2:30 pm / CH 445

John Napier and the Invention of Logarithms

The existence of a table of logarithms is not viewed with nearly as much excitement today as it was at the beginning of the 17^{th }century, or even in the first half of the 20^{th} century. E.W. Hobson, 1910-1931 Sadleirian Professor of Mathematics at Cambridge University, identified Napier’s work as one of the two greatest scientific discoveries that the world has ever seen. The talk will center on John Napier, the nature of his original work on logarithms, and the methods by which his tables were produced.

2:30 pm / CH 445

Quenched asymptotics for Brownian motion in a Gaussian potential

Recall that the notion of generalized function is introduced for the functions that are not defined point-wise, and is given as a linear functional over test functions. The same idea applies to random fields.

In this talk, we study the long term asymptotics for the quenched exponential moment of V(B(s)) where B(s) is d-dimensional Brownian motion, V(.) is a generalized Gaussian field. We will discuss the solution to an open problem posed by Carmona and Molchanov with an answer different from what was conjectured; the quenched laws for Brownian motions in Newtonian-type potentials, and in the potentials driven by white noise or by fractional white noise.

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"Inference and Control of Gene Regulatory Networks"

Abstract: we develop a comprehensive framework for optimal perturbation control of dynamic networks. The aim of the perturbation is to drive the network away from an undesirable steady-state distribution and to force it to converge towards a desired steady-state distribution. The proposed framework does not make any assumptions about the topology of the initial network, and is thus applicable to general topology networks. We subsequently demonstrate that there exists at most one optimal perturbation that forces the network into the desirable steady-state distribution. We investigate the robustness of the optimal perturbation control to errors in the probability transition matrix, and demonstrate that the proposed optimal perturbation control is robust to data and inference errors in the probability transition matrix of the initial network. Finally, we apply the proposed optimal

perturbation control method to the Human melanoma gene regulatory network in order to force the network from an initial steady-state distribution associated with melanoma and into a desirable steady-state distribution corresponding to a benign cell.

]]>Abstract: we develop a comprehensive framework for optimal perturbation control of dynamic networks. The aim of the perturbation is to drive the network away from an undesirable steady-state distribution and to force it to converge towards a desired steady-state distribution. The proposed framework does not make any assumptions about the topology of the initial network, and is thus applicable to general topology networks. We subsequently demonstrate that there exists at most one optimal perturbation that forces the network into the desirable steady-state distribution. We investigate the robustness of the optimal perturbation control to errors in the probability transition matrix, and demonstrate that the proposed optimal perturbation control is robust to data and inference errors in the probability transition matrix of the initial network. Finally, we apply the proposed optimal

perturbation control method to the Human melanoma gene regulatory network in order to force the network from an initial steady-state distribution associated with melanoma and into a desirable steady-state distribution corresponding to a benign cell.

CANCELLED

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2:30 pm / CH 445

On the chaotic character of some parabolic stochastic PDEs

Abstract: We present a family of non-linear noisy heat equations that have

"intermittent" and/or "chaotic" behavior. Among other things, we shall see

that a characteristic feature of many such noisy PDEs is that they develop

"shocks." All terms in quotations will be made precise during the talk.

This is based on joint work with Daniel Conus, Mathew Joseph, and Shang-Yuan Shiu.

2:30 pm / CH 445

Mathematical problems in the theory of molecular motors

The molecular motors are the special systems driven by the external periodic

random fields and can transform the molecular chaos into the deterministic

motion. This is one of the most popular subjects in the modern cell biology,

chemical kinetics etc. The analysis of the corresponding mathematical problems

includes the combination of the spectral theory, homogenization, complex

analysis, large deviations technique etc.

The talk will contain the basic definitions, models and recent asymptotical results

by me and Prof. B.Vainberg.]]>

2:30 pm / CH 445

Meager Composants of Continua

A continuum X is a compact, connected metric space. If p is in X then the meager composant M_p of p is defined by

M_p = {x ∈ X | there exists a nowhere dense subcontinuum A such that p, x ∈ A}.

In my talk I will discuss the relationship between the meager composants and certain topological properties such as X being locally connected, non-Suslinean or indecomposable. Several examples will be given.

