A self-avoiding walk is a path on a graph that visits each site at most once. The model was introduced by chemists Flory and Orr as a model for polymer molecules. It leads to rich mathematical theories and is difficult to understand because of its non-markovian character. Two basic questions: about the number of self-avoiding walks of a fixed length and about an average distance between the endpoints - are still far from being answered.
Denote the number of the self-avoiding walks of length $n$ in $Z^d$ by $c_n$. The beginning of rigorous results on the self-avoiding walk is the power-law behaviour of $c_n$: $1/n c_n^{1/n}$ tends to $\mu$ (Hammersley, Morton '54). This mu is called a connective constant. One of the classical results is an upper bound on $c_n$ (Hammersley, Welsh '62). Another important result is convergence of $c_{n+2}/c_n$ to $\mu^2$ (Kesten '63). Recently, it was proven that the connective constant on the honeycomb lattice is equal to $\sqrt{2 + \sqrt{2}}$ (Duminil-Copin, Smirnov '11). Their proof can be generalised to a weighted self-avoiding walk on $Z^2$ we allow self-touchings but penalise of them (G. '14).
There is even less known results on an average between the endpoints. Recently it was proven that a self-avoiding walk on $Z^d$ is sub-ballistic (Duminil-Copin, Hammond '12). There is also a result on the delocalisation of a self-avoiding walk that in particular shows that a it doesn't come back to the origin (Duminil-Copin, G., Hammond, Manolescu '13).
A novel approach to describe the variability of the statistics of intracranial EEG (icEEG) data is proposed that is an adaptation of the diffusion map framework. Diffusion maps, which extend principal components analysis and provide a nonlinear approach, provide dimensionality reduction of the data as well as pattern recognition that can be used to distinguish different states of a patient, for example, interictal and preseizure states. A new algorithm, which is an extension of diffusion maps, is developed to construct coordinates that generate efficient geometric representations of the complex structures in the icEEG data. Numerical results show that the proposed approach provides a distinction between interictal and preseizure states.
Furthermore, the algorithm is also applied to classify magnetic resonance images (MRI) of brains of patients with Alzheimer's Disease and those without Alzheimer's Disease. The method is adapted to the MRI and accounts for the variability in calibration of the MRI of different patients.
Additionally, the icEEG data of the epilepsy patients are used to test the existence of a relationship between distant parts of the default mode network (DMN), a resting state network defined by fMRI studies. Magnitude squared coherence, mutual information, cross-approximate entropy, and the coherence of the gamma power time-series were estimated, for one hour icEEG recordings of background activity from 9 patients, to evaluate the relationship between two test areas. These two test areas were within the DMN (anterior cingulate and orbital frontal, denoted as T1 and posterior cingulate and mesial parietal, denoted as T2), and one control area (denoted as C) was outside the DMN. The goal was to test if the relationship between T1 and T2 was stronger than the relationship between each of these areas and C. A low level of relationship was observed among the 3 areas tested. The relationships among T1, T2, and C did not demonstrate support for the DMN. The results obtained underscore the considerable difference between electrophysiological and hemodynamic measurements of brain activity and possibly suggest a lack of neuronal involvement in the DMN.
The Langlands Program (based on visionary conjectures formulated by Robert Langlands) is one of the most fascinating and challenging topics in twenty-first century mathematics. This talk will give an introduction to one part of the program : the local Langlands conjecture. This conjecture is about the representation theory of reductive p-adic groups. The conjecture asserts that this representation theory can be understood via a certain Galois group. The talk is intended for non-experts and will be expository i.e. no previous knowledge of the subject will be assumed and all are welcome.
Chromosomes are packed in the nuclei of cells into a very particular structure called chromatin. As the details of this dynamic structure are elusive to direct observation due to diffraction limits, we can only observe them indirectly through so called chromosome contact matrices. I will discuss the topic of chromosome contact matrices and some mathematical problems associated with their analysis (normalization, decomposition, finding domain boundaries). No background knowledge will be required.
I will try to explain the idea of quantum computing. I will start from the basics and then present the simple quantum Deutsch-Jozsa algorithm. If there will be time I will sketch the ideas behind the famous Shor factorization algorithm.
A large literature was devoted in the last years to the derivation of fracture models from more regular ones, mainly within the framework of Gamma-convergence. Such a variational approximation has turned out to be an efficient analytical tool and has provided a numerical strategy to analyze the behaviour of those energies and their minimizers. The first result in that sense was obtained by Ambrosio and Tortorelli for fracture energies depending on the length of the crack, with regularizing functionals which were in fact damage energies. In this spirit we will present a recent approximation result for fracture energies depending also on the amplitude of the crack (joint work with S. Conti and M. Focardi).
We will present new and elementary approach to showing the smoothness of solutions of partial differential equations with fractional dissipation. Embarrassingly, since 1934, the regularity theorists cannot agree whether solutions to Navier-Stokes remain regular or blow up, even when lured by a one-million-dollar prize. Hence the regularity theory of PDEs has been dubbed their dark side. One of the most promising direction of attacking the Navier-Stokes problem is to consider the vorticity equation. This approach contributed to a serious interest in PDEs with dissipation given by a fractional laplacian, which will be the starting point of our talk. After introducing the concept of a fractional laplacian and commenting on applicability of PDEs involving it, we will focus on a new and ingenious technique by Kiselev, Nazarov, Volberg and Shterenberg. It allows to obtain a clear picture of the smoothness vs. blowup regimes for fractional Burgers equation. Next, we will comment on its applicability to the critical 2D dissipative quasi-geostrophic equation. Time permitting, an alternative method will be presented. We will finish with a variation of the method by Kiselev et als. that allows to disprove a blowup conjecture for the critical fractional Smoluchowski-Poisson equation. This last part is based on a joint work with Rafael Granero.
