Seminars and Colloquia by Series

Model Complexity Optimization

Series
Other Talks
Time
Friday, January 16, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Klaus 2447
Speaker
Alexey Chervonenkis – Russian Academy of Science and Royal Holloway University of London
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.

Integral points on higher-dimensional varieties

Series
Job Candidate Talk
Time
Friday, January 16, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Aaron Levin – Scuola Normale Superiore Pisa
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.

Conditions of the uniform convergence of empirical averages to their expectations

Series
Stochastics Seminar
Time
Thursday, January 15, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Alexey Chervonenkis – Russian Academy of Sciences and Royal Holloway University of London
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.

Stimulus space topology and geometry from neural activity

Series
Job Candidate Talk
Time
Thursday, January 15, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Carina Curto – Mathematics Department, New York University
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.

Some recent results in topological graph theory

Series
Graph Theory Seminar
Time
Thursday, January 15, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Hein van der Holst – Eindhoven University of Technology
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.)

Schur's problems on means of algebraic numbers

Series
School of Mathematics Colloquium
Time
Thursday, January 15, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Igor Pritzker – Oklahoma State University
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.

Can you compute the asymptotics of the Apery sequence?

Series
Research Horizons Seminar
Time
Wednesday, January 14, 2009 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Stavros Garoufalidis – School of Mathematics, Georgia Tech
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.

Dilute Quantum Gases

Series
Math Physics Seminar
Time
Monday, January 12, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Robert Seiringer – Princeton University
We present an overview of mathematical results on the low temperature properties of dilute quantum gases, which have been obtained in the past few years. The discussion includes, for instance, results on the free energy in the thermodynamic limit, and on Bose-Einstein condensation, Superfluidity and quantized vortices in trapped gases. All these properties are intensely being studied in current experiments on cold atomic gases. We will give a brief description of the mathematics involved in understanding these phenomena, starting from the underlying many-body Schroedinger equation.

Electro-Optics for Beach Zone Observation

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 12, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Frank Crosby – Naval Surface Warfare Center, Panama City
Several imaging innovations have been designed to find hidden objects in coastal areas of entry, such as beaches and ports. Each imaging device is designed to exploit particular distinguishing characteristics. This talk with cover using a tunable multi-spectral camera for polarization based detection and object identification with a flash LIDAR camera that produces three-dimensional imagery.

Some random matrix problems in high-dimensional statistics

Series
Job Candidate Talk
Time
Thursday, January 8, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Noureddine El Karoui – UC Berkeley
It is now increasingly common in statistical practice to encounter datasets in which the number of observations, n, is of the same order of magnitude as the number of measurements, p, we have per observation. This simple remark has important consequences for theoretical (and applied) statistics. Namely, it suggests on the theoretical front that we should study the properties of statistical procedures in an asymptotic framework where p and n both go to infinity (and p/n has for instance a finite non-zero limit). This is drastically different from the classical theory where p is held fixed when n goes to infinity. Since a number of techniques in multivariate statistics rely fundamentally on sample covariance matrices and their eigenvalues and eigenvectors, the spectral properties of large dimensional covariance matrices play a key role in such "large n, large p" analyses. In this talk, I will present a few problems I have worked on, concerning different aspects of the interaction between random matrix theory and multivariate statistics. I will discuss some fluctuation properties of the largest eigenvalue of sample covariance matrices when the population covariance is (fairly) general, talk about estimation problems for large dimensional covariance matrices and, time permitting, address some applications in a classic problem of mathematical finance. The talk will be self-contained and no prior knowledge of statistics or random matrix theory will be assumed.

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