr 29 14:00 Yulin Gong (University of Bristol): Observability and Semiclassical Control for Schrödinger Equations on Non-compact Hyperbolic Surfaces
In this talk, we study the observability of the Schrödinger equation on $X$, a noncompact covering space of a compact hyperbolic surface $M$. Using a generalized Bloch theory, wave functions on $X$ are identified as sections of a unitary flat Hilbert bundle over $M$. We extend the semiclassical analysis to unitary flat Hilbert bundles and generalize Dyatlov and Jin's semiclassical control to all flat unitary Hilbert bundles over $M$, with uniform constants independent of the choice of bundle. Furthermore, if the Riemannian cover $X \rightarrow M$ is a normal cover with a virtually Abelian deck transform group $\Gamma$, we apply the generalized Bloch theorem to derive the observability from all $\Gamma$-periodic open subsets of $X$. We will also discuss the application of uniform semiclassical control in spectral geometry. This is joint work with Xin Fu (Westlake University) and Yunlei Wang (Louisiana State University).
Venue: MCS2068
