Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, March 29, 2016.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, March 29, 2016

11:00 am in 345 Altgeld Hall,Tuesday, March 29, 2016

Waldhausen K-theory and topological coHochschild homology

Kathryn Hess (Ecole Polytechnique Federale de Lausanne)

Abstract: I will present joint work with Brooke Shipley, in which we have defined a model category structure on the category of $\Sigma^{\infty}X_+$-comodule spectra such that the K-theory of the associated Waldhausen category of homotopically finite objects is naturally weakly equivalent to the usual Waldhausen K-theory of $X$, $A(X)$. I will describe the relation of this comodule approach to $A(X)$ to the more familiar description in terms of $\Sigma^\infty \Omega X_+$-module spectra. I will also explain the construction and properties of the topological coHochschild homology of $X$, which is a potentially interesting approximation to $A(X)$.

12:00 pm in Altgeld Hall 243,Tuesday, March 29, 2016

On /\-positioning of arcs between parallel support planes

Yevgenya Movshovich (EIU)

Abstract: The following result for an arc, called by J. Wetzel, the /\-property, was proved in Theorem 5.1 of "Besicovitch triangles cover unit arcs", Geom. Dedicata, vol. 123, (2006), ] by P. Coulton and Y. Movshovich: Any simple plane polygonal finite arc g has two parallel support lines and three parameters r < t < u; so that g(t) lies on one line, while g(r) and g(u) lie on the other. When showing that a convex set contains all unit arcs, the /\-property allow us to study only 3 and 4-segment arcs, shaped as letters S and W or a staple. There were two announcements on extending the result of Theorem 5.1 from polygonal to simple arcs: one by Y. M. (Geometry Seminar, UIUC, 2009) and the other by R. Alexander, J. E. Wetzel, W. Wichiramala in their recently submitted paper "The /\-property of a simple arc". In this talk we prove Theorem 5.1 omitting all three requirements on a rectifiable arc: polygonal, simple and plane.

1:00 pm in 345 Altgeld Hall,Tuesday, March 29, 2016

Generic behavior of measure preserving transformations and representations of $L^0(T)$

Mahmood Etedadialiabadi (UIUC)

Abstract: We consider the behavior of generic measure preserving transformations of a Borel measure space, i.e., of generic elements of $Aut(\mu)$. In particular, we are interested in the closed group generated a generic transformation and whether this group is isomorphic to $L^0(\mathbb{T})$. Del Junco--Lemanczyk showed that a generic $T\in Aut(\mu)$ satisfies a certain orthogonality condition on the maximal spectral types of powers of $T$. We introduce an analogous condition (DL-condition) for unitary representations of $L^0(\mathbb{T})$. Solecki showed that one can identify each unitary representation of $L^0(\mathbb{T})$ with a sequence of measures. We will show that the DL-condition is equivalent to an orthogonality condition on this sequence of measures. Also, we will show that generic functions in $L^0(\mathbb{T})$ are independent in a precise sense, in particular, generic $f,g\in L^0(\mathbb{T})$ are almost disjoint.

1:00 pm in 347 Altgeld Hall,Tuesday, March 29, 2016

Holomorphic maps from the unit ball to type iv classical domains

Ming Xiao (UIUC Math)

Abstract: We will discuss holomorphic proper and isometric maps from the unit ball to type IV classical domains. Some rigidity and irrigidity phenomena of such mappings will be discussed. It is a joint work with Yuan Yuan.

1:00 pm in 243 Altgeld Hall,Tuesday, March 29, 2016

Smoothing estimates for dispersive partial differential equations

Erin Compaan (University of Illinois)

Abstract: This talk will present some recent results on regularity and dynamics of two dispersive systems. The first part of the talk will address a coupled KdV-KdV system on the circle. For this problem, the nonlinear part of the evolution is shown to reside in a smoother space than the initial data. The gain in smoothness depends delicately on certain arithmetic properties of a coupling parameter. In the second part of the talk, similar smoothing estimates for the Klein-Gordon-Schrodinger system on a Euclidean space of dimension d> 1 will be presented. The talk will conclude with a discussion of implications of these smoothing estimates for the global dynamics of the system.

