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Mathematics & Statistics

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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.

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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.

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How the Genome Folds

Erez Lieberman Aiden

Harvard Society of Fellows/

Visiting Faculty at Google

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.

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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.

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Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall’s work 25 years later

Andrei Martínez-Finkelshtein

Universidad de Almería

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.

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\(n\)-ary algebras: from Physics to Mathematics

Abdenacer Makhlouf

University of Haute Alsace

France

3:00pm-4:00pm

PHY 130

Mohamed Elhamdadi

**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.

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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.

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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”.

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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.

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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.

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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.

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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.

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Cyclic Sieving and Cluster Multicomplexes

Brendon Rhoades

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.

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Searching for Structure in Graph Theory: Chromatic Index and Immersion

Jessica McDonald

Simon Fraser University

Burnaby, British Columbia

Canada

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.

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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.

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Freak Waves in the Ocean

Victor L'vov

Weizmann Institute of Science

Rehovot, Israel

3:00pm-4:00pm

PHY 130

Arcadii Grinshpan

**Abstract**

Ships are disappearing all over the world’s oceans at a rate of about one every week. These drownings often happen in mysterious circumstances. With little evidence researchers usually put the blame on human errors or poor maintenance. But an alarming series of drownings and near drownings including world class vessels has pushed the search for better reasons than the regular ones: freak (or rogue, monster, killing) waves.

A freak wave in the ocean is a catastrophic event when energy and momentum of the wave field spontaneously concentrate in a localized area of space generating of short wave train consisting of several waves with energy and momentum density in order of magnitude exceeding the background level. Freak waves could be disastrous for ships, drilling platforms, lighthouses and other coastal constructions.

I will present observations of freak waves, their effect on ships and discuss possible mechanisms of their creation and evolutions.

^{* Rick Mabry received his Ph.D. from USF in 1985 under the direction of Professor A. G. Kartsatos.↵}