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for events the day of Tuesday, October 18, 2016.

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Tuesday, October 18, 2016

11:00 am in 345 Altgeld Hall,Tuesday, October 18, 2016

2-monads in homotopy theory

Angélica Osorno (Reed College)

Abstract: The classifying functor from categories to topological spaces provides a way of constructing spaces with certain properties or structure from categories with similar properties of structure. An important example of this is the construction of infinite loop spaces from symmetric monoidal categories. The particular kinds of extra structure can typically be encoded by monads on the category of small categories. In order to provide more flexibility in the kinds of morphisms allowed, one can work with the associated 2-monad in the 2-category of categories, functors, and natural transformations. In this talk I will give the categorical setup required, and I will give examples of interest to homotopy theorists. I will also outline how this method of working can give general statements about strictifications and comparisons of homotopy theories. This is partially based on work with two different sets of collaborators: Nick Gurski, Niles Johnson, and Marc Stephan; Bert Guillou, Peter May, and Mona Merling.

1:00 pm in 345 Altgeld Hall,Tuesday, October 18, 2016

Model theory for $C^*$-algebras

Alessandro Vignati (York University Math)

Abstract: Expanding the foundational work of Ben Yaacov, Berenstein, Henson, and Utvyatsov on model theory for metric structure, a series of influential papers by Farah, Hart, and Sherman started developing the model theory for $C^*$-algebras. In this talk, after a brief introduction, I will give a short survey on the known results concerning model theoretical notions such as saturation, quantifier elimination, and existentially closed models, when viewed in this setting. This is the joint work of many, included in collaborative papers with Eagle, Eagle-Farah-Kirchberg and Eagle-Goldbring.

2:00 pm in 241 Altgeld Hall,Tuesday, October 18, 2016

Higher order mollifications of the zeta function

Nicolas Robles (UIUC )

Abstract: We will discuss the process of mollifying the Riemann zeta-function and how this can be used to obtain mean value moments and proportions of zeros. We will also describe how to mollify $1/(\zeta + \zeta' + \zeta'' +...)$ and discuss some properties of the associated exponential sums.

3:00 pm in 243 Altgeld Hall,Tuesday, October 18, 2016

Rational Curves on Large Degree Hypersurfaces in Positive Characteristic

Matthew Woolf (UIC)

Abstract: One of the most foundational results about complex hypersurfaces is that there is a dichotomy between hypersurfaces of small degree, which are covered by rational curves, and hypersurfaces of large degree, on which rational curves are rare. Over a field of positive characteristic, however, one can construct smooth hypersurfaces of arbitrarily large degree which are unirational. In this talk, I will discuss joint work with Eric Riedl showing that nonetheless, a general hypersurface of large degree does not have many rational curves.

3:00 pm in 241 Altgeld Hall,Tuesday, October 18, 2016

Supersaturation in the Boolean lattice and Kleitman's conjecture

Adam Zsolt Wagner (Illinois Math)

Abstract: A central theorem in combinatorics is Sperner's Theorem, which determines the maximum size of a family that does not contain a 2-chain. Erdős later extended this result and determined the largest family not containing a k-chain. Erdős and Katona and later Kleitman asked how many such chains must appear in families whose size is larger than the corresponding extremal result. This question was resolved for 2-chains by Kleitman in 1966, who showed that amongst families of size M, the number of 2-chains is minimized by a family whose sets are taken as close to the middle layer as possible. He also conjectured that the same conclusion should hold for all k, not just 2. The best result on this question is due to Das, Gan and Sudakov who showed that Kleitman's conjecture holds for families whose size is at most the size of the k+1 middle layers of the hypercube. Our main result is that for every fixed k and $\varepsilon > 0$, if n is sufficiently large then Kleitman's conjecture holds for families of size at most $(1-\varepsilon)2^n$, thereby establishing Kleitman's conjecture asymptotically. Our proof is based on ideas of Kleitman and Das, Gan and Sudakov. This is joint work with József Balogh.

4:00 pm in 314 Altgeld Hall,Tuesday, October 18, 2016

Advanced Mathematical Pathways for Underserved Students

Daniel Zaharopol (Executive Director, Art of Problem Solving Foundation; Director, Bridge to Enter Advanced Mathematics)

Abstract: Those of us in science and math careers had many experiences to help us get here. From our own independent projects, to a teacher who took a special interest and gave us extra problems, to clubs and competitions, we were shaped not just by the standard school curriculum but in how we went beyond it. Now imagine someone who wants to be a scientist or a mathematician entering their freshman year of college---but who doesn't have that same preparation. College is a difficult transition for everyone, but on top of the usual challenges, an underserved student will have far fewer academic experiences that demanded college-level thinking, and they might be entering a very different culture. Far too many low-income and underrepresented students drop out at this point, contributing to the gaps we see in attainment at both the undergraduate and graduate levels.

Everyone talks about the need for greater diversity in math and science, but what can we actually do about it? I will first look at the kind of thinking that our fields demand of us and the preparation many of us received. Then, I'll share the progress made at Bridge to Enter Advanced Mathematics (BEAM), a program I started in New York City which provides this pathway for underserved students. In particular, I will talk about what it means (and takes) to teach deep, rich, proof-based mathematics to young students with disadvantaged backgrounds, and what is required to coach them into taking up other opportunities in the future.