Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, November 19, 2013.

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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, November 19, 2013

11:00 am in 241 Altgeld Hall,Tuesday, November 19, 2013

#### Schur's partition theorem and mixed mock modular forms

###### Karl Mahlburg (LSU Math)

Abstract: I will discuss families of partitions with gap conditions that were introduced by Schur and Andrews, and describe their intrinsic connections to combinatorial q-series and automorphic forms. The generating functions for these families naturally lead to fundamental identities for theta functions and Hickerson's universal mock theta function. This provides a very general answer to a conjecture of Andrews, in which he predicted the modularity of the generating function for Schur's partitions. As a final application, we prove the striking result that the universal mock theta function can be expressed as a conditional probability in a certain natural probability space with an in finite sequence of independent events.

1:00 pm in 141 Coordinated Science Lab (CSL),Tuesday, November 19, 2013

#### Persistent Homology: Basics, Applications, Software

###### Paul Bendich   [email] (Duke Math)

Abstract: [This is a tutorial style 2 hour talk] Topological Data Analysis (TDA) is now around 15 years old, and it is becoming a widely-used technique in data analysis, often in combination with more traditional methods. One of the key tools in TDA is the persistence diagram, which provides a compact description of the multi-scale topological information carried by a point cloud or other embedded object. In this informal tutorial, I will start by going over the very basics of persistent homology, showing how one can go from a point cloud, or a space equipped with a function, and produce a persistence diagram. Depending on audience interest, I can then cover some or all of the following: -some simple (and not-so-simple) applications which show how persistence diagrams, in combination with statistical ideas, can extract interesting information from unusual datasets. -the geometry of the space of persistence diagrams: why it's strange, and the possible implications for statistics with diagrams. -different algorithms for computing persistence diagrams, some of which are correct and slow, some of which are correct (in certain circumstances) and fast, some of which are fast and approximate with good guarantees. -a demonstration of our new software package RCA, which gives a very fast (and correct!) computation of persistence diagrams for components and loops.

1:00 pm in 345 Altgeld Hall,Tuesday, November 19, 2013

#### Constructible totally disconnected locally compact Polish groups and applications

###### Phillip Wesolek (UIC)

Abstract: The class of constructible totally disconnected locally compact (t.d.l.c.) Polish groups is the collection of t.d.l.c. Polish groups built from profinite and discrete groups via group extension and countable increasing union. These groups appear often in the study of t.d.l.c. Polish groups. We show this class satisfies surprisingly robust closure properties. We go on to give an application to the study of $p$-adic Lie groups. In particular, we show every $p$-adic Lie group decomposes into constructible and topologically simple groups via group extensions. This result is analogous to the solvable by semi-simple decomposition for connected Lie groups.

2:00 pm in Altgeld Hall,Tuesday, November 19, 2013

#### Two singularity phenomena

###### Max Glick (U. Minnesota)

Abstract: I will discuss two properties that may arise in the study of the singularities of a discrete dynamical system. The first is singularity confinement, introduced by Grammaticos, Ramani, and Papageorgiou. A singularity is said to be confined if it can be bypassed by a higher iterate of the map. This property is well-studied and believed to be closely tied to integrability. Â  The second property arises from trying to move away from the singularity by applying the inverse map. It is surprisingly common that after a large but predictable number of steps another singularity, mirroring the first, is reached. This behavior was observed by R. Schwartz for the pentagram map. I will demonstrate numerous other examples including Dodgson condensation, Adler's polygon recutting, and some new geometrically-defined systems.

2:00 pm in Altgeld Hall 347,Tuesday, November 19, 2013

#### On some properties of mean field spin glasses

###### Antonio Tuca Auffinger (U Chicago Math)

Abstract: Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems." As mathematical objects, they provide several fascinating structures and conjectures. In this talk, we overview some recent progress in mean field models that include the famous Sherrigton-Kirkpatrick model and the bipartite model. We will focus on properties of the energy landscape and of the functional order parameter. We will explain how these properties help to shed more light in the mysterious and beautiful solution proposed 30 years ago by G. Parisi. Based on joint works with Wei-Kuo Chen.