Assuming that both temperature and pressure are continuous functions, we can conclude that there are always two antipodal points on Earth with exactly the same pressure and temperature. This is the two-dimensional version of the celebrated Borsuk-Ulam Theorem which states that for any continuous map from the n-dimensional sphere to n-dimensional real Euclidean space there is always a pair of antipodal points on the sphere that are identified by the map. Our quest to unravel topological mysteries in the Middle Earth of quantum spaces will begin with gentle preparations in the Shire of elementary topology. Then, after riding swiftly through the Rohan of C*-algebras and Gelfand-Naimark Theorems and carefully avoiding the Mordor of incomprehensible technicalities, we shall arrive in the Gondor of compact quantum groups acting on unital C*-algebras. It is therein that the generalized Borsuk-Ulam-type statements dwell waiting to be proven or disproven. Time permitting, we shall pay tribute to the ancient quantum group SUq(2), and show how the proven non-trivializability of the SUq(2)-principal instanton bundle is a special case of two different noncommutative Borsuk-Ulam-type conjectures. (Based on joint work with Paul F. Baum and Ludwik Dąbrowski.)
Classification is of fundamental importance in mathematics: to what extent can we distinguish two complicated mathematical objects by a simpler invariant? To every unital C*-algebra A, one may assign its Elliott invariant. This consists of two abelian groups K0(A) and K1(A), the tracial state space T(A), and a pairing map between tracial states and states on K0(A). Elliott conjectured that for any two separable simple unital nuclear C*-algebras A and B, there exists a *-isomorphism between A and B if and only if there exists an isomorphism of their Elliott invariants. The conjecture has been proven for many subclasses of separable simple unital nuclear C*-algebras, but is now known not to hold in full generality, leading to reformulations of the conjecture. In this talk I will introduce the main concepts and milestones in the classification programme and discuss some of the interesting open problems.
Algebraic geometry is the study of solutions to systems of polynomial equations, so-called varieties. In this talk I discuss various famous examples of algebraic varieties, such as projective spaces, and curves and surfaces. We go on to think about moduli spaces of some of these objects, and examine their most interesting features using straightforward and explicit equations.
In order to study a given lattice in a Lie group, one would like to use both the Lie-theoretic methods for closed subgroups and techniques of discrete groups. The latter work well (usually) provided that a group is finitely generated. How to ensure this is the case? A beautiful geometric argument attributed to Poincaré shows that finite generation holds for lattices in SL(2,R) (or more generally, for groups acting properly cocompactly on a hyperbolic space): using it one is able to write explicitly a finite system of generators. One would expect that showing it for some other Lie groups would amount to a clever guess of system of generators for a given lattice (and, probably technical, establishing the desired properties). For SL(n,R) with n>2 the question remained unsolved up to 1967, when Kazhdan introduced the famous property (T): an interplay between analysis and group theory. A shockingly short and direct argument answers the aforementioned question, but the answer is by no means explicit (which may be even more shocking). Later on, Kazhdan's property (T) was used by Margulis in order to provide explicit (i.e. non-random) constructions of expander graphs - graphs that are both sparse and highly connected, a very useful topic in engineering. During the talk I would like to present all the aforementioned constructions, and if time permits, some modern applications of Kazhdan's property (T).
Poncelet's closure theorem also known as Poncelet's porism is known to be one of the most astonishing and beautiful theorems in classical projective geometry. In its easiest form, it states that, if there is a polygon with n-sides inscribed in one circle $O$ and circumscribed around another circle $o$, then for each point $p \in O$ one can construct such a polygon with vertex $p$. The theorem was formulated and proven by Jean-Victor Poncelet during his captivity in Russia in the years 1812–1814. The proof was synthetic but based on not quite formal arguments commonly used at that time. A modern proof is realized in terms of algebraic geometry and interprets the theorem in the language of elliptic curves. In our talk we shall survey the history of the theorem and modern approaches to its proof as well as possible generalizations and surprising relations to seemingly unrelated questions in number theory.
We discuss first the regularity properties of two-dimensional periodic water waves traveling under the influence of gravity. The situation considered is that of rotational waves with a rough - that is discontinuous or unbounded - vorticity distribution. Such waves have a remarkable property: the wave surface and all the other streamlines are real-analytic graphs. Using this optimal regularity property, I will show that the symmetry of the wave surface can be characterized in terms of the underlying flow. Secondly, I will point out that the wave profile can be recovered from the velocity field on a vertical axis of symmetry of the wave surface.
We will introduce notion of Hopf Algebra and present the concept of quantum groups. We will give several examples of "classical" quantum groups and explain how one can study their representation theory, and present some related classification results.
The homeomorphic solutions to a nonlinear Beltrami equation (with the ellipticity bounded by the square root of 2 near the infinity) generate a two-dimensional submanifold of the local Sobolev space of differentiability 1 and integrability 2. We will discuss quasiconformal maps and sketch the proof that uses normal family arguments and the uniqueness of solutions to the Beltrami equation.
I will discuss the method of characteristics for first order partial differential equations. Starting from the basics we will gradually enter the complex world of branching and colliding characteristics. Finding a way through it by means of measure theory will lead to new results regarding hyperbolic differential equations.
One of the main parts of the theory of dynamical systems deals with iterates of continuous maps of a compact space to itself. If this space is one-dimensional, general phenomena (like, for example, chaos) can be present, while the simplicity of the space allows us to use strong tools. I will describe various aspects of one-dimensional dynamics, both topological and smooth.