2:30pm / 445

Mathematical approaches to Image Restoration and Segmentation

This talk will start with an introduction to image processing, from the well-known Total Variation minimizing denoising. The talk will consider the image inpainting problem and colorization. The term ``colorization'' was introduced by Wilson Markle who first processed the gray scale moon image from the Apollo mission. This term was used to describe the process of adding color to grayscale movies or TV broadcasting programs. A couple of variational colorization models will be presented which demonstrate different effects. In another direction, the Reproducing Kernel Hilbert Space approach will be presented for an effective colorization application. A link to image segmentation will be made through a medical image application.

]]>This talk will start with an introduction to image processing, from the well-known Total Variation minimizing denoising. The talk will consider the image inpainting problem and colorization. The term ``colorization'' was introduced by Wilson Markle who first processed the gray scale moon image from the Apollo mission. This term was used to describe the process of adding color to grayscale movies or TV broadcasting programs. A couple of variational colorization models will be presented which demonstrate different effects. In another direction, the Reproducing Kernel Hilbert Space approach will be presented for an effective colorization application. A link to image segmentation will be made through a medical image application.

2:30 pm / CH 405 (PLEASE NOTE THE NONSTANDARD LOCATION)

Title: Mathematics of the Rubik's Cube

Abstract: One of the most popular puzzles ever invented, the Rubik's cube, hides mathematical ideas of remarkable complexity. The 43 quintillion configurations possible on a standard 3x3x3 cube can be organized in an interesting algebraic structure, the Rubik's cube group. Using ideas and tools from modern algebra, we will describe the structure of the Rubik's cube group and outline a method for solving any configuration.

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2:30 pm / CH 445

On the Excursion Probabilities of Gaussian Random Fields

In this talk, we consider the excursion probabilities of two types of Gaussian random fields: those with stationary increments and smooth sample functions, and those with anisotropic and non-smooth (or fractal) sample functions. For the first type Gaussian random fields, we compute the expectation of the Euler characteristic of the excursion set and show that it provides an approximation to the excursion probability with exponentially small error. This verifies the ``Expected Euler Characteristic Heuristic'' for Gaussian random fields with stationary increments. For the second type of Gaussian random fields, we prove an asymptotic result which extends those of Pickands (1969), Piterbarg (1996), Chan and Lai (2006). This talk is based on joint works with Dan Cheng.2:30 pm / CH 443

Title: Status of Boltzmann-Sinai Hypothesis: Checkmarked

Abstract. The Boltzmann-Sinai Hypothesis dates back to 1963 as Sinai's modern formulation of Ludwig Boltzmann's statistical hypothesis in physics, actually as a conjecture: Every hard ball system on a flat torus is (completely hyperbolic and) ergodic (i. e. "chaotic", by using a nowadays fashionable, but a bit profane language) after fixing the values of the obviously invariant kinetic quantities.

In the half century since its inception quite a few people have worked on this conjecture, made substantial steps in the proof, created useful concepts and technical tools, or proved the conjecture in some special cases, sometimes under natural assumptions. Quite recently I was able to complete this project by putting the last, missing piece of the puzzle to its place, getting the result in full generality.

In the talk I plan to present the brief history of the proof by sketching the most important concepts and technical tools that the proof required. The talk should be accessible to a wide audience, including graduate students.

2:30 pm / CH 443

Title: The Corona Theorem

Abstract: Carleson's Corona Theorem from the 1960's has served as a major motivation for many results in complex function theory, operator theory and harmonic analysis. In this talk we will discuss some generalizations of this result to certain vector valued functions and connections with geometry and to function spaces on the unit ball in several complex variables.

]]>Abstract: Carleson's Corona Theorem from the 1960's has served as a major motivation for many results in complex function theory, operator theory and harmonic analysis. In this talk we will discuss some generalizations of this result to certain vector valued functions and connections with geometry and to function spaces on the unit ball in several complex variables.