2:00 pm in 243 Altgeld Hall,Tuesday, November 19, 2013

#### Escape paths of Besicovitch triangles (revisited)

###### Yevgenya Movshivich (EIU Math)

Abstract: An escape path of an oval is the shortest path that does not fit in the interior of the oval. In 1965, A. S. Besicovitch conjectured that a certain symmetric unit $z$-arc is an escape path for the equilateral triangle of side $\sqrt{28/27}$. The conjecture was proven in "Besicovitch triangles cover unit arcs", Geom. Dedicata, 123 (2006) by P. Coulton and Y. M. for a family of Besicovitch isosceles triangular covers of unit arcs. The base angle, alpha, there ranged from about 52.2 degrees to 60 degrees. The low limit of this range was changed to 45 degrees in “Besicovitch triangles extended”, Geom. Dedicata, 159 (2012), by Y. M. Having just one escape unit arc, means that this cover of unit arcs is minimal (tight). In the Spring 2008 in two separate talks by P. Coulton and by Y. Movshovich, it was announced that a family of non-isosceles triangular covers of unit arcs (that contained all Besicovitch isosceles triangular covers) were found and the isosceles covers had infinitely many escape unit paths. A few months later we discovered that the pure non-isosceles covers are not minimal, all unit arcs fit in their interior, thus they have no escape unit paths. At the same time, each isosceles cover had a $Z$-arc as its only escape unit path. We will present a geometric argument supporting this last statement and conjecture on the sizes of the non-isosceles triangular covers of unit arcs that would make them minimal.

3:00 pm in 241 Altgeld Hall,Tuesday, November 19, 2013

#### Degrees of vertices and sets in uniform hypergraphs

###### Benjamin M. Reiniger   [email] (UIUC Math)

Abstract: I will discuss generalizations of results on the degree sequences of graphs to hypergraphs. For the degree sequence of a hypergraph, this will involve mostly work joint with Behrens, Erbes, Ferrara, Hartke, Spinoza, and Tomlinson. I will also discuss some preliminary work on the codegree functions of uniform hypergraphs.

3:00 pm in 347 Altgeld Hall,Tuesday, November 19, 2013

#### Distributed Algorithms for Resource Coordination in Networked Systems Described by Directed Graphs

###### Alejandro Dominguez-Garcia (UIUC ECE)

Abstract: We consider a multicomponent system in which a certain amount of a resource must be collectively provided by the system components, and address two different problems: (P1) the constrained fair-splitting problem, in which the amount of resource that each component can provide is limited by capacity constraints, but there is no cost associated to the amount of resource provided; and (P2) the constrained optimal dispatch problem, in which there are constraints on component upper and lower capacity, and each component has associated a quadratic cost. In our setup, the system components can only receive/transmit information from/to a subset of components; however, the information exchange between a pair of components is not necessarily bidirectional. The amount of resource to be collectively provided by the system components is known to an (outside) entity who cannot directly communicate with all components, but only with a limited number in its immediate neighborhood. In this talk, we will discuss low-complexity iterative algorithms that allow the components to solve problems P1 and P2 while adhering to the communication constraints described above. We will establish convergence of the proposed algorithms and demonstrate their performance via numerical examples. Additionally, we will discuss modifications of these algorithms that ensure almost sure convergence to the correct solution of problems P1 and P2 even in the presence of packet drops in the communication links. (Joint work with C. N. Hadjicostis and N. H. Vaidya)

3:00 pm in 243 Altgeld Hall,Tuesday, November 19, 2013

#### A classification of extremal Lagrangian planes

###### Benjamin Bakker (Courant Institute)

Abstract: Classically, an extremal class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. Settling a conjecture of Hassett and Tschinkel, we prove the natural generalization to higher dimensions: for a holomorphic symplectic variety $M$ deformation equivalent to a Hilbert scheme of $n$ points on a K3 surface, an extremal effective curve class $R$ sweeps out a Lagrangian $n$-plane if and only if certain intersection-theoretic criteria are met, including $(R,R)=-(n+3)/2$. The proof uses recent work of Bayer and Macri to represent effective cycles in moduli spaces of sheaves using Bridgeland stability conditions.

4:00 pm in 243 Altgeld Hall,Tuesday, November 19, 2013

#### Introduction to Grothendieck Topologies: Part II

###### Juan S. Villeta-Garcia (UIUC Math)

Abstract: We will continue our discussion of Grothendieck topologies, focusing on the etale site, and its associated cohomology. We'll begin with an introduction to etale morphisms and why we care about them. We will draw our examples from the cohomology of curves. The exposition will be basic and aimed at beginners (such as the speaker). Professors are welcome to attend.