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Mathematics & Statistics

# Colloquia — Spring 2012

## Friday, April 27, 2012

Title
Speaker

Time
Place

The Schwarzian norm and distortion of conformal mappings
Peter Duren
University of Michigan
3:00pm-4:00pm
PHY 130
Dmitry Khavinson

Abstract

The Schwarzian derivative of a locally univalent analytic function is $$Sf=\left(f''/f'\right)'-\frac12\left(f''/f'\right)^2.$$ The Schwarzian norm of a function defined in the unit disk $$\mathbb{D}$$ is given by $$\|Sf\|=\sup\limits_{z\in\mathbb{D}}\left(1-|z|^2\right)^2|Sf(z)|.$$ We begin by discussing some basic properties of the Schwarzian derivative and its classical applications to problems of differential equations and conformal mapping. A long-celebrated theorem of Nehari says that if $$\|Sf\|\le 2$$ then $$f$$ is (globally) univalent in $$\mathbb{D}$$. A weaker bound $$\|Sf\|\le 2\left(1+\delta^2\right)$$ for some $$\delta > 0$$ is known to imply the sharp lower bound $$d(\alpha,\beta)\ge\pi/\delta$$ on the hyperbolic distance between any pair of distinct points $$\alpha$$ and $$\beta$$ in $$\mathbb{D}$$ where $$f(\alpha)=f(\beta)$$. We will outline a proof that any upper bound on the Schwarzian norm is equivalent to both an upper bound and a lower bound on two-point distortion. It seems remarkable that such precise geometric information is encoded in the Schwarzian norm. Finally, we propose to describe a generalization of Nehari's theorem to harmonic mappings, or rather their lifts to a minimal surface.

## Friday, April 20, 2012

Title
Speaker

Time
Place

Exact solution of the six-vertex model: the Riemann-Hilbert approach
Pavel Bleher
Indiana University-Purdue University Indianapolis
3:00pm-4:00pm
PHY 130
Razvan Teodorescu

Abstract

We will review the Riemann-Hilbert approach to the exact solution of the sixvertex model with domain wall boundary conditions in different phase regions. This is a joint project with Karl Liechty.

## Thursday, April 19, 2012

Title
Speaker

Time
Place

How the Genome Folds
Erez Lieberman Aiden
Harvard Society of Fellows/
11:00am-12:00pm
PHY 120
Nagle Lecture Committee

Abstract

I describe Hi-C, a novel technology for probing the three-dimensional architecture of whole genomes. Developed together with collaborators at the Broad Institute and UMass Medical School, Hi-C couples proximity-dependent DNA ligation and massively parallel sequencing.

Our lab employs Hi-C to construct spatial proximity maps of the human genome. Using Hi-C, it is possible to confirm the presence of chromosome territories and the spatial proximity of small, gene-rich chromosomes. Hi-C maps also reveal an additional level of genome organization that is characterized by the spatial segregation of open and closed chromatin to form two genome-wide compartments. At the megabase scale, the conformation of chromatin is consistent with a fractal globule, a knot-free conformation that enables maximally dense packing while preserving the ability to easily fold and unfold any genomic locus. The fractal globule is distinct from the more commonly used globular equilibrium model. Our results demonstrate the power of Hi-C to map the dynamic conformations of whole genomes.

## Friday, April 13, 2012

Title
Speaker

Time
Place

Construction of Riemann surfaces by using parallel translations
Kiyoshi Ohba
Ochanomizu University
Japan
3:00pm-4:00pm
PHY 130
Masahiko Saito

Abstract

A Riemann surface is a connected orientable surface with a complex structure. The problem of how to parametrize the complex structures on a fixed topological surface originated with B. Riemann. In this talk, I will introduce some methods of constructing Riemann surfaces by cutting the complex plane along line segments and pasting by parallel translations, and observe visually the deformation of complex structures of Riemann surfaces.

## Friday, March 30, 2012

Title
Speaker

Time
Place

Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall’s work 25 years later
Andrei Martínez-Finkelshtein
Almería, SPAIN
3:00pm-4:00pm
PHY 120
E. Rakhmanov

Abstract

In 1986 J. Nuttall published in Constructive Approximation a paper where he studied the behavior of the denominators (“generalized Jacobi polynomials”) and the remainders of the Padé approximants to a special class of algebraic functions with 3 branch points. I will try to look at this problem 25 years later from a modern perspective. On one hand, the generalized Jacobi polynomials constitute an instance of the so-called Heine-Stieltjes polynomials, i.e. they are solutions of linear ODE with polynomial coefficients. On the other, they satisfy complex orthogonality relations, and thus are suitable for the Riemann-Hilbert asymptotic analysis. Along with the names mentioned in the title, this talk features also a special appearance by Riemann surfaces, quadratic differentials, compact sets of minimal capacity, special functions and other characters.

