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Mathematics & Statistics

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Scaling limits in conformal aggregation models

Alan Sola

University of Cambridge

3:00pm-4:00pm

CMC 130

Catherine Bénéteau

**Abstract**

In the one-parameter Hastings-Levitov growth process, aggregation of particles in the plane is modeled using compositions of random conformal maps. This procedure produces random sets in the plane, called clusters, and by varying the parameter value, one obtains clusters resembling those observed in physical processes, such as Eden growth and diffusion-limited aggregation (DLA).

It is of interest to understand the asymptotic behavior of clusters as the size of individual particles becomes small, and the speed of aggregation is rapid. Computer simulations produce intriguing pictures, and seem to reveal a rich structure in the HL process, but due to the strong dependence of the cluster's growth on its past, it seems to be a difficult problem to translate numerical observations into mathematically rigorous statements.

Reporting in part on joint work with F. Johansson Viklund (Columbia) and A. Turner (Lancaster), I will discuss some recent progress in the study of small-particle scaling limits in some variants of the HL growth process.

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The Hermitian two matrix model with quartic potential

Arno Kuijlaars

Katholieke Universiteit Leuven

3:00pm-4:00pm

CMC 130

Vilmos Totik

**Abstract**

I will discuss eigenvalues of random matrices from the Hermitian two-matrix model with an even quartic potential. The mean limiting eigenvalue distribution is governed by a vector equilibrium problem for three measures with external fields and an upper constraint. Varying the parameters one observes phase transitions at the closing or opening of a gap in the limiting spectrum. The talk is based on joint work with Maurice Duits (Stockholm) and Man Yue Mo (Bristol).

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On the Sz.-Nagy--Foias functional calculus

László Kérchy

Visiting Professor

Indiana University Bloomington

3:00pm-4:00pm

CMC 130

Vilmos Totik

**Abstract**

The Sz.-Nagy--Foias functional calculus, operating with bounded analytic functions on the open unit disc, proved to be an efficient tool in the study of Hilbert space contractions. After making an overview of some basic structure theorems yielded by this calculus, we introduce the quasianalytic spectral set of a contraction. Roughly speaking this is the largest measurable set on the unit circle satisfying the condition that if a decreasing sequence of functions is asymptotically non-vanishing on this set then, the corresponding functions of the contraction form a decreasing sequence of operators which is also asymptotically non-vanishing. We show how this spectral invariant can be used in the study of the hyperinvariant subspace problem.

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Exact solutions to a variable-coefficient modified Korteweg-de Vries equation

Yi Zhang

Zhejiang Normal University

Zhejiang, China

3:00pm-4:00pm

CMC 130

Wen-Xiu Ma

**Abstract**

In this talk, we will discuss a variable-coefficient modified Korteweg-de Vries (vc-mKdV) equation. With the help of symbolic computation, the bilinear forms for the vc-mKdV equation are obtained when the coefficient functions obey the Painlevé-integrable conditions. The \(N\)-soliton solution and Riemann-theta function periodic wave solutions are derived through the Hirota bilinear method. Finally, we will talk about Wronskian determinant solutions of the equation by means of the Wronskian technique.

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Progress and problems in graph packing

Gexin Yu

Assistant Professor

College of William and Mary

4:00-5:00pm

CMC 130

Brendan Nagle

**Abstract**

Two graphs pack if one graph can be placed on the vertex set of the other to make a simple graph. Many extremal graph theory problems can be viewed as graph packing problems. In this talk, we will survey some important progress toward the Bollobas-Eldridge-Catlin Conjecture and present some new problems on graph packing.

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Enhancements of Counting Invariants

Sam Nelson

Claremont McKenna College

Claremont, CA

3:00pm-4:00pm

CMC 130

Mohamed Elhamdadi

**Abstract**

Counting invariants are computable invariants of knots and links defined as cardinalities of sets of homomorphisms between certain algebraic objects associated to knots and links. An enhancement of a counting invariant takes advantage of extra information or structure to strengthen the counting invariant. In this talk we will see various examples of counting invariants and enhancements.

