Past

In this talk we will discuss a new approach to non rationality of projective varieties based on HMS. Examples will be discussed.

I will elaborate on some recent developments on the theory of special functions which are relevant to the calculation of Feynman integrals in perturbative quantum field theory, highlighting the connections with some recent ideas in pure mathematics.

Speaker: Daniel Woodhouse (North)
Title: Generalizing Leighton's Graph Covering Theorem
Abstract: Before he ran off and became a multimillionaire, exploiting his knowledge of network optimisation, the computer scientist F. Thomas Leighton proved an innocuous looking result about finite graphs. The result states that any pair of finite graphs with isomorphic universal covers have isomorphic finite covers. I will explain what all this means, and why this should be of tremendous interest to group theorists and topologists.

Speaker: Benjamin Fehrman (South)
Title: Large deviations for particle processes and stochastic PDE
Abstract: In this talk, we will introduce the theory of large deviations through a simple example based on flipping a coin.  We will then define the zero range particle process, and show that its diffusive scaling limit solves a nonlinear diffusion equation.  The large deviations of the particle process about its scaling limit formally coincide with the large deviations of a certain ill-posed, singular stochastic PDE.  We will explain in what sense this relationship has been made mathematically precise.

Configuration spaces of points in Euclidean space or on a manifold are well studied in algebraic topology. But what if the points have some positive thickness? This is a natural setting from the point of view of physics, since this the energy landscape of a hard-spheres system. Such systems are observed experimentally to go through phase transitions, but little is known mathematically.

In this talk, I will focus on two special cases where we have started to learn some things about the homology: (1) hard disks in an infinite strip, and (2) hard squares in a square or rectangle. We will discuss some theorems and conjectures, and also some computational results. We suggest definitions for "homological solid, liquid, and gas" regimes based on what we have learned so far.

This is joint work with Hannah Alpert, Ulrich Bauer, Robert MacPherson, and Kelly Spendlove.

Outbreaks and epidemics from Ebola to influenza and measles are often in the news. Statistical analysis and modelling are frequently used to understand the transmission dynamics of epidemics as well as to inform and evaluate control measures, with real-time analysis being the most challenging but potentially most impactful. Examples will be drawn from diseases affecting both humans and animals.

Understanding the mechanisms of mutagenesis is important for prevention and treatment of numerous diseases, most prominently cancer. Large sequencing datasets revealed a substantial number of mutational processes in recent years, many of which are poorly understood or of completely unknown aetiology. These mutational processes leave characteristic sequence patterns in the DNA, often called "mutational signatures". We use bioinformatics methods to characterise the mutational signatures with respect to different genomic features and processes in order to unravel the aetiology and mechanisms of mutagenesis. 

In this talk, I will present our results on how mutational processes might be modulated by DNA replication. We developed a linear-algebra-based method to quantify the magnitude of replication strand asymmetry of mutational signatures in individual patients, followed by detection of these signatures in early and late replicating regions. Our analysis shows that a surprisingly high proportion (more than 75 %) of mutational signatures exhibits a significant replication strand asymmetry or correlation with replication timing. However, distinct groups of signatures have distinct replication-associated properties, capturing differences in DNA repair related to replication, and how different types of DNA damage are translated into mutations during replication. These findings shed new light on the aetiology of several common but poorly explained mutational signatures, such as suggesting a novel role of replication in the mutagenesis due to 5-methylcytosine (signature 1), or supporting involvement of oxidative damage in the aetiology of a signature characteristic for oesophageal cancers (signature 17). I will conclude with our ongoing work of wet-lab validations of some of these hypotheses and usage of computational methods (such as genetic algorithms) in guiding the development of experimental protocols.

G. Dimitrov and L. Katzarkov introduced in their paper from 2016 the counting of non-commutative curves and their (semi-)stability using T. Bridgeland's stability conditions on triangulated categories. To some degree one could think of this as the non-commutative analog of Gromov-Witten theory. However, its full meaning has not yet been fully discovered. For example there seems to be a relation to proving Markov's conjecture. 

For the talk, I will go over the definitions of stability conditions, non-commutative curves and their counting. After developing some tools relying on working with exceptional collections, I will consider the derived category of representations on the acyclic triangular quiver and will talk about the explicit computation of the invariants for this example.

