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Series: PDE Seminar

Living systems are subject to constant evolution through the two processes of mutations and selection, a principle discovered by Darwin. In a very simple, general, and idealized description, their environment can be considered as a nutrient shared by all the population. This allows certain individuals, characterized by a 'phenotypical trait', to expand faster because they are better adapted to the environment. This leads to select the 'best fitted trait' in the population (singular point of the system). On the other hand, the new-born population undergoes small variance on the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait?
We will give a mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the 'best fitted trait' and eventually compute various forms of branching points, which represent the cohabitation of two different populations.
The concepts are based on the asymptotic analysis of the above mentioned parabolic equations, one appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that describe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed.
This work is based on collaborations with O. Diekmann, P.-E. Jabin, S. Mischler, S. Cuadrado, J. Carrillo, S. Genieys, M. Gauduchon and G. Barles.

Series: Other Talks

<p>You are cordially invited to attend a reception that will follow the seminar to chat informally with faculty and students. Refreshments will be provided.</p>

The existing machine learning paradigm considers a simple scheme: given a set of training examples find in a given collection of functions the one that in the best possible way approximates the unknown decision rule. In such a paradigm a teacher does not play an important role. In human learning, however, the role of a teacher is very important: along with examples a teacher provides students with explanations, comments, comparisons, and so on. In this talk I will introduce elements of human teaching in machine learning. I will consider an advanced learning paradigm called learning using hidden information (LUHI), where at the training stage a teacher gives some additional information x^* about training example x. This information will not be available at the test stage. I will consider the LUHI paradigm for support vector machine type of algorithms, demonstrate its superiority over the classical one and discuss general questions related to this paradigm. For details see FODAVA, Foundations of Data Analysis and Visual Analytics

Series: Job Candidate Talk

After introducing and reviewing the situation for rational and integral points on curves, I will discuss various aspects of integral points on higher-dimensional varieties. In addition to discussing recent higher-dimensional results, I will also touch on connections with the value distribution theory of holomorphic functions and give some concrete open problems.

Series: Other Talks

It is shown (theoretically and empirically) that a reliable result can be gained only in the case of a certain relation between the capacity of the class of models from which we choose and the size of the training set. There are different ways to measure the capacity of a class of models. In practice the size of a training set is always finite and limited. It leads to an idea to choose a model from the most narrow class, or in other words to use the simplest model (Occam's razor). But if our class is narrow, it is possible that there is no true model within the class or a model close to the true one. It means that there will be greater residual error or larger number of errors even on the training set. So the problem of model complexity choice arises – to find a balance between errors due to limited number of training data and errors due to excessive model simplicity. I shall review different approaches to the problem.

Series: Job Candidate Talk

Series: Stochastics Seminar

The uniform convergence of empirical averages to their expectations for a set of bounded test functions will be discussed. In our previous work, we proved a necessary and sufficient condition for the uniform convergence that can be formulated in terms of the epsilon-entropy of certain sets associated to the sample. In this talk, I will consider the case where that condition is violated. The main result is that in this situation strong almost sure oscillations take place. In fact, with probability one, for a given oscillation pattern, one can find an admissible test function that realizes this pattern for any positive prescribed precision level.

Series: Job Candidate Talk

We construct our understanding of the world solely from neuronal activity generated in our brains. How do we do this? Many studies have investigated how the electrical activity of neurons (action potentials) is related to outside stimuli, and maps of these relationships -- often called receptive fields -- are routinely computed from data collected in neuroscience experiments. Yet how the brain can understand the meaning of this activity, without the dictionary provided by these maps, remains a mystery. I will present some recent results on this question in the context of hippocampal place cells -- i.e., neurons in rodent hippocampus whose activity is strongly correlated to the animal's position in space. In particular, we find that topological and geometric features of the animal's physical environment can be derived purely from the activity of hippocampal place cells. Relating stimulus space topology and geometry to neural activity opens up new opportunities for investigating the connectivity of recurrent networks in the brain. I will conclude by discussing some current projects along these lines.

Series: Graph Theory Seminar

Each graph can be embedded in 3-space. The problem becomes more interesting if we put restrictions on the type of embedding. For example, a linkless embedding of a graph is one where each pair of vertex-disjoint circuits has linking number equal to zero. The class of all graphs that have a linkless embedding is closed under taking minors. Robertson, Seymour, and Thomas gave the forbidden minors for this class of graphs. Open remained how to find a linkless embedding in polynomial time. In the talk we start with discussing an algorithm to find a linkless embedding.Instead of embedding the graph in 3-space, we could also consider mapping properties of certain superstructures of the graph in 3-space, and, indeed, if this superstructure has not the right mapping properties in 3-space, see whether it has the right one in 4-space, etc. Recently, we introduced for a graph G a new graph parameter \sigma(G), which is defined as the smallest d such that superstructures of G have a zero intersection mapping in d-space. The nicest property of this graph parameter is its independence of the superstructure and thus depends on the graph only. For d=2 and d=3, \sigma(G) \leq d if and only if G is outerplanar and planar, respectively. The graphs G with \sigma(G)\leq 4 are exactly those that have a linkless embedding. In the second part of the talk we will discuss this new graph parameter. (This part is joint work with R. Pendavingh.)

Series: School of Mathematics Colloquium

Issai Schur (1918) considered a class of polynomials with integer coefficients and simple zeros in the closed unit disk. He studied the limit behavior of the arithmetic means s_n for zeros of such polynomials as the degree n tends to infinity. Under the assumption that the leading coefficients are bounded, Schur proved that \limsup_{n\to\infty} |s_n| \le 1-\sqrt{e}/2. We show that \lim_{n\to\infty} s_n = 0 as a consequence of the asymptotic equidistribution of zeros near the unit circle. Furthermore, we estimate the rate of convergence of s_n to 0. These results follow from our generalization of the Erdos-Turan theorem on discrepancy in angular equidistribution of zeros. We give a range of applications to polynomials with integer coefficients. In particular, we show that integer polynomials have some unexpected restrictions of growth on the unit disk. Schur also studied problems on means of algebraic numbers on the real line. When all conjugate algebraic numbers are positive, the problem of finding \liminf_{n\to\infty} s_n was developed further by Siegel and many others. We provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line.

Series: Research Horizons Seminar

The Apery sequence is a sequence of natural numbers 1,5,73,1445,...which is used to prove the irrationality of zeta(3). Can you compute its asymptotic expansion to all orders of 1/n? The talk will not assume a lot, but promises to compute, and also justify.