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Paul Montel’s Problem and Complex Approximation Theory

Arthur Danielyan

University of South Florida

4:00pm-5:00pm

CHE 302

Yuncheng You

**Abstract**

We first give a brief survey of the development of complex approximation theory starting from the Weierstrass theorem, and discuss the solutions of several problems proposed by P. Montel (in 1910), M. A. Lavrentiev (1936), A. Davie (1972), L. Zalcman (1982), and other people. New elementary approaches to some fundamental theorems (due to P. Fatou, F. and M. Riesz, A. Beurling, E. Bishop, L. Carleson, W. Rudin, and others) will be presented. In certain cases the new idea transforms the commonly accepted point of view to classical theorems and properties of functions. For example, the boundary uniqueness property of analytic or univalent functions turns out to be a particular case (a corollary) of the property of the existence of boundary values of the same functions.

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On Controllability and Optimality in Geometry

Mauricio Godoy Molina

Post-doctoral Fellow

University of Göttingen

3:05pm-3:55pm

CMC 108

Boris Shekhtman

**Abstract**

Control theory deals with problems related to steering underdetermined dynamical systems. Two of the most important problems in this area are the existence of admissible trajectories connecting any two states of a system (controllability), and finding the best ones with respect to some optimality criterion. As expected, the information one can obtain for these problems stated in full generality is very little, thus one needs interesting examples to work and analyze. Fortunately differential geometry provides a plethora of problems that can be formulated in these terms. In this talk I will discuss some results about the controllability of the system of manifolds rolling and the geometry of optimal curves in certain step two sub-Riemannian manifolds.

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Categorification in topology and algebra

Radmila Sazdanovic

University of Pennsylvania

3:00pm-4:00pm

CMC 130

Nataša Jonoska

**Abstract**

We will introduce the notion of categorification and discuss several examples. In particular, we will focus on Khovanov homology and chromatic homology theories categorifying the chromatic polynomial for graphs and relations between them. We develop a diagrammatic categorification of the polynomial ring \(Z[x]\), based on a geometrically defined algebra and explain how it generalizes to categorification of orthogonal polynomials, including Chebyshev polynomials and the Hermite polynomials.

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Spreading processes on graphs and PageRank

Paul Horn

Post-doctoral Fellow

Harvard University

3:30pm-4:30pm

NES 104

Brendan Nagle

**Abstract**

Disease breaks out on a graph! As medicine is expensive, it is unrealistic to send medicine to all vertices in preparation to fight the outbreak, but we still desire to ensure that the disease dies out quickly on the medicated vertices and escapes the medicated set with low probability. Under a variant of the contact process, a classical model of the spread of disease, we show how to accomplish such a goal on an arbitrary host graph. In particular, we show that the probability that the disease escapes a given medicated set can be bounded in terms of the PageRank of the complement. We additionally look at a broad generalization of this, where multiple interacting processes spread on a graph, where a new vectorized version of PageRank becomes critical to our understanding of the process.

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A probabilistic approach to Hilbert's Sixteenth Problem

Erik Lundberg

Golomb Assistant Professor of Mathematics

Purdue University

4:00pm-5:00pm

CHE 101

Yuncheng You

**Abstract**

The degree \(d\) of a polynomial equals the number of zeros it has (counted with multiplicity), but the number of *real* zeros can range anywhere from zero to \(d\), and determining that number is difficult even case by case. The situation is worse for the zero set of a polynomial in several variables, and studying the connected components of its real section is related to Hilbert's sixteenth problem. In this talk, I will give probabilistic answers to some questions concerning the topology, volume, and arrangement of components of the zero set (in projective space) for a Gaussian ensemble of homogeneous polynomials.

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**Note**

Can one hear the heat of a body? Mathematics of thermoacoustic tomography

Peter Kuchment

Texas A&M

3:00pm-4:00pm

CMC 130

Igor Chitikov

*This colloquium has been cancelled.*

**Abstract**

Tomography has been a constant source of wonderful mathematical problems for the last three decades. It involves many mathematical techniques, most notably coming from integral geometry (Radon transform), PDEs, harmonic analysis, microlocal analysis, spectral theory, and certainly numerical analysis. In the past decade, brand new “hybrid” methods of medical imaging have been emerging, a distinguishing feature of which is that they combine different physical types of signals (e.g., electromagnetic and ultrasound). The most developed among them, albeit still not in clinics, is the so called thermoacoustic (also known as photoacoustic or optoacoustic) tomography (TAT). The mathematical problems arising are rather complex and require a wide variety of analytic tools. The lecture will survey the main problems of TAT and recent progress in this area. No prior knowledge of the subject will be assumed. Time permitting, a wider variety of the emerging hybrid techniques will be briefly discussed.