In the 1920s Weyl proved the first non-trivial estimate for the Riemann zeta function on the critical line: \zeta(1/2+it) << (1+|t|)^{1/6+\epsilon}. The analogous bound for a Dirichlet L-function L(1/2,\chi) of conductor q as q tends to infinity is still unknown in full generality. In a breakthrough around 2000, Conrey and Iwaniec proved the analogue of the Weyl bound for L(1/2,\chi) when \chi is assumed to be quadratic of conductor q.  Building on the work of Conrey and Iwaniec, we show (joint work with Matt Young) that the Weyl bound for L(1/2,\chi) holds for all primitive Dirichlet characters \chi. The extension to all moduli q is based on aLindelöf-on-average upper bound for the fourth moment of Dirichlet L-functions of conductor q along a coset of the subgroup of characters modulo d when q^*|d, where q^* is the least positive integer such that q^2|(q^*)^3.

In biological cells, genomic DNA is complexed with proteins, forming so-called chromatin structure, and packed into the nucleus. Not only the nucleotide (A, T, G, C) sequence of DNA but also the 3D structure affects the genomic function. For example, certain regions of DNA are tightly packed with proteins (heterochromatin), which inhibits expression of genes coded there. The structure sometimes changes drastically depending on the state (e.g. cell cycle or developmental stage) of the cell. Hence, the structural dynamics of chromatin is now attracting attention in cell biology and medicine. However, it is difficult to experimentally observe the motion of the entire structure in detail. To combine and interpret data from different modes of observation (such as live imaging and electron micrograph) and predict the behavior, structural models of chromatin are needed. Although we can use molecular dynamics simulation at a microscopic level (~ kilo base-pairs) and for a short time (~ microseconds), we cannot reproduce long-term behavior of the entire nucleus. Mesoscopic models are wanted for that purpose, however hard to develop (there are fundamental difficulties).

In this seminar, I will introduce our recent theoretical/computational studies of chromatin structure, either microscopic (molecular dynamics of DNA or single nucleosomes) or abstract (polymer models and reaction-diffusion processes), toward development of such a mesoscopic model including local "states" of DNA and binding proteins.

References:

T. Kameda, A. Awazu, Y. Togashi, "Histone Tail Dynamics in Partially Disassembled Nucleosomes During Chromatin Remodeling", Front. Mol. Biosci., in press (2019).

Y. Togashi, "Modeling of Nanomachine/Micromachine Crowds: Interplay between the Internal State and Surroundings", J. Phys. Chem. B 123, 1481-1490 (2019).

E. Rolls, Y. Togashi, R. Erban, "Varying the Resolution of the Rouse Model on Temporal and Spatial Scales: Application to Multiscale Modelling of DNA Dynamics", Multiscale Model. Simul. 15, 1672-1693 (2017).

S. Shinkai, T. Nozaki, K. Maeshima, Y. Togashi, "Dynamic Nucleosome Movement Provides Structural Information of Topological Chromatin Domains in Living Human Cells", PLoS Comput. Biol. 12, e1005136 (2016).

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Everybody is familiar with the concept of eigenvalues of a matrix. In this talk, we consider the nonlinear eigenvalue problem. These are problems for which the eigenvalue parameter appears in a nonlinear way in the equation. In physics, the Schroedinger equation for determining the bound states in a semiconductor device, introduces terms with square roots of different shifts of the eigenvalue. In mechanical and civil engineering, new materials often have nonlinear damping properties. For the vibration analysis of such materials, this leads to nonlinear functions of the eigenvalue in the system matrix.

One particular example is the sandwhich beam problem, where a layer of damping material is sandwhiched between two layers of steel. Another example is the stability analysis of the Helmholtz equation with a noise excitation produced by burners in a combustion chamber. The burners lead to a boundary condition with delay terms (exponentials of the eigenvalue).

We often receive the question: “How can we solve a nonlinear eigenvalue problem?” This talk explains the different steps to be taken for using Krylov methods. The general approach works as follows: 1) approximate the nonlinearity by a rational function; 2) rewrite this rational eigenvalue problem as a linear eigenvalue problem and then 3) solve this by a Krylov method. We explain each of the three steps.

