Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, November 15, 2016.

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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, November 15, 2016

12:00 pm in 243 Altgeld Hall,Tuesday, November 15, 2016

#### Polynomial-time Nielsen--Thurston classification

###### Mark Bell (Illinois Math)

Abstract: We will discuss a new polynomial-time algorithm, joint with Richard Webb, for computing the Nielsen--Thurston type of a mapping class. The procedure works by considering the maps action on the curve graph. To be able to compute this action, we need to be able to construct geodesics through the curve graph. However, this graph is locally infinite and so standard pathfinding algorithms struggle. We will discuss a new refinement of the techniques of Leasure, Shackleton, Watanabe and Webb for overcoming this local infiniteness that allows such geodesics to be found in polynomial time.

1:00 pm in 7 Illini Hall,Tuesday, November 15, 2016

#### When Newton Met Banach

###### Hadrian Quan   [email] (UIUC Math)

Abstract: The regular value theorem states that for a smooth map $f:\mathbb{R}^N\rightarrow \mathbb{R}$, the preimage of a 'regular value’ of $f$ is a smooth manifold. Sard’s theorem states that such regular values are the typical points in the image of $f$. If we replaced our $f$ with a smooth mapping $F: X\rightarrow Y$ between two Banach spaces $X$ and $Y$, then versions of both Sard’s theorem and the Regular value theorem still hold. I will state and prove both of these results using Newton’s Method from Calc I as a guiding principle. Along the way I’ll try and stress how building manifolds and moduli spaces is not too different from finding solutions to equations.

1:00 pm in 345 Altgeld Hall,Tuesday, November 15, 2016

#### Foundations of cologic

###### Alex Kruckman (Indiana University Math)

Abstract: The existence of a robust categorical dual to first-order logic is hinted at in (at least) four independent bodies of work: (1) Projective Fraïssé theory [Solecki & coauthors, Panagiotopolous]. (2) The cologic of profinite groups (e.g. Galois groups), which plays an important role in the model theory of PAC fields [Cherlin - van den Dries - Macintyre, Chatzidakis]. (3) Ultracoproducts and coelementary classes of compact Hausdorff spaces [Bankston]. (4) Universal coalgebras and coalgebraic logic [Rutten, Kurz - Rosicky, Moss, others]. In this talk, I will propose a natural syntax and semantics for such a dual "cologic", in which "coformulas" express properties of partitions of "costructures", dually to the way in which formulas express properties of tuples from structures. I will show how the basic theorems and constructions of first-order logic (completeness, compactness, ultraproducts, Henkin constructions, Löwenheim-Skolem, etc.) can be dualized, and I will discuss some possible extensions of the framework. Note: For those who attended Midwest Model Theory day at UIC, the first half of the talk will essentially be a repeat, but I will focus on different material in the second half.

2:00 pm in 241 Altgeld Hall,Tuesday, November 15, 2016

#### Bounds on multiplicative character sums over a finite field

###### Chieu Minh Tran (UIUC )

Abstract: We will discuss a folkloric result on weak bounds for multiplicative character sums whose summation domain is a non-smooth variety over a finite field. We will briefly talk about a proof based on ell-adic cohomology, the Grothendieck-Lefschetz trace formula and Deligne's theorem in Weil II. Then, we will talk about a more elementary proof using model-theoretic techniques and a Weil-style bound for multiplicative character sums on a curve. If time permits, we will also explain the model-theoretic context where we need this result.

3:00 pm in 243 Altgeld Hall,Tuesday, November 15, 2016

#### Cartier descent and p-curvature in mixed characteristic.

###### Chris Dodd (UIUC)

Abstract: I'll review the classical notions of the title in positive characteristic, and then explain some recent progress in "lifting" these notions to mixed characteristic; with applications to p-adic differential equations.

3:00 pm in 241 Altgeld Hall,Tuesday, November 15, 2016

#### Capture time in the game Cops and Robbers

###### Benjamin Reiniger (Illinois Institute of Technology)

Abstract: The game of Cops and Robbers involves a team of $k$ cops trying to catch a single robber on a graph $G$. The players alternate turns moving along edges of $G$. We consider the minimum number of turns needed for the cops to catch the robber, called the $k$-capture time of $G$. We determine the capture times exactly for trees and asymptotically for grids and cubes, and we give bounds on the capture time of planar graphs when the number of cops is 3 or $\sqrt{n}$. This is joint work with Anthony Bonato, Xavier Pérez-Giménez, and Paweł Prałat.

4:00 pm in 445 Altgeld Hall,Tuesday, November 15, 2016

#### A Theory for Measures of Tail Risk

###### Dr. Fangda Liu (Central University of Finance and Economics, China)

Abstract: In modern risk management, a crucial consideration is how to describe and quantify a risk according to its tail behaviour. Prominent, yet elementary, examples of tail risk measures are Value-at-Risk (VaR) and the Expected Shortfall (ES), which are two popular classes of regulatory risk measures in banking and insurance. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures which quantify the tail risk, that is, the behavior of a risk beyond a certain quantile. We establish a connecting between a tai risk measure and its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further tail shortfall risk measure and elicitability. In particular, there is no elicitable tail convex risk measure rather than the essential supremum, and under a continuity condition, the only elicitable and positively homogeneous monetary tail risk measures are the VaRs.