2:30 pm / CH 443

Functional Central Limit Theorems for Heavy Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows

(joint with Takashi Owada)

Abstract:

We establish a new class of functional central limit theorems for

partial sums of certain symmetric stationary infinitely divisible processes with

regularly varying Levy measures. The limit process is a new class of

symmetric stable self-similar processes with stationary increments,

that coincides on a part of its parameter space with a previously

described process. The normalizing sequence and the limiting process

are determined by the ergodic theoretical properties of the flow

underlying the integral representation of the process. These

properties can be interpreted as determining how long is the memory of

the stationary infinitely divisible process. We also

establish functional convergence, in a strong distributional sense,

for conservative pointwise dual ergodic maps preserving an infinite

measure.]]>

2:30 pm / CH 443

`Positivity Preserving Semigroups and Heat Kernel and Green's Function Bounds`

`Abstract: Exploiting the notions of `*positivity** preserving* linear operators and *operator domination,* We establish Gaussian upper bounds for the heat kernels of Schrodinger-type operators on bounded Lipschitz domains with (possibly, non-local) Robin-type boundary conditions. Green's function estimates are also discussed.

]]>2:30 pm / CH 443

Theory of gene trees and species trees

A species tree provides a representation of the bifurcating structure that relates a set of species. A gene tree, which need not be identical to the species tree, relates the ancestry of a set of genetic lineages sampled from the species at a particular point in the genome. Conditional on a species tree, gene trees can be viewed as generated probabilistically from a distribution that naturally extends standard perspectives on within-population evolution. However, the gene tree distribution on the space of possible trees has unexpected combinatorial properties that produce surprising consequences for methods of reconstructing evolutionary trees. We explore these properties theoretically, using simulations, and with an example from data on North American pines.

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2:30 pm / CH 443

The many forms of E8

Abstract:

E8 has been a source of fascination in the theory of Lie groups for more than 100 years, and recently it has featured in several unrelated news stories. But it can be hard for non-experts to decipher the news, since there are various objects that are all called E8. This talk will describe the main ones, which ones go with which applications, and recent progress in understanding the various forms.]]>

2:30 pm / CH 445

Title: Smoothness properties for some infinite-dimensional heat kernel measures

Abstract: Smoothness is a fundamental principle in the study of measures on infinite-dimensional spaces, where an obvious obstruction to overcome is the lack of an infinite-dimensional Lebesgue or volume measure. Canonical examples of smooth measures include those induced by a Brownian motion, both its end point distribution and as a real-valued path. Heat kernel measure is the law of a Brownian motion on a curved space, and as such is the natural analogue of Gaussian measure there. We will discuss some recent smoothness results for these measures on certain natural classes of infinite-dimensional groups, including in some degenerate settings. This is joint work with F. Baudoin, D. Dobbs, and M. Gordina.

]]>Abstract: Smoothness is a fundamental principle in the study of measures on infinite-dimensional spaces, where an obvious obstruction to overcome is the lack of an infinite-dimensional Lebesgue or volume measure. Canonical examples of smooth measures include those induced by a Brownian motion, both its end point distribution and as a real-valued path. Heat kernel measure is the law of a Brownian motion on a curved space, and as such is the natural analogue of Gaussian measure there. We will discuss some recent smoothness results for these measures on certain natural classes of infinite-dimensional groups, including in some degenerate settings. This is joint work with F. Baudoin, D. Dobbs, and M. Gordina.

2:30 pm / CH 445

Title: Arithmetic groups and Geometric group theory

Abstract: The geometry of arithmetic groups -- such as SL(n,Z) -- allows us to deduce some of their algebraic properties. As examples, we'll talk about how we can use geometry to see that SL(n,Z) is finitely generated and finitely presented. We'll talk about other finiteness properties of arithmetic groups that extend the notions of finite generation and presentability, we'll talk about what's known for arithmetic groups, and give at least one example of what is still unknown. The topics in this talk provide good illustrations for the sorts of questions that geometric group theory hopes to answer for general finitely generated groups.]]>Cancelled

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3:30 pm / CH 205

"Microscopic derivation of the Ginzburg-Landau functional"

Abstract: I give an overview about rather recent joint results with R. Frank, R. Seiringer and J. P. Solovej.]]>

Nidhal Bouaynaya (University of Arkansas, Little Rock)

2:30 pm / CH 445
Sparse Time-varying Genetic Network Recovery using Constrained Kalman Filtering

