Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, April 22, 2014.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, April 22, 2014

11:00 am in 321 Altgeld Hall (common room),Tuesday, April 22, 2014

#### Meet and Greet

###### Professor Neil Turok (Founder of AIMS and Director of the Perimeter Institute)

Abstract: Neil Turok founded the African Institute for Mathematical Sciences (AIMS), in 2003, and launched its Next Einstein Initiative (NEI) in 2008. He has made major contributions to cosmology, in particular to our understanding of the Big Bang and the very early universe. On Monday Neil will talk about AIMS and the NEI at the MillerCom event at 4pm, and at noon he will give a talk on Gravity Waves(see the Calendar entries for more information). The hour event on Tuesday will be an opportunity for the mathematics community to meet and talk with Neil. Tea will be served.

1:00 pm in 347 Altgeld Hall,Tuesday, April 22, 2014

#### Rigorous justification of the modulation approximation to the full water wave problem

###### Nathan Totz (Duke)

Abstract: We consider solutions to the infinite depth water wave problem neglecting surface tension which are to leading order wave packets with small $O(\epsilon)$ amplitude and slow spatial decay that are balanced. Multiscale calculations formally suggest that such solutions have modulations that evolve on $O(\epsilon^{-2})$ time scales according to a version of a cubic NLS equation depending on dimension. Justifying this rigorously is a real problem, since standard existence results do not yield solutions to the water wave problem that exist for long enough to see the NLS dynamics. Nonetheless, given initial data suitably close to such a wave packet in $L^2$ Sobolev space, we show that there exists a unique solution to the water wave problem which remains within $o(\epsilon)$ to the formal approximation on the natural NLS time scales. The key ingredient in the proof is a formulation of the evolution equations for the water wave problem developed by Sijue Wu (U Mich.) with either no quadratic nonlinearities (in 2D) or mild quadratic nonlinearities that can be eliminated using the method of normal forms (in 3D).

1:00 pm in 243 Altgeld Hall,Tuesday, April 22, 2014

#### On the geometry of the flip graph

###### Valentina Disarlo (Indiana U Math)

Abstract: Given an orientable finite type punctured surface, its flip graph is the graph whose vertices are the ideal triangulations of the surface (up to isotopy) and two vertices are joined by an edge if the two corresponding triangulations differ by a flip, i.e. the replacement of one diagonal of the a quadrilateral by the other one. The combinatorics of this graph is crucial in works of Thurston and Penner's decorated Teichmuller theory. In this talk we will explore the geometric properties of this graph, proving that it provides a coarse model of the mapping class group in which the mapping class groups of the subsurfaces are convex. Moreover, we will provide bounds on the growth of the diameter of the flip graph modulo the mapping class group, providing a partial answer to an open problem in combinatorics.

1:00 pm in 345 Altgeld Hall,Tuesday, April 22, 2014

#### Minimal metrics on topological groups

###### Christian Rosendal (UIC)

Abstract: We discuss the problem of when a metrisable topological group G has a canonically defined geometry in a neighbourhood of 1. This naturally leads to the concept of minimal metrics on G, that we characterise in terms of a linear growth condition on powers of group elements.

2:00 pm in Altgeld Hall 347,Tuesday, April 22, 2014

#### Monotone solutions of linear differential equations with applications

###### Hans W. Volkmer (University of Wisconsin-Milwaukee Math)

Abstract: We review conditions on p and q such that the linear differential equation y''+p(x)y+q(x)y=0 admits (essentially) unique positive increasing and positive decreasing solutions. These solutions are used to solve the Kolmogorov backward equation. Some examples involving Brownian motion with affine drift will be discussed.

2:00 pm in 241 Altgeld Hall,Tuesday, April 22, 2014

#### Centralizers of generic measure-preserving automorphisms (part 3)

Abstract: We will finish the proof by Melleray and Tsankov of the fact that for the generic measure-preserving automorphism $T$ of a standard probability space $(X,\mu)$, the centralizer of $T$ inside $\text{Aut}(X,\mu)$ is as small as possible.

2:00 pm in 241 Altgeld Hall,Tuesday, April 22, 2014

#### Centralizers of generic measure-preserving automorphisms (part 4)

Abstract: In the first half of the talk, Mahmood will finish the proof by Melleray and Tsankov of the fact that for the generic measure-preserving automorphism $T$ of a standard probability space $(X,\mu)$, the centralizer of $T$ inside $\text{Aut}(X,\mu)$ is as small as possible.

3:00 pm in 243 Altgeld Hall,Tuesday, April 22, 2014

#### Counting curves on K3 surfaces: the Katz-Klemm-Vafa formula

###### Rahul Pandharipande (ETH Zurich)

Abstract: I will explain our recent proof (with R. Thomas) of the KKV formula governing higher genus curve counting in arbitrary classes on K3 surfaces. The subject intertwines Gromov-Witten, Noether-Lefschetz, and Donaldson-Thomas theories. A tour of these ideas will be included in the talk.

3:00 pm in 241 Altgeld Hall,Tuesday, April 22, 2014

#### New developments on the theory of partially ordered sets

###### Peter Hamburger   [email] (Western Kentucky University)

Abstract: In 1971 Bogart and Trotter conjectured that every finite poset on at least 3 elements has a pair whose removal does not decrease the dimension by more than 1. This conjecture has become known as the Removable Pair Conjecture. In 1992 Brightwell and Scheinerman introduced fractional dimension of posets, and they made a similar conjecture for fractional dimension. In1994 Felsner and Trotter suggested a weakening of the conjecture: is there an absolute positive constant $c$ not bigger than 1 so that any poset with 3 or more elements contains a pair whose removal decreases the fractional dimension by at most $2-c$? We prove the Brightwell-Scheinerman conjecture, which is of course equivalent to the Felsner-Trotter conjecture with $c=1$. Fishburn (1970) characterized those finite posets that contain an induced standard example $2+2=S_2$ as a subposet. However the existence of a standard example $S_k, (k>2)$, as an induced subposet has not been settled. We show that a poset with no induced subposet $S_k, (k>2)$, has dimension that is sublinear in terms of the number of elements. Biró-Füredi-Jahanbekam conjectured that if the dimension of a poset is slightly less than half the number of points, it contains large standard examples. We disprove this conjecture. We show that a statement similar to the Biró-Füredi-Jahanbekam conjecture holds for fractional dimension in bipartite posets. This is joint work with Csaba Biró, Department of Mathematics, University of Louisville, and Attila Pór, Department of Mathematics, Western Kentucky University

4:00 pm in Altgeld Hall,Tuesday, April 22, 2014

#### Regularity and Piecewise Polynomial Functions

###### Michael DiPasquale (UIUC Math)

Abstract: The algebra $C^r(\mathcal{P})$ of piecewise polynomial functions continuously differentiable of order $r$ over a polytopal complex $\mathcal{P}$ is a fundamental object in approximation theory. One of the fundamental questions in spline theory is to compute the dimension of the vector space $C^r_k(\mathcal{P})$ of splines of degree at most $k$. In the 1980s Billera pioneered an algebraic approach to spline theory using tools from homological and commutative algebra. We show how this approach, particularly the notions of the Hilbert polynomial and Castelnuovo-Mumford regularity, has interesting things to say about computing the dimension of $C^r_k(\mathcal{P})$.