Apr 27 (Mon)
14:00 MCS2068 PureJulian Sahasrabudhe (Cambridge): Spectral laws of random matrices
Let \(A\) be an \(n\times n\) matrix where \(n\) is large and each entry is distributed identically and independently at random. What is the approximate distribution of the eigenvalues of \(A\) in the complex plane? This basic question, which goes back to the pioneering work of Wigner in the 1950s, has proven to be surprisingly rich with several beautiful mathematical connections. In this talk I will discuss something of the history of this problem, in addition to some recent advances.
Venue: MCS2068
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 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
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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: 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): TBA
Usual Venue: MCS2068
Contact: michael.r.magee@durham.ac.uk
Apr 27 14:00 Julian Sahasrabudhe (Cambridge): Spectral laws of random matrices
Let \(A\) be an \(n\times n\) matrix where \(n\) is large and each entry is distributed identically and independently at random. What is the approximate distribution of the eigenvalues of \(A\) in the complex plane? This basic question, which goes back to the pioneering work of Wigner in the 1950s, has proven to be surprisingly rich with several beautiful mathematical connections. In this talk I will discuss something of the history of this problem, in addition to some recent advances.
Venue: MCS2068
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
No upcoming seminars have been scheduled (not unusual outside term time).