]]>We tackle the problem of recovering time-varying (gene) networks from a series of undersampled and noisy observations. Gene regulatory networks evolve over time in response to functional requirements in the cell and environmental conditions. Collected genetic profiles from dynamic biological processes, such as cell development, cancer progression and treatment recovery, underlie genetic interactions that rewire over the course of time. We formulate the problem of estimating time-varying networks in a state-space framework. We show that, due to the small number of measurements, the system is unobservable; thus making the application of the standard Kalman filter ineffective. We remedy the problem by performing simultaneous compression and state estimation. The sparsity property of gene regulatory networks is incorporated as a constraint in the Kalman filter, leading to a compressed Kalman estimate and reducing the number of required observations for effective tracking of the network. Moreover, we improve the estimation accuracy by taking into account the entire sample set for each time instant estimate of the network through a forward backward smoothing procedure. The proposed constrained and smoothed Kalman filter is shown to yield good tracking results for varying small and medium-size networks.

2:30pm / CH 443

Title: Control of Systems Governed by Partial differential equations

]]>Abstract: Many control problems, such as active control of acoustic noise and vibration suppression, involve systems modelled by a partial differential equation. There are essentially two approaches to controller design for systems modelled by partial differential equations. In the first approach, the full model of the system is used in controller design. The designed controller is generally infinite-dimensional and is often subsequently reduced before implementation. This approach is generally not feasible, since a closed-form expression for the solution is not available. For most practical examples, a finite-dimensional approximation of the system is obtained and the controller is designed using this finite-dimensional approximation. The hope is that the controller has the desired effect when implemented on the original system. That this method is not always successful was first documented more than 30 years ago. A controller that stabilizes a reduced-order model need not necessarily stabilize the original model; or some other aspect of the system performance may be unacceptable. Systems with infinitely many eigenvalues either on or asymptotic to the imaginary axis are notorious candidates for problems.

In this talk, some issues associated with approximation of systems for the purpose of controller design are discussed along with conditions under which satisfactory controllers can be obtained using approximations.

2:30 pm / CH 443

Abstract: The theories of continuous extension of continuous functions and continuous metrics develop in parallel. The Tietze- Urysohn Extension Theorem asserts that every continuous real-valued mapping of a closed subset of a normal space extends continuously to the entire space. Hausdorff (1930) proved an analogous theorem for metrics. Dugundji (1951) proved that if A is a closed subset of a paracompact Hausdorff space X then there is a continuous, linear, norm-preserving, regular extension operator from C*(A) to C*(X). In the 1990's Bessaga and mathematicians from the Lviv school proved the analogue of Dugundji's theorem for metrics. Kuratowski (1956) showed how to topologize the space of continuous, bounded, real-valued functions with bounded variable domains in a metric space. In the 1980's Fillipov motivated by problems in ordinary differential equations began studying continuous extension operators for spaces of functions with variable domains. In this talk we will discuss recent progress in the study of continuous extension operators for functions and metrics with variable domains.

2:30 pm / CH 443

Title: The Wiener chaos and the Central Limit Theorem

Abstract:

The famous Wiener-Ito Theorem says that any square-integrable function of paths of a Brownian motion can uniquely be written as an orthogonal series of multiple integrals with respect to Brownian motion. Such series is known as the Wiener-Ito chaos decomposition of a Brownian functional. The celebrated fourth moment theorem of Nualart and Peccati (2005) states that, for homogeneous Wiener chaoses of a fixed order, the convergence of the second moments of chaoses to 1 and of their fourth moments to 3 is sufficient for the convergence in distribution of chaoses to the standard normal law, that is, the Central Limit Theorem holds.

In this talk we will explain the concept of Wiener chaos in some detail, and present surprisingly simple criteria for the asymptotic independence in the space of Wiener chaos, from which the above mentioned fourth moment theorem follows. We will show several other applications of our asymptotic independence criteria to the limit theory. That includes the asymptotic behavior of short and long range dependent stationary Gaussian time series. This talk is based on a joint work with Ivan Nourdin.

2:30 pm / CH 445

The Vacuum State of Quantum Many-Body Systems

A quantum many-body system is a field or particle system with interactions described in the framework of quantum mechanics. The vacuum state, or ground state, is the state of minimal energy of such a system. Mathematically, therefore, to find this state is to solve optimization problem. In this talk I will point to some of the special features of this optimization problem: locality, entanglement, and universality.