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Mathematics & Statistics

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Roles that temperature plays in self-assembly

Shinnosuke Seki

Åalto University

School of Science

Department of Information and Computer Science

Åalto, Finland

3:00pm-4:00pm

CMC 130

Nataša Jonoska

**Abstract**

Self-assembly can be found anywhere in our daily life: growth of snow crystal, heart pulse, white-black pattern of zebra, neural network of brains, and so on. Complex patterns or behaviors emerge from quite simple rules among its components without any central authority or external control. From the viewpoint of computer science, let us ask: given a final goal, how simple can a system be from which the final goal emerges spontaneously? This question will lead us to an exciting and fruitful field of algorithmic self-assembly.

Winfree's abstract Tile Assembly Model (aTAM) is a model of molecular self-assembly of DNA complexes known as tiles, which float freely in solution and attach one at a time to a growing “seed” assembly based on specific binding sites on their four sides if the sum of binding energies of bonds thus formed are strong enough compared to the threshold called the temperature. Almost all studies have focused on the case when temperature was \(2\), that is, only \(3\) energy levels \(0\), \(1\), \(2\), were meaningful. Our recent works expand the focus on arbitrary temperature, and investigate roles of temperature in self-assembly.

In this talk, audiences are first introduced to the world of self-assembly and to aTAM. The second part aims at presenting our recent proposal of a polynomial-time algorithm that, given the \(n \times n\) square, finds the aTAM with minimum number of tile types to self-assemble the square. This is a positive answer to an open question proposed by Adleman, et al. in their STOC 2002 paper. This investigation reveals the relationship between the size of aTAM and environmental temperature. The question of whether the number of tile types are the temperature can be minimized simultaenously is also addressed.

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Quadratic differentials: Recent applications to Complex Analysis, Potential Theory and Differential Equations

Alexander Yu. Solynin

Department of Mathematics and Statistics

Texas Tech University

Lubbock, TX

4:00pm-5:00pm

CMC 130

Dmitry Khavinson

**Abstract**

A quadratic differential on a Riemann surface is a \((2;0)\)-form \(Q(z)\,dz^2\) defined by a meromorphic function \(Q(z)\). First appeared in extremal problems of geometric function theory, quadratic differentials have found applications in numerous areas of mathematics and theoretical physics. In this talk, I will discuss a general extremal problem related to quadratic differentials. Then I will discuss several applications of the theory of quadratic differentials to particular extremal problems in Complex Analysis and Potential Theory. In the third part of my talk I will discuss recent applications of quadratic differentials to the study of properties of polynomial solutions of some classical Differential Equations.

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Constructions of Paley type group schemes

Yuqing Chen

Wright State University

3:00pm-4:00pm

CMC 130

Xiang-dong Hou

**Abstract**

Paley type group schemes are special 2-class association schemes which give rise to skew Hadamard designs or Paley type strongly regular graphs. Classical examples of such schemes include quadratic residues of finite fields of odd characteristics. In this talk I will present new constructions of Paley type group schemes in finite fields. This is a joint work with Tao Feng of Zhejiang University.

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Bifurcation of Non-Autonomous Stochastic Equations

Bixiang Wang

New Mexico Institute of Mining and Technology

3:00pm-4:00pm

CMC 130

Yuncheng You

**Abstract**

This talk is concerned with bifurcation of random dynamical systems generated by non-autonomous stochastic equations. We first introduce definitions of pathwise random almost periodic and almost automorphic solutions for stochastic equations, which are corresponding counterparts of non-autonomous deterministic systems. We then discuss pitchfork bifurcation of random periodic (almost periodic, almost automorphic) solutions of equations with multiplicative noise. We also demonstrate that additive white noise could destroy bifurcation of non-autonomous deterministic equations. Finally, we discuss bifurcation of random periodic solutions of a class of stochastic parabolic equations on bounded domains.

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Self Assembly of Aperiodic Hierarchical Tilings

Chaim Goodman-Strauss

University of Arkansas

3:00pm-4:00pm

CMC 130

Nataša Jonoska

**Abstract**

In 1982 Daniel Schechtman discovered the first quasicrystalline materials, in which the atomic structure is apparently hierarchical and non-periodic, just as in many aperiodic tilings, such as those formed by the Penrose tiles. Despite much activity, it is still not clear how local interactions can assemble such hierarchical structures—in fact, a result by Dworkin and Shei has long been interpreted as saying no such local interactions could suffice and that global information is required. Recently Jonoska and Karpenko introduced a method involving passing signals; here we discuss self-assembly of a hierarchal tiling using purely local rules, in Winfree's abstract Tile Assembly Model.

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Graph expanders

Zoran Šunić

Texas A&M University

3:00pm-4:00pm

CMC 130

Milé Krajčevski

**Abstract**

We first motivate our discussion of expanders by an example of their use in coding theory. Then we survey some known constructions due to Margulis (coming from representation theory), Reingold, Vadhan, Wigderson (zig-zag product of graphs), and others. We end with some open questions.

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Group of diffeomorphisms of the unit circle and sub-Riemannian geometry

Irina Markina

Department of Mathematics

University of Bergen

Bergen, NORWAY

4:00pm-5:00pm

CMC 130

Catherine Bénéteau

**Abstract**

We consider the group of sense-preserving diffeomorphisms of the unit circle and its central extension — the Virasoro-Bott group as sub-Riemannian manifolds. Shortly, a sub-Riemannian manifold is a smooth manifold \(M\) with a given sub-bundle \(D\) of the tangent bundle, and with a metric defined on the sub-bundle \(D\). The different sub-bundles on considered groups are related to some spaces of normalized univalent functions. We present formulas for geodesics for different choices of metrics. The geodesic equations are generalizations of Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDEs. We show that any two points in these groups can be connected by a curve tangent to the chosen sub-bundle. We also discuss the similarities and peculiarities of the structure of sub-Riemannian geodesics on infinite and finite dimensional manifolds.

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Sampling from the Dirichlet space

Daniel Seco

Universitat Autònoma de Barcelona

Barcelona, SPAIN

3:00pm-4:00pm

CMC 130

Catherine Bénéteau

**Abstract**

Sampling is the process of recovering a function in a given space of functions \(E\) from its values at a sequence of points. Sequences for which this can be done with upper and lower norm control are called sampling sequences. The control is based on the norm of the sequence in a given space of sequences \(l\).

In this talk we present the problem of characterizing sampling sequences when we take the space of analytic functions on the disk whose derivatives are square integrable (the Dirichlet space) at sequences of a naturally associated space.