Dr. Yuval Yifrach (Universität Zürich)
Title T.B.A.
13:30 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Dr. Denis Nesterov (ETH Zürich)
Title T.B.A.
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Adrish Banerjee (Indian Institute of Technology, Kanpur)
Algebraic Design of DNA Codes with Biological and Combinatorial Constraints abstract
Abstract:
<p>DNA-based archival storage is vulnerable to synthesis and sequencing errors driven by long homopolymer runs, GC-content imbalance, and secondary-structure formation. At the same time, codeword design must also guard against nonspecific hybridization, captured by the Reversible (R) and Reversible-Complement (RC) constraints. In this talk, we present an algebraic framework that satisfies all of these requirements simultaneously. DNA block sets \\(\\mathcal{A} \\subseteq \\Sigma_{\\text{DNA}}^t\\) are built by an incremental algorithm that admits a candidate block only if no occurrence of it, within any four-block concatenation, creates a disjoint pair of Secondary-Complement/Reverse-Secondary-Complement (SC/RSC) substrings of length \\(\\ell\\); this local, checkable test on 4-fold concatenations suffices to guarantee that arbitrarily long concatenations of \\(\\mathcal{A}\\) remain \\((\\ell + 1)\\)-free-structure. Specializing this construction to \\(t = 2\\), \\(\\ell = 3\\) yields the four-element block set \\(\\mathcal{A}_4 = \\{AC, CA, TC, CT\\}\\), whose concatenations are further shown to be 3-free-homopolymer and exactly GC-balanced. A bijective, distance-preserving map \\(\\psi : \\mathbb{Z}_4 \\to \\mathcal{A}_4\\) then lifts linear codes over \\(\\mathbb{Z}_4\\) into DNA codes, with a simple generator-matrix symmetry condition guaranteeing the R and RC constraints. Using this map, four families are constructed: Modified Simplex DNA codes (Types 1 and 2), Modified Hamming DNA codes, and Reed--Muller-type DNA codes, each with non-vanishing rate or relative minimum distance as the length grows, and several members meeting the quaternary Singleton and Gilbert--Varshamov bounds.</p>
15:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Amina Abdurrahman (IHES)
Title T.B.A.
15:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Prof. Dr. Olaf Steinbach (TU Graz)
Space-time tensor-product finite element methods for parabolic problem abstract
Abstract:
We study space-time Galerkin--Petrov formulations for parabolic evolutionproblems and their relation to classical implicit time-stepping schemes.Although such schemes are stable in the usual time-stepping sense, theirinterpretation as space-time operator equations may lead to conditionalstability, with constants depending on the relation between temporal andspatial mesh sizes. We revisit this phenomenon for the continuous Galerkinmethod of Aziz and Monk, which yields the Crank--Nicolson scheme in thelowest-order case, and provide a detailed space-time error analysis forsolutions of both high and low regularity. In particular, the space-timeframework allows us to analyze the deteriorated behaviour of classicaltime-stepping methods for nonsmooth initial data. By applying integrationby parts in time, we derive an adjoint space-time formulation thatincorporates the initial condition in a natural variational way. In thelowest-order case, this formulation leads to a Rannacher-type smoothingof the initial data. The theoretical results are complemented bynumerical experiments. Joint work with R. Löscher and M. Reichelt.
16:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Prof. Dr. Marianna Russkikh (University of Notre Dame)
From Dimers to Maximal Surfaces in Minkowski Space R^{2,1} abstract
Abstract:
We discuss a class of graph embeddings into Minkowski space R^{2,2} = C^{1,1}, called t-surfaces, which arise in the study of the planar dimer model. A t-surface consists of a (perfect) t-embedding together with its associated origami map. Perfect t-embeddings were recently introduced as a key tool for proving that the gradient of the dimer height function converges to that of the Gaussian Free Field in a canonically associated metric, under suitable technical assumptions. After introducing the notion of a t-embedding, we describe a construction of perfect t-embeddings for regular hexagons of the hexagonal lattice. For which the corresponding t-surfaces converge to space-like maximal surfaces in Minkowski space R^{2,1}. As a consequence, this construction yields a new proof of the convergence of fluctuations of the dimer height function to the Gaussian Free Field in the conformal structure induced by the limiting maximal surface.
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43