Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, April 23, 2019.

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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 23, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, April 23, 2019

#### On Hardy-Rellich-type inequalities

###### Fritz Gesztesy (Baylor University)

Abstract: We will illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisely, using this factorization method, we will derive a general inequality and demonstrate how particular choices of the parameters contained in this inequality yield well-known inequalities, such as the classical Hardy and Rellich inequalities, as special cases. Actually, other special cases yield additional and apparently less well-known inequalities. We will indicate that our method is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators. If time permits, we might illustrate a very recent new and most elementary proof in the one-dimensional context. This talk will be accessible to students. This is based on joint work with Lance Littlejohn, Isaac Michael, and Michael Pang.

1:00 pm in 345 Altgeld Hall ,Tuesday, April 23, 2019

#### Expansions of the real field which does not introduce new smooth functions

###### Alex Savatovsky (Universität Konstanz)

Abstract: We will give some conditions under which an expansion of the real field does not define new smooth functions. We will give a very rough sketch of the proof and discuss generalizations.

2:00 pm in 243 Altgeld Hall,Tuesday, April 23, 2019

#### Partitions of hypergraphs under variable degeneracy constraints

###### Michael Stiebitz (TU Ilmenau)

Abstract: We use the concept of variable degeneracy of a hypergraph in order to unify the seemingly remote problems of determining the point partition numbers and the list chromatic number of hypergraphs. Our hypergraphs may have multiple edges, but no loops. Given a hypergraph $G$ and a sequence $f = (f_1, f_2, \dots , f_p)$ of $p \ge 1$ vertex functions $f_i : V(G) → \mathbb N_0$ such that $f_1(v) + f_2(v) + · · · + f_p(v) \ge d_G(v)$ for all $v \in V(G)$, we want to find a sequence $(G_1, G_2, \dots , G_p)$ of vertex disjoint induced subhypergraphs containing all vertices of $G$ such that each hypergraph G_i is strictly $f_i$-degenerate, that is, for every non-empty subhypergraph $G' \subseteq G_i$ there is a vertex $v \in V (G')$ such that $d_{G'}(v) < f_i(v)$. The main result says that such a sequence of hypergraphs exists if and only if $(G, f)$ is not a so-called hard pair. Hard pairs form a recursively defined family of configurations, obtained from three basic types of configurations by the operation of merging a vertex. For simple graphs this result was obtained by O. Borodin, A. V. Kostochka, and B. Toft in 2000. As a simple consequence of our result we obtain a Brooks-type result for the list chromatic number of digraphs due to A. Harautyunyan and B. Mohar. In a digraph coloring the aim is to color the vertices of a directed graph $D$ such that each color class induces an acyclic digraph of $D$, that is, a directed graph not containing any directed cycle. This coloring concept was introduced by V. Neumann-Lara in the 1980s.

3:00 pm in 243 Altgeld Hall,Tuesday, April 23, 2019

#### Virtual Euler characteristics of Quot scheme of surfaces

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

Abstract: Let S be a nonsingular projective surface. Quot schemes of quotients on S with supports of dimensions 0 and 1 always have 2-term obstruction theories (and therefore also have natural virtual fundamental classes). I will explain what we know about the virtual Euler characteristics in this theory: theorems, conjectures, and a lot of examples. Joint work with Dragos Oprea.

4:00 pm in 245 Altgeld Hall,Tuesday, April 23, 2019

#### The challenge of modeling dryland vegetation pattern formation using ideas from dynamical systems

###### Mary Silber (University of Chicago)

Abstract: A beautiful example of spontaneous pattern formation appears in the distribution of vegetation in some dry-land environments. Examples from Africa, Australia and the Americas reveal that vegetation, at a community scale, may spontaneously form into stripe-like bands, alternating with striking regularity with bands of bare soil, in response to aridity stress. A typical length scale for such patterns is 100 m; they are readily surveyed by modern satellites (and explored from your armchair in Google maps). These ecosystems represent some of Earth’s most vulnerable under threats to desertification, and some ecologists have suggested that the patterns, so easily monitored by satellites, may have potential as early warning signs of ecosystem collapse. I will describe efforts based in simple mathematical models, inspired by decades of physics research on pattern formation, to understand the morphology of the patterns, focusing particularly on topographic influences. I will take a critical look at the role of mathematical models in developing potential remote probes of these ecosystems. How does mathematical modeling influence what we see? Does it suggest what we should monitor? Could it lead us astray?