I will present recent results concerning a class of nonlinear parabolic systems of partial differential equations with small cross-diffusion (see doi.org/10.1051/m2an/2018036 and arXiv:1906.08060). Such systems can be interpreted as a perturbation of a linear problem and they have been proposed to describe the dynamics of a variety of large systems of interacting particles. I will discuss well-posedness, regularity, stability and convergence to the stationary state for (strong) solutions in an appropriate Banach space. I will also present some applications and refinements of the above-mentioned results for specific models.

I will present joint work with A. Pillay on the theory of NIP formulas in arbitrary groups, which exhibit a local formulation of the notion of finitely satisfiable generics (as defined by Hrushovski, Peterzil, and Pillay). This setting generalizes ``local stable group theory" (i.e., the study of stable formulas in groups) and also the case of arbitrary NIP formulas in pseudofinite groups. Time permitting, I will mention an application of these results in additive combinatorics.

We will discuss what it means to study the homology of a group via the construction of the classifying space. We will look at some examples of this construction and some of its main properties. We then use this to define and study the homology of the mapping class group of oriented surfaces, focusing on the approach used by Harer to prove his Homology Stability Theorem.

In this talk, I will introduce the notion of a sheaf on a topological space. I will then explain why "topological spaces" are an artificial limitation on enjoying life (esp. cohomology) to the fullest and what to do about that (answer: sites). Sheaves also fail our needs, but they have a suitable natural upgrade (i.e. stacks).
This talk will be heavily peppered with examples that come from the world around you (music, torsors, etc.).
 

Congruence monoids in the ring of integers are given by certain unions of arithmetic progressions. To each congruence monoid, there is a canonical way to associate a semigroup C*-algebra. I will explain this construction and then discuss joint work with Xin Li on K-theoretic invariants. I will also indicate how all of this generalizes to congruence monoids in the ring of integers of an arbitrary algebraic number field.

Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given 
instance. This lecture illustrates how these two fields complement each other, especially in the light of emerging new applications. We start with a gem from
the 19th century, namely the 3264 conics that are tangent to five given conics in the plane. Thereafter we turn to current problems in statistics, with focus on 
maximum likelihood estimation for linear Gaussian covariance models.
 

The Approximate Message Passing (AMP) algorithm is a powerful iterative method for reconstructing undersampled sparse signals. Unfortunately, AMP is sensitive to the type of sensing matrix employed and frequently encounters convergence problems. One case where AMP tends to fail is compressed sensing MRI, where Fourier coefficients of a natural image are sampled with variable density. An AMP-inspired algorithm constructed specifically for MRI is presented that exhibits a 'state evolution', where at every iteration the image estimate before thresholding behaves as the ground truth corrupted by Gaussian noise with known covariance. Numerical experiments explore the practical benefits of such effective noise behaviour.
 

Boundary integral equations (BIEs) are well established for solving scattering at bounded infinitely thin objects, so-called screens, which are modelled as “open surfaces” in 3D and as “open curves” in 2D. Moreover, the unknowns of these BIEs are the jumps of traces across $\Gamma$. Things change considerably when considering scattering at multi-screens, which are arbitrary arrangements of thin panels that may not be even locally orientable because of junction points (2D) or junction lines (3D). Indeed, the notion of jumps of traces is no longer meaningful at these junctions. This issue can be solved by switching to a quotient space perspective of traces, as done in recent work by Claeys and Hiptmair. In this talk, we present the extension of the quotient-space approach to the Galerkin boundary element (BE) discretization of first-kind BIEs. Unlike previous approaches, the new quotient-space BEM relies on minimal geometry information and does not require any special treatment at junctions. Moreover, it allows for a rigorous numerical analysis.
 

Recent advances in experimental imaging techniques have allowed us to observe the fine details of how droplets behave upon impact onto a substrate. However, these are highly non-linear, multiscale phenomena and are thus a formidable challenge to model. In addition, when the substrate is deformable, such as an elastic sheet, the fluid-structure interaction introduces an extra layer of complexity.

We present two modeling approaches for droplet impact onto deformable substrates: matched asymptotics and direct numerical simulations. In the former, we use Wagner's theory of impact to derive analytical expressions which approximate the behavior during the early time of impact. In the latter, we use the open source volume-of-fluid code Basilisk to conduct simulations designed to give insight into the later times of impact.

We conclude by showing how these methods are complementary, and a combination of both can give a thorough understanding of the droplet impact across timescales.