Speaker: Adam Sheffer, CalTech
Title: Geometric Incidences and the Polynomial Method
Abstract: While the topic of geometric incidences has existed for several decades, in recent years it has been experiencing a renaissance due to the introduction of new polynomial methods. This progress involves a variety of new results and techniques, and also interactions with fields such as algebraic geometry and harmonic analysis.
Title: Counting lattice points inside a d-dimensional polytope via Fourier analysis
Abstract: Given a convex body $B$ which is embedded in a Euclidean space $R^d$, we can ask how many lattice points are contained inside $B$, i.e. the number of points in the intersection of $B$ and the integer lattice $Z^d$. Alternatively, we can count the lattice points inside B with weights, which sometimes creates more nicely behaved lattice-point enumerating functions.
Speaker: Gil Kalai, HU
Title: Algebraic-topological invariants of hypergraphs and extremal combinatorics
We will discuss some algebraic invariants of hypergraphs and some connection to algebraic topology. We will present some conjectural (rather speculative) relations with two central problems in extremal combinatorics: The Turan (4,3) conjecture and the Erdos-Rado sunflower conjecture.
Speaker: Micha Sharir (Tel Aviv University)
Title: Eliminating depth cycles for lines and triangles, with applications to bounding incidences
The talk presents three related results.
We first consider the problem of eliminating all depth cycles in a set of n lines in 3-space.
For two lines l_1, l_2 in 3-space (in general position), we say that l_1 lies below l_2 if the
unique vertical line that meets both lines meets l_1 at a point below the point where it meets l_2.
This depth relationship typically has cycles, which can be eliminated if we cut the lines into
Speaker: Lukas Kühne (University of Bonn)
Title: Heavy hyperplanes in multiarrangements and their freeness
One of the central topics among the theory of hyperplane arrangements is their freeness. Terao's conjecture tries to link the freeness with the combinatorics of an arrangement. One of the few categories of arrangements which satisfy this conjecture consists of 3-dimensional arrangements with an unbalanced Ziegler restriction. This means that the arrangement contains a lot of hyperplanes intersecting in one single line
Title: Inp-minimal ordered groups.
Abstract. The main goal of the talk is to present the proof of the theorem stating that inp-minimal (left)-ordered groups are abelian. This generalizes a previous result of P. Simon for bi-ordered inp-minimal groups.
Title: Fixed points of finite groups on modules
Abstract: Suppose G is a finite group, p is a prime, S is a Sylow p-subgroup of G, and V is a G-module over a field of characteristic p. In some situations, an easy calculation shows that the fixed points of G on V are the same as the fixed points of the normalizer
of S in G. Generalizations of this result have been obtained previously to study the structure of G for p odd. We plan to describe a new generalization for p = 2. (This is part of joint work with J. Lynd that removes the classification of finite simple groups
Title: Stability patterns in representation theory and applications
Many natural sequences of objects come equipped with group actions, e.g. the symmetric group on n letters acting on a space X_n. This leads to fundamental instability of invariants, such as homology, arising from the representation theory of the sequence of groups. Representation stability is a new and increasingly important set of ideas that describe a sense in which such sequence of representations (of different groups) stabilizes.