Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, December 2, 2014.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    November 2014          December 2014           January 2015
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1       1  2  3  4  5  6                1  2  3
2  3  4  5  6  7  8    7  8  9 10 11 12 13    4  5  6  7  8  9 10
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Tuesday, December 2, 2014

11:00 am in 241 Altgeld Hall,Tuesday, December 2, 2014

#### The distribution of $k$-free numbers and the derivative of the Riemann zeta-function

###### Xianchang Meng (UIUC Math)

Abstract: Under the Riemann Hypothesis, we connect the distribution of $k$-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of $\zeta(s)$. Moreover, with additional assumptions, we prove the existence of a limiting distribution of $e^{-\frac{y}{2k}}M_k(e^y)$ and study the tail of the limiting distribution, where $M_k(x)=\sum_{n\leq x}\mu_k(n)-\frac{x}{\zeta(k)}$ and $\mu_k(n)$ is the characteristic function of $k$-free numbers. Finally, we make a conjecture about the maximum order of $M_k(x)$ by heuristic analysis on the tail of the limiting distribution.

11:00 am in 243 Altgeld Hall,Tuesday, December 2, 2014

#### Stable homology of the moduli space of flat manifold bundles

###### Sam Nariman   [email] (Stanford University)

Abstract: We discuss homological stability for diffeomorphism groups with discrete topology, in particular we will see that the group homology of the diffeomorphism group of the g-fold connect sum of S^n x S^n, as a discrete group is independent of g in a range, provided that n>2. This was motivated by a conjecture posed by Morita about discrete surface diffeomorphism groups. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford-Morita-Miller classes can be detected on a flat #^g S^n x S^n bundle. If time permits, I will report on the progress about the surface case.

1:00 pm in Altgeld Hall 243,Tuesday, December 2, 2014

#### A new solution to the von Neumann-Day problem for finitely presented groups

###### Yash Lodha (Cornell University)

Abstract: In this talk, I will describe a finitely presented subgroup of Monod’s group of piecewise projective homeomorphisms of the real line. This in particular provides a new example of a finitely presented group which is nonamenable and yet does not contain a nonabelian free subgroup. The example is moreover torsion free. A portion of this is joint work with Justin Moore. Recent work on the higher finiteness properties of this group will be discussed if time permits. View talk at http://youtu.be/PMNYe4OOw1A

2:00 pm in 347 Altgeld Hall,Tuesday, December 2, 2014

#### Local and Global Well-posedness of Stochastic PDE with Locally Monotone Coefficients

###### Wei Liu (Jiangsu Normal University)

Abstract: In this talk we will present some recent results on the well-posedness of SPDE with locally monotone coefficients, which generalize the classical variational framework established by Pardoux, Krylov and Rozovskii etc. The main results are applicable to a large class of SPDE models in fluid mechnics such as stochastic Burgers type equations, stochastic Navier-Stokes equations, stochastic 2D hydrodynamical systems and stochastic power law fluid models.

2:00 pm in 243 Altgeld Hall,Tuesday, December 2, 2014

#### Invariant Peano curves of rational maps and mating of polynomials

###### Daniel Meyer (Jacobs University, Bremen)

Abstract: Let $f: S^2\to S^2$ be a rational map, such that each critical point has finite orbit. Then each sufficiently high iterate $F=f^n$ has in invariant Peano curve $\gamma: S^1\to S^2$ (onto) such that $F(\gamma(z))= \gamma(z^d)$, here $d=\deg F$. The result is analogue to a well-known result by Cannon-Thurston, which give group invariant Peano curves in the setting of Kleinian groups. The result also yields that the rational map $F$ may be expressed as the so-called mating of two polynomials. Similar mating have recently been used by Le Gall in the random setting to construct random surfaces.

3:00 pm in 345 Altgeld Hall,Tuesday, December 2, 2014

#### Compact spaces as quotients of projective Fraisse limits

###### Aristotelis Panagiotopoulos (UIUC )

Abstract: Projective Fraisse structures were introduced by T. Irwin and S. Solecki and they were used to provide a very useful construction of a certain compact space known as the pseudo-arc. We develop a theory of projective Fraisse limits in the Irwin-Solecki spirit which moreover support a dual structure. Let $K$ be a totally disconnected, second countable, compact space. We prove that a subgroup $G$ of $\mathrm{Homeo}(K)$ is closed in the compact-open topology if and only if it is the automorphism group of some dual topological Fraisse limit $\boldsymbol{K}$ on domain $K$. As an application we prove that every second countable, compact space is the the quotient of topological Fraisse limit $\boldsymbol{K}$ with a closed equivalence relation on $K$ that is definable in $\boldsymbol{K}$.

3:00 pm in 347 Altgeld Hall,Tuesday, December 2, 2014

#### Redundancy allocation in reliability systems

###### Peng Zhao (Jiangsu Normal University)

Abstract: In this talk, we discuss the problem of optimal allocation of standby [active] redundancy at component level versus system level. We will present some interesting comparison results of reliability systems in the sense of various stochastic orderings for both the matching spares case and non-matching spares case, respectively.

3:00 pm in 241 Altgeld Hall,Tuesday, December 2, 2014

#### Maximal triangle-free graphs

###### Šárka Petříčková   [email] (UIUC Math)

Abstract: Paul Erdős suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. In this talk, we will discuss two results: 1) The number of maximal triangle-free graphs is $2^{n^2/8+o(n^2)}$ (Balogh-P., 2014). 2) Almost every maximal triangle-free graph $G$ admits a vertex partition $X\cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set (Balogh-Liu-P.-Sharifzadeh, 2014+). Main tools include the Ruzsa-Szemerédi triangle removal lemma, the Erdős-Simonovits stability theorem, a recent structural characterization of the independent sets in hypergraphs, and a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_3$'s.