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for events the day of Tuesday, April 3, 2018.

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Tuesday, April 3, 2018

1:00 pm in 345 Altgeld Hall,Tuesday, April 3, 2018

Finite versus infinite: An intricate shift

Yann Pequignot (UCLA)

Abstract: The Borel chromatic number — introduced by Kechris, Solecki, and Todorcevic (1999) — generalizes the chromatic number on finite graphs to definable graphs on topological spaces. While the $G_0$ dichotomy states that there exists a minimal graph with uncountable Borel chromatic number, it turns out that characterizing when a graph has infinite Borel chromatic number is far more intricate. Even in the case of graphs generated by a single function, our understanding is actually very poor.
 The Shift Graph on the space of infinite subsets of natural numbers is generated by the function that removes the minimum element. It is acyclic but has infinite Borel chromatic number. In 1999, Kechris, Solecki, and Todorcevic asked whether the Shift Graph is minimal among the graphs generated by a single Borel function that have infinite Borel chromatic number. I will explain why the answer is negative using a representation theorem for $\Sigma^1_2$ sets due to Marcone.

1:00 pm in 347 Altgeld Hall,Tuesday, April 3, 2018

Some new results on maximal averages and $L^p$ Sobolev regularity of Radon transforms

Michael Greenblatt (University of Illinois at Chicago)

Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where $p > 2$. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as $|(x,y)|^{-t}$ for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to prove sharp (up to endpoints) $L^p$ to $L^p_a $ Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2. These Radon transform results have higher-dimensional generalizations which will also be described.

3:00 pm in 241 Altgeld Hall,Tuesday, April 3, 2018

Packing chromatic number of subdivisions of cubic graphs

Xujun Liu (Illinois Math)

Abstract: A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. For a graph $G$, let $D(G)$ denote the graph obtained from $G$ by subdividing every edge. The questions on the value of the maximum of $\chi_p(G)$ and of $\chi_p(D(G))$ over the class of subcubic graphs $G$ appear in several papers. Gastineau and Togni asked whether $\chi_p(D(G))\leq 5$ for any subcubic $G$, and later Brešar, Klavžar, Rall and Wash conjectured this, but no upper bound was proved. Recently the authors proved that $\chi_p(G)$ is not bounded in the class of subcubic graphs $G$. In contrast, in this paper we show that $\chi_p(D(G))$ is bounded in this class, and does not exceed $8$. Joint work with József Balogh and Alexandr Kostochka.

3:00 pm in 243 Altgeld Hall,Tuesday, April 3, 2018

Equal sums of higher powers of binary quadratic forms, I

Bruce Reznick (UIUC)

Abstract: We will describe all non-trivial solutions to the equation $f_1^d(x,y) + f_2^d(x,y) = f_3^d(x,y) + f_4^d(x,y)$ for quadratic forms $f_j \in \mathbb C[x,y]$. No particular prerequisites are needed and tools will be derived during the talk. Lots of fun stuff. The content of the second talk, next week, will be shaped by the reaction to this one.

4:00 pm in 243 Altgeld Hall,Tuesday, April 3, 2018

Irving Reiner lectures: Lectures on Quantum Schubert Calculus II

Leonardo C. Mihalcea (Virginia Tech )

Abstract: The quantum cohomology ring of a complex projective manifold X is a deformation of the ordinary cohomology ring of X. It was defined by Kontsevich in the mid 1990’s in relation to physics and enumerative geometry. Its structure constants - the Gromov-Witten invariants - encode numbers such as how many conics pass through 3 general points in the Grassmann manifold of 2-planes in the 4-space. The quantum cohomology ring is best understood when X has many symmetries, or good combinatorial properties, and these lectures will focus to the case when X is a Grassmann manifold or a flag manifold. The subject is quite rich, and intensively studied, with connections to algebraic combinatorics, algebraic and symplectic geometry, representation theory, and integrable systems. My goal is to introduce the audience to some of the basic ideas and techniques in the subject, such as how to calculate effectively in the quantum cohomology rings, and what are the geometric ideas behind the calculations, all illustrated by examples. The lectures are intended for graduate students, in particular I am not assuming prior knowledge of quantum cohomology. I plan to include the following topics. The ‘quantum = classical’ phenomenon of Buch, Kresch and Tamvakis: how a ‘quantum’ calculation can be performed in the ‘classical’ cohomology of an auxiliary space - this leads to formulas based on Knutson and Tao’s puzzles; the technique of curve neighborhoods and the quantum Chevalley formula: what are these, and how they help to get recursive formulas for the equivariant Gromov-Witten invariants; the quantum Schubert Calculus of Grassmannians: a presentation for the quantum ring and polynomial representatives for Schubert classes; quantum K-theory: what is it, what we know, and why is everything so much harder in this case. If time permits, I may briefly mention the connection between quantum cohomology and Toda lattice (B. Kim’s theorem), and the ‘quantum=affine’ phenomenon (D. Peterson’s conjecture, proved by T. Lam and M. Shimozono).

4:30 pm in 314 Altgeld Hall,Tuesday, April 3, 2018

Music in the Math Department: Mozart, Bach, Haken

Rudolf Haken (University of Illinois School of Music)

Abstract: Free concert by Rudolf Haken, Professor of Viola, UI School of Music. 5 & 6-string violas