## Friday, March 23, 2012

Title
Speaker

Time
Place

$$n$$-ary algebras: from Physics to Mathematics
Abdenacer Makhlouf
University of Haute Alsace
France
3:00pm-4:00pm
PHY 130

Abstract

Lie algebras and Poisson algebras have played an extremely important role in mathematics and physics for a long time. Their generalizations, known as $$n$$-Lie algebras and Nambu algebras also arise naturally in physics in many different contexts. In this talk, I will review some basics on $$n$$-ary algebras, present some key constructions and discuss the representation theory and cohomology theory.

## Friday, March 9, 2012

Title
Speaker

Time
Place

Multi-Dimensional Obliquely Reflected BSDEs Arising from Real Options
Shanjian Tang
Fudan University
China
3:00pm-4:00pm
PHY 130
Yuncheng You

Abstract

The definition of real options is recalled. The price of real options is associated to an optimal switching problem for backward stochastic differential equations (BSDEs). Interestingly, the associated dynamical programming equation for such kind of optimal switching problem is a multi-dimensional BSDE with oblique reflection.

## Friday, February 24, 2012

Title
Speaker

Time
Place

Fourier bases on fractal Hilbert Spaces
Keri Kornelson
University of Oklahoma
Norman, OK
3:00pm-4:00pm
PHY 130
Catherine Bénéteau

Abstract

The study of Bernoulli convolution measures, which are supported on Cantor subsets of the real line, dates back to the 1930's, and experienced a resurgence with the connection between the measures and iterated function systems. We will use this IFS approach to consider the question of Fourier bases on the $$L^2$$ spaces with respect to Bernoulli convolution measures.

There are some interesting phenomena that arise in this setting, e.g., one can sometimes scale or shift the Fourier frequencies of an orthonormal basis by an integer and obtain another ONB.

We also describe properties of the unitary operator mapping between two such Fourier bases. This operator exhibits a fractal-like self-similarity, leading us to call it an “operator-fractal”.

## Monday, February 13, 2012

Title
Speaker

Time
Place

An introduction to handlebody-knot theory
Atsushi Ishii
Tsukuba University
Japan
3:05pm-3:55pm
PHY 108
Masahiko Saito

Abstract

A handlebody-knot is a handlebody embedded in the $$3$$-sphere. A handlebody-link is a disjoint union of handlebodies embedded in the $$3$$-sphere. A handlebody-knot is a $$1$$-component handlebody-link. Two handlebody-links are equivalent if one can be transformed into the other by an isotopy of the $$3$$-sphere. I will explain how two handlebody-links can be distinguished. We may decompose handlebody-links to distinguish them, or we may use invariants. This talk is an introduction to handlebody-knot theory.

## Friday, February 10, 2012

Title
Speaker

Time
Place

Some Combinatoric and Comedic Consequences of the Proof of the Pizza Conjecture
Rick Mabry*
Louisiana State University in Shreveport
Shreveport, LA
4:00pm-5:00pm
PHY 130
Athanassios Kartsatos

Abstract

The recent proof of the “Pizza Conjecture” by Deiermann and Mabry was accomplished by wishful thinking. Unable to solve the original problem, the would-be solvers generalized the problem in a fairly extreme way and hoped for the best. The resulting problem was ultimately solved by reducing a nice, continuous (calculus) problem to a gruesome, discrete (combinatorial) one. The solution made some news, and, in a comedy of errors, cheesy comments flooded the internet in the aftermath. This talk will highlight some mathematical, gastronomical, and comical slices of what ensued, and look at a more recent pizza theorem and its combinatorial solution.

Title
Speaker

Time
Place

Sparse Ramsey Hosts
Kevin Milans
University of South Carolina
Columbia, SC
3:00pm-4:00pm
PHY 130
Brendan Nagle

Abstract

In Ramsey Theory, we study conditions under which every partition of a large structure yields a part with additional structure. For example, Van der Waerden's theorem states that every $$s$$-coloring of the integers contains arbitrarily long monochromatic arithmetic progressions, and the Hales-Jewett Theorem guarantees that every game of tic-tac-toe in high dimensions has a winner. Ramsey's Theorem implies that for any target graph $$G$$, every $$s$$-coloring of the edges of some sufficiently large host graph contains a monochromatic copy of $$G$$. In Ramsey's Theorem, the host graph is dense (in fact complete). We explore conditions under which the host graph can be sparse and still force a monochromatic copy of $$G$$.

