Upcoming seminars

Upcoming Seminars
(in 2 days)
Defense: Dissertation
Ramsey Theorem and Ramsey Tur\'an Type Results for Hypergraphs
Vindya Bhat, Emory University
Contact: Vindya Bhat, vbhat@emory.edu
Venue: Mathematics and Science Center, Room W302
{This thesis defense includes Ramsey Theorem type results and Ramsey-Tur\'an type results. Both topics involve finding substructures within hypergraphs under certain conditions.} \\ \noindent{\textbf{Ramsey Theorem type results:}} \\ The Induced Ramsey Theorem (1975) states that for $c,r \geq 2$ and every $r$-graph $G$, there exists an $r$-graph $H$ such that every $c$-coloring of the edges of $H$ contains a monochromatic induced copy of $G$. A natural question to ask is what other subgraphs $F$ (besides edges) of $G$ can be partitioned and have the $F$-Ramsey property. We give results on the $F$-Ramsey property of two types of objects: hypergraphs or partial Steiner systems. We find that while the restrictions on the Ramsey properties of hypergraphs are lifted by any linear ordering of the vertex set, the Ramsey properties for partial Steiner systems (with vertex set linearly ordered or unordered) are quite restricted. \\ \noindent{\textbf{Ramsey-Tur\'an type results:}} \\ Tur\'an's Theorem (1941) states that for $1 < k \leq n$, every graph $G$ on $n$ vertices not containing a $K_{k+1}$ has at most $|E(T_{k}(n))|$ edges, where $T_{k}(n)$ is the graph on $n$ vertices obtained by partitioning $n$ vertices into $k$ classes of each size $\lfloor{\frac{n}{k}}\rfloor$ or $\lceil{\frac{n}{k}}\rceil$ and joining two vertices if and only if they are in two different classes. In 1946, Erd\H{o}s and Stone showed that any sufficiently large dense graph will contain $T_k(n)$. Nearly 75 years later, in spite of considerable interest and effort, no generalization of Tur\'an's Theorem or Erd\H{o}s-Stone Theorem for hypergraphs is known. Instead, we consider a variant of this question where we restrict to quasi-random hypegraphs and prove some partial results in this direction.
(in 2 days)
Seminar: Combinatorics
Proof of the Middle Levels Conjecture
Torsten Muetze, Georgia Tech
Contact: Dwight Duffus, dwight@mathcs.emory.edu
Venue: Mathematics and Science Center, Room W302
Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length $2n+1$ that have exactly $n$ or $n+1$ entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ in exactly one bit. The middle levels conjecture asserts that this graph has a Hamilton cycle for every $ n \ge 1$. This conjecture originated probably with Havel, Buck and Wiedemann, but has also been (mis)attributed to Dejter, Erdos, Trotter and various others, and despite considerable efforts it remained open during the last 30 years. In this talk I present a proof of the middle levels conjecture. In fact, I show that the middle layer graph has $2^{2^{\Omega(n)}}$ different Hamilton cycles, which is best possible.
(in 3 days)
Seminar: Algebra
Tropical schemes and the Berkovich analytification
Noah Giansiracusa, UGA
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Venue: Mathematics and Science Center, Room W304
In “