Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, November 27, 2018.

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Tuesday, November 27, 2018

1:00 pm in 345 Altgeld Hall,Tuesday, November 27, 2018

The groups $\mathbb{Z}$ and $\mathbb{Q}$ with predicates for being square-free

Neer Bhardwaj (UIUC)

Abstract: I will begin with a brief survey of model-theoretic tameness properties of some of the most natural abelian groups. Then I move on to describe our work where we consider two expansions each of groups of integers and rational numbers with a predicate for being square-free. We prove that one of the structures is model theoretically wild while the other three structures are model theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences. This is joint work with Minh Chieu Tran.

2:00 pm in 243 Altgeld Hall,Tuesday, November 27, 2018

How many edges guarantee a monochromatic ordered path?

Mikhail Lavrov (Illinois Math)

Abstract: An ordered graph is a simple graph with an ordering on its vertices, in which an ordered path $P_n$ is a path on $n$ edges whose vertices are in increasing order. In this talk, we will investigate the ordered size Ramsey number $\tilde r(P_r, P_s)$. This is the minimum $m$ for which some $m$-edge graph $H$ exists, such that every red-blue coloring of some the edges of $H$ contains either a red $P_r$ or a blue $P_s$.

I will show upper and lower bounds on $\tilde r(P_r, P_s)$ which are tight up to a polylogarithmic factor, and discuss connections to other Ramsey numbers for paths.

This is joint work with József Balogh, Felix Clemen, and Emily Heath.

2:00 pm in 345 Altgeld Hall,Tuesday, November 27, 2018

On the structure of preferential attachment networks with community structure.

Bruce Hajek (ECE)

Abstract: An extensive theory of community detection has developed within the past few years. The goal is to discover clusters of vertices in a graph based on the edges of the graph. In this talk we focus on the problem of community detection for the Barabasi-Albert preferential attachment model with communities, defined by Jonathan Jordan. In such model, vertices are sequentially attached to the graph, with preference to attach more edges to existing vertices with larger degrees, multiplied by affinities based on community membership. It is shown that the model has sufficient structure to formulate approximate belief propagation algorithms for community detection. (Details at arxiv 1801.06818.) Based on joint work with S. Sankagiri.

3:00 pm in 243 Altgeld Hall,Tuesday, November 27, 2018

Effective generation problems for families of varieties

Yajnaseni Dutta (Northwestern University)

Abstract: In birational geometry, a great deal of interest lies in the study of space of global sections (i.e. linear system) associated to a line bundle and when it produces morphisms to projective spaces (i.e. are globally generated). For instance, Takao Fujita, in 1988, conjectured that there is an effective bound on the twists of ample line bundles to obtain global generation of canonical bundles. Even though the conjecture remains unsolved as of today, partial progress was made by Angehrn-Siu, Ein-Lazarsfeld, Heier, Helmke, Kawamata, Reider, Ye-Zhu et al. In this talk I will focus on similar global generation problems for pushforwards of canonical and pluricanonical bundles under certain morphisms f: Y --> X. The canonical bundle case was first approached by Kawamata and the proof combines the Hodge theoretic properties of the morphism and the birational geometric techniques from the known cases of the Fujita conjecture. I will show how birational techniques and invariants of positivity of line bundles allow us to avoid Hodge theory to make a global generation statement on an open subset (of X) over which f behaves particularly nicely, partially proving a conjecture proposed by Popa and Schnell. This, in particular, ensures an effective non-vanishing of global sections of the pushforward. As an upshot I will discuss positivity properties of such pushforwards and an effective vanishing theorem. Finally, If time permits I will give a crash course on Kawamata's method and explain why Hodge theory is indispensable to hope for further improvements in this direction.

4:00 pm in 314 Altgeld Hall,Tuesday, November 27, 2018

What we still don't know about addition and multiplication

Carl Pomerance (Dartmouth College)

Abstract: How could there be something we don’t know about arithmetic? It would seem that subject was sewn up in third grade. But here’s a problem: What is the most efficient method for multiplication? No one knows. And here is another: How many different numbers appear in a large multiplication table? There are many more such problems, largely unsolved, and for which we could use some help!

A reception will be held from 5-6 pm in 239 Altgeld Hall following this first lecture.