We write $$H \stackrel{s}{\to} G$$ if every $$s$$-edge-coloring of $$H$$ contains a monochromatic copy of $$G$$. The $$s$$-color Ramsey number of $$G$$ is the minimum $$k$$ such that some $$k$$-vertex graph $$H$$ satisfies $$H \stackrel{s}{\to} G$$. The degree Ramsey number of $$G$$ is the minimum $$k$$ such that some graph $$H$$ with maximum degree $$k$$ satisfies $$H \stackrel{s}{\to} G$$. Chvátal, Rödl, Szemerédi, and Trotter proved that the Ramsey number of bounded-degree graphs grows only linearly, sharply contrasting the exponential growth that generally occurs when the bounded-degree assumption is dropped. We are interested in the analogous degree Ramsey question: is the $$s$$-color degree Ramsey number of $$G$$ bounded by some function of $$s$$ and the maximum degree of $$G$$? We resolve this question in the affirmative when $$G$$ is restricted to a family of graphs that have a global tree structure; this family includes all outerplanar graphs. We also investigate the behavior of the $$s$$-color degree Ramsey number as $$s$$ grows. This talk includes results from three separate projects that are joint with P. Horn, T. Jiang, B. Kinnersley, V. Rödl, and D. West.

## Wednesday, February 8, 2012

Title
Speaker

Time
Place
Speaker

Automata generating free products of groups of order $$2$$
Dmytro Savchuk
SUNY Binghamton
Binghamton, NY
3:00pm-4:00pm
PHY 118
Nataša Jonoska

Abstract

We construct a family of automata with $$n$$ states, $$n>3$$, acting on a rooted binary tree that generate the free products of cyclic groups of order $$2$$. Groups generated by automata is a fascinating class of groups that includes counterexamples to several famous conjectures in group theory. I will start from discussing the definition and main properties of these groups. Then I will give a short exposition of the history of the question, explain the construction and main ideas behind the proof, which involve the notion of a dual automaton.

This is a joint result with Yaroslav Vorobets of Texas A&M University.

## Monday, February 6, 2012

Title
Speaker

Time
Place

Cyclic Sieving and Cluster Multicomplexes
University of Southern California
Los Angeles, CA
3:05pm-3:55pm
PHY 108
Brian Curtin

Abstract

Let $$X$$ be a finite set, $$C=\langle c\rangle$$ be a finite cyclic group acting on $$X$$, and $$X(q)\in N[q]$$ be a polynomial with nonnegative integer coefficients. Following Reiner, Stanton, and White, we say that the triple $$(X,C,X(q))$$ exhibits the *cyclic sieving phenomenon* if for any integer $$d>0$$, the number of fixed points of $$c^d$$ is equal to $$X(\zeta^d)$$, where $$\zeta$$ is a primitive $$|C|^{\mathrm{th}}$$ root of unity. We explain how one can use representation theory to prove instances of the cyclic sieving phenomenon involving the action of tropical Coxeter elements on (complexes closely related to) cluster complexes. The representation theory involves cluster monomial bases of geometric realizations of finite type cluster algebras.

## Friday, February 3, 2012

Title
Speaker

Time
Place

Searching for Structure in Graph Theory: Chromatic Index and Immersion
Jessica McDonald
Simon Fraser University
Burnaby, British Columbia
3:00pm-4:00pm
PHY 130
Brendan Nagle

Abstract

Searching for structure is a fundamental theme in graph theory. The celebrated Goldberg-Seymour Conjecture is an example of this — it asserts that all multigraphs with “high” chromatic index contain a “dense” subgraph. In this talk we discuss edge-colouring in this context, and use the method of Tashkinov trees to gain new insights. Namely, we extend a classical characterization result of Vizing, and prove an approximation bound towards the Goldberg-Seymour Conjecture. We also consider the important containment relation of immersion. In particular, motivated by the Graph Minors Project of Robertson and Seymour and by Hadwiger's Conjecture, we explore conditions under which graphs and digraphs contain clique immersions. The results we obtain are in analogue to the clique subdivision theorem of Bollobas-Thomason and Komlos-Szemeredi.

## Wednesday, February 1, 2012

Title
Speaker

Time
Place

Duality and equivalence of graphs in surfaces
Iain Moffatt
University of South Alabama
Mobile, AL
3:00pm-4:00pm
PHY 108
Masahiko Saito

Abstract

This talk revolves around two fundamental constructions in graph theory: duals and medial graphs. There are a host of well-known relations between duals and medial graphs of graphs drawn in the plane. (This can also be thought of in terms of knot diagrams and their graphs.) By considering these relations we will be led to the working principle that duality and equality of plane graphs are equivalent concepts. It is then natural to ask what happens when we change our notion of equality. In this talk we will see how isomorphism of abstract graphs corresponds to an extension of duality called twisted duality, and how twisted duality extends the fundamental relations between duals and medial graphs from graphs in the plane to graphs in other surfaces. We will then go on to see how this group action leads to a deeper understanding of the properties of, and relationships among, various graph polynomials, including the chromatic polynomial, the Penrose polynomial, and topological Tutte polynomials.

## Friday, January 20, 2012

Title
Speaker

Time
Place

Freak Waves in the Ocean
Victor L'vov
Weizmann Institute of Science
Rehovot, Israel
3:00pm-4:00pm
PHY 130