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Longtime behavior of ill-posed dissipative systems

Michele Coti Zelati

Indiana University Bloomington

1:00pm-2:00pm

CMC 108

Yuncheng You

**Abstract**

Infinite-dimensional dynamical systems arise quite naturally in the study of the longtime dynamics of complex phenomena, typically featuring a nontrivial and often chaotic asymptotic behavior. Many interesting problems cannot be studied by well-defined semiflows, as the possible lack of uniqueness of solutions only allows to define a family of set-valued solution operators. Perhaps, the most famous example is the three-dimensional Navier-Stokes system. However, many applications can also be found in atmospheric science and numerical mathematics. I will describe a few possible theoretical approaches to these problems and some applications.

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Linear Codes from Commutative Algebraic Perspectives

Stefan Tohaneanu

Post-Doctoral Fellow/Assistant Professor

University of Western Ontario

3:30pm-4:30pm

NES 108

Boris Shekhtman

**Abstract**

Let \(\mathcal C\) be a linear code of block-length \(n\), dimension \(k\), and minimum distance \(d\). Let \(A\) be a \(k\times n\) matrix which is a generating matrix for \(\mathcal C\). Assuming that \(A\) has no proportional columns, nor any zero column, we can think of the columns of \(A\) as homogeneous coordinates of \(n\) distinct points in \(\mathbb P^{k-1}\). In this way, \(d\), the minimum distance, has the geometrical interpretation \(d=n-\mathrm{hyp}(\Gamma)\), where \(\Gamma\) is the set of points constructed above, and \(\mathrm{hyp}(\Gamma)\) is the maximum number of points of \(\Gamma\) contained in some hyperplane. To find \(\mathrm{hyp}(\Gamma)\) is a problem referred to as “the exact fitting problem”, and for \(k=3\) or for any \(k\geq 4\) but with \(\Gamma\) having more geometrical structure, we give a formula relating \(\mathrm{hyp}(\Gamma)\) to the index of nilpotency of some ideal constructed from \(\Gamma\). In the second part of the talk we show that the minimal graded free resolution of the ideal of the points \(\Gamma\) gives lower bounds for the minimum distance \(d\). The talk is intended for a general audience, with just a minimal background in Linear Algebra and Abstract Algebra (ideals, rings, modules, the ring of polynomials in several variables).

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Growth of random groups of intermediate growth and Gap Conjecture

Rostislav Grigorchuk

Texas A&M University

3:00pm-4:00pm

CMC 130

Dmytro Savchuk

**Abstract**

In 1968, Milnor raised the question if there are finitely generated groups of intermediate growth (between polynomial and exponential). In 1983, the speaker answered this question by providing a continuum of such groups with different types of growth. He also introduced a topology in the space of groups which happened to be a useful tool for many studies in modern group theory. While some of constructed groups have growth bounded from above by a function of type \(e^{n^α}\) for some \(α < 1\), there are groups whose growth is as close to the growth of exponential function as requested. Also there are groups whose growth function behaves differently on different parts of the domain (which is the set of natural numbers): sometimes in the intermediate way, and sometimes in the exponential way. Such groups are called “oscillating”. We will show that in topological sense (i.e., in the sense of Baire category) a typical group from our construction is oscillating. Then we will show that if we convert our construction into a family of random groups of intermediate growth by using an appropriate ergodic probability measure, then a typical with respect to this measure group is not oscillating and has growth bounded by a function of the type \(e^{n^α}\), \(α < 1\). We will also discuss the so called Gap Conjecture concerning growth of groups and formulate some results in the direction of its confirmation. Being proved this conjecture would give a far going generalization of the famous theorem of Gromov on characterization of groups of polynomial growth.