2:00 pm in Altgeld Hall 241,Tuesday, March 29, 2016

Zagier polynomials, their asymptotics and exact formulas

Atul Dixit (IIT, Gandhinagar, India)

Abstract: In 1998 Don Zagier introduced the modified Bernoulli numbers \begin{equation*} B_{n}^{*}=\sum_{r=0}^{n}\binom{n+r}{2r}\frac{B_r}{n+r}\hspace{4mm} (n>0), \end{equation*} and showed that they satisfy amusing variants of some properties of the ordinary Bernoulli numbers. In particular, he studied the asymptotic behavior of $B_{2n}^{*}$, and also obtained an exact formula for them, the motivation for which came from the representation of $B_{2n}$ in terms of the Riemann zeta function $\zeta(2n)$. Recently Victor H. Moll, Christophe Vignat and I generalized the modified Bernoulli numbers to Zagier polynomials $B_{n}^{*}(x)$. Thesepolynomials also have a rich structure, and we have shown that a theory parallel to that of Bernoulli polynomials exists for the Zagier polynomials. In this talk we will talk on an exact formula for $B_{2n}^{*}(x), 0

3:00 pm in 241 Altgeld Hall,Tuesday, March 29, 2016

Subtended Angles

Zoltan Furedi (UIUC and Renyi Instute of Math., Budapest, Hungary)

Abstract: We consider a combinatorial geometry problem, where in the solution we use the most fashionable tools (a combination of probabilistic and algebraic methods). Suppose that $d\ge2$ and $m$ are fixed. What is the largest $n$ such that, given any $n$ distinct angles $0<\theta_1,\theta_2,\dots,\theta_n<\pi$, we can realise all these angles by placing $m$ points in $\mathbb R^d$? We say an angle $\theta$ is realised if there exist points $A$, $B$ and $C$ such that $A\widehat BC=\theta$. E.g., 3 points in general can represent only one prescribed angle, (two obtuse angles cannot be realized by 3 points). Some of our results: Suppose that $m\ge 5$ and $n\le 2m-4$. Then, given any $n$ distinct angles, there is an arrangement of $m$ points in the plane realising of these angles. Moreover, these points may be chosen in convex position. Here the value of $2m-4$ is the best possible: There exists a set of $2m-3$ (distinct) angles such that no arrangement of $m$ points in any dimension realises all angles in the set. The results presented are joint work with G. Szigeti and Balister, Bollobas, Leader, and Walters.

3:00 pm in Altgeld Hall 243,Tuesday, March 29, 2016

Applications of Monodromy

Daniel Brake (University of Notre Dame)

Abstract: Monodromy action plays an important role in a number of mathematical theories. Stemming from a fundamental principle in complex analysis, the Cauchy integral formula, monodromy loops give all sorts of information about the interior of a region given boundary data. The uses include computing whether a pole is contained in the interior, and determining the breakup of the sheets coming together at a pole. As a consequence, monodromy is used in numerical algebraic geometry to decompose a pure-dimensional set into its irreducible components. This talk will give an overview of monodromy, and some new connections to algebraic geometry. In particular, we will discuss how to use it to compute some local properties of algebraic varieties, as in the Numerical Local Irreducible Decomposition, and a new method for computing real and complex tropical curves.

4:00 pm in 243 Altgeld Hall,Tuesday, March 29, 2016

The geometry of polynomial interpolation and GC sets

Nathan Fieldsteel (UIUC Math)

Abstract: Suppose $f$ is an unknown polynomial of degree $d$ in $1$ variable, and that you know the values it takes on a finite set of points $X$ in $\mathbb{R}$. It turns out that, as long as $X$ has enough points, the values of $f$ on $X$ completely determine $f$ itself. But if you ask the analogous question for $2$ or more variables, things are not so clear. Whether or not the values of $f$ on a finite set $X$ in $\mathbb{R}^2$ allows one to recover the polynomial $f$ depends on the geometry of $X$. This will be a survey talk on the geometry of polynomial interpolation and related questions. This talk will be very accessible.

4:00 pm in 245 Altgeld Hall,Tuesday, March 29, 2016

Topology of the microconnectome

Kathryn Hess (Ecole Polytechnique Federale de Lausanne )

Abstract: The Blue Brain Project team recently provided the network graph for a neocortical microcircuit comprising 8 million connections between 31,000 neurons (https://bbpnmc.epfl.ch). Since traditional graph-theoretical methods may not be sufficient to understand the immense complexity of such a biological network, we explored whether methods from algebraic topology could provide a new perspective on its structural and functional organization. Structural topological analysis revealed that directed graphs representing connectivity among neurons in the microcircuit deviated significantly from different varieties of randomized graph. In particular, the directed graphs contained on the order of 10^7 simplices (groups of neurons with all-to-all directed connectivity). Some of these simplices contained up to eight neurons, making them the most extreme neuronal clustering motif ever reported. Functional topological analysis of simulated neuronal activity in the microcircuit revealed novel spatio-temporal metrics that provide an effective classification of functional responses to qualitatively different stimuli. This study represents the first algebraic topological analysis of structural connectomics and connectomics-based spatio-temporal activity in a biologically realistic neural microcircuit. The methods used in the study show promise for more general applications in network science. (Joint work with P. Dlotko, R. Levi, H. Markram, E. Muller, M. Nolte, M. Scolamiero, K. Turner)