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for events the day of Thursday, March 2, 2017.

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           1  2  3  4             1  2  3  4                      1
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Thursday, March 2, 2017

11:00 am in Altgeld Hall,Thursday, March 2, 2017

Sums in short intervals and decompositions of arithmetic functions

Brad Rodgers (University of Michigan)

Abstract: In this talk we will discuss some old and new conjectures about the behavior of sums of arithmetic functions in short intervals, along with analogues of these conjectures in a function field setting that have been proved in recent years. We will pay particular attention to some surprising phenomena that comes into play, and a decomposition of arithmetic functions in a function field setting that helps elucidate what's happening.

12:00 pm in 243 Altgeld Hall,Thursday, March 2, 2017

Hyperbolic volumes of random links

Malik Obeidin   [email] (University of Illinois)

Abstract: What does a random link look like? There have been a few different proposed models for sampling from the set of links -- in this talk, I will describe a model based on random link diagrams in the plane. Such diagrams can be sampled uniformly on a computer due to the work of Gilles Schaeffer, so one can experiment with various invariants of links with the topology software SnapPy. I will present data showing what happens with some of the different invariants SnapPy can compute, and I will outline a proof that the hyperbolic volume of the complement of a random alternating link diagram is asymptotically a linear function of the number of crossings. In contrast, for nonalternating links, I will show why the diagrams we get generically represent satellite (and hence nonhyperbolic) links.

1:00 pm in 345 Altgeld Hall,Thursday, March 2, 2017

Henson's universal triangle-free graphs have finite big Ramsey degrees

Natasha Dobrinen (University of Denver)

Abstract: A triangle-free graph on countably many vertices is universal triangle-free if every countable triangle-free graph embeds into it. Universal triangle-free graphs were constructed by Henson in 1971, which we will denote as $\mathcal{H}_3$. Being an analogue of the random graph, its Ramsey properties are of interest. Henson proved that for any partition of the vertices in $\mathcal{H}_3$ into two colors, there is either a copy of $\mathcal{H}_3$ in one color (furthermore, only leaving out finitely many vertices in the first color), or else the other color contains all finite triangle-free graphs. In 1986, Komj\'{a}th and R\"{o}dl proved that the vertices in $\mathcal{H}_3$ have the Ramsey property: For any partition of the vertices into two colors, one of the colors contains a copy of $\mathcal{H}_3$. In 1998, Sauer showed that there is a partition of the edges in $\mathcal{H}_3$ into two colors such that every subcopy of $\mathcal{H}_3$ has edges with both colors. He also showed that for any coloring of the edges into finitely many colors, there is a subcopy of $\mathcal{H}_3$ in which all edges have at most two colors. Thus, we say that the big Ramsey degree for edges in $\mathcal{H}_3$ is two. It remained open whether all finite triangle-free graphs have finite big Ramsey degrees; that is, whether for each finite triangle-free graph G there is an integer n such that for any finitary coloring of all copies of G, there is a subcopy of $\mathcal{H}_3$ in which all copies of G take on no more than n colors. We prove that indeed this is the case.

2:00 pm in 243 Altgeld Hall,Thursday, March 2, 2017

Singular integrals on Heisenberg curves

Sean Li (U Chicago)

Abstract: In 1977, Calderon proved that the Cauchy transform is bounded as a singular integral operator on the L_2 space of Lipschitz graphs in the complex plane. This subsequently sparked much work on singular integral operators on subsets of Euclidean space. It is now known that the boundedness of singular integrals of certain odd kernels is intricately linked to a rectifiability structure of the underlying sets. We study this connection between singular integrals and geometry for 1-dimensional subsets of the Heisenberg group where we find a similar connection. However, the kernels studied turn out to be positive and even, in stark contrast with the Euclidean setting. Joint work with V. Chousionis.

3:00 pm in 243 Altgeld Hall,Thursday, March 2, 2017

Multidimensional Persistent Homology

Hal Schenck (UIUC Math)

Abstract: A fundamental tool in topological data analysis is persistent homology, which allows detection and analysis of underlying structure in large datasets. Persistent homology (PH) assigns a module over a principal ideal domain to a filtered simplicial complex. While the theory of persistent homology for filtrations associated to a single parameter is well-understood, the situation for multifiltrations is more delicate; Carlsson-Zomorodian introduced multidimensional persistent homology (MPH) for multifiltered complexes via multigraded modules over a polynomial ring. We use tools of commutative and homological algebra to analyze MPH, proving that the MPH modules are supported on coordinate subspace arrangements, and that restricting an MPH module to the diagonal subspace $V(x_i-x_j | i \ne j)$ yields a PH module whose rank is equal to the rank of the original MPH module. This gives one answer to a question asked by Carlsson-Zomorodian. This is joint work with Nina Otter, Heather Harrington, Ulrike Tillman (Oxford).