We determine the minimal forbidden minors characterising the class of countable graphs that embed into some compact surface. We will also discuss Thomas’s conjecture that the Robertson—Seymour Graph Minor Theorem extends to countably infinite graphs. [https://arxiv.org/abs/2301.11042]

We show that every 2-coloring of the natural numbers and any finite coloring of the rationals contains monochromatic sets of the form $\{x, y, xy, x+y\}$. We also discuss generalizations and obstructions to extending this result to arbitrary finite coloring of the naturals. This is partially based on joint work with Marcin Sabok.

Motivated by the desire to construct large independent sets in random graphs, Karp and Sipser modified the usual greedy construction to yield an algorithm that outputs an independent set with a large cardinal called the Karp-Sipser core. When run on the Erdős-Rényi $G(n,c/n)$ random graph, this algorithm is optimal as long as $c < e$. We will present the proof of a physics conjecture of Bauer and Golinelli (2002) stating that at criticality, the size of the Karp-Sipser core is of order $n^{3/5}$. Along the way we shall highlight the similarities and differences with the usual greedy algorithm and the $k$-core algorithm.
Based on a joint work with Nicolas Curien and Thomas Budzinski.

A $k$-block in a graph is a set of $k$ vertices every two of which are joined by $k$ vertex disjoint paths. By a result of Weissauer, graphs with no $k$-blocks admit tree-decompositions with especially useful structure. While several constructions show that it is probably very difficult to characterize induced subgraph obstructions for bounded tree width, a lot can be said about graphs with no $k$-blocks. On the other hand, forbidding induced subgraphs places significant restrictions on the structure of a $k$-block in a graphs. We will discuss this phenomenon and its consequences in the study of tree-decompositions in classes of graphs defined by forbidden induced subgraphs.

We introduce a new method - ESPIRA (Estimation of Signal Parameters via Iterative Rational Approximation) \cite{DP22,  DPP21} - for the recovery of complex exponential  sums
$$
f(t)=\sum_{j=1}^{M} \gamma_j \mathrm{e}^{\lambda_j t},
$$
that are determined by a finite number of parameters: the order $M$, weights $\gamma_j \in \mathbb{C} \setminus \{0\}$ and nodes  $\mathrm{e}^{\lambda_j} \in \mathbb{C}$ for $j=1,...,M$.  Our new recovery procedure is based on the observation that Fourier coefficients (or DFT coefficients) of exponential sums have a special rational structure.  To  reconstruct this structure in a stable way we use the AAA algorithm  proposed by Nakatsukasa et al.   We show that ESPIRA can be interpreted as a matrix pencil method applied to Loewner matrices. 

During the talk we will demonstrate that ESPIRA outperforms Prony-like methods such as ESPRIT and MPM for noisy data and for signal approximation by short exponential sums.  

Bibliography
N. Derevianko,  G.  Plonka, 
Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients, Anal.  Appl.,  20(3),  2022,  543-577.

N. Derevianko,  G. Plonka,  M. Petz, 
From ESPRIT to ESPIRA: Estimation of signal parameters by iterative rational approximation,   IMA J. Numer. Anal.,  43(2),  2023, 789--827.  

Y. Nakatsukasa, O. Sète,   L.N. Trefethen,  The AAA algorithm for rational approximation.
SIAM J. Sci. Comput., 40(3),   2018,  A1494–A1522.  

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time. 

Students presenting are:

Adrian Martini, supervisor Alison Ethridge

Fang Rui Lim, supervisor Rama Cont

Thomas Groves, supervisor Dmitry Beylaev

Sarah-Jean Meyer, supervisor Massimiliano Gubinelli

One can think of the stabilisation of an ∞-category as the ∞-category of objects that admit infinite deloopings. For example, the ∞-category of spectra is the stabilisation of the ∞-category of homotopy types. Costabilisation is the opposite notion of stabilisation, where we are interested in objects that allow infinite desuspensions. It is easy to see that the costabilisation of the ∞-category of homotopy types is trivial. Fix a prime number p. In this talk I will show that the costablisation of the ∞-category of T(h)-local spectral Lie algebras is equivalent to the ∞-category of T(h)-local spectra, where T(h) denotes a p-local telescope spectrum of height h. A key ingredient of the proof is to relate spectral Lie algebras to (spectral) Eₙ algebras via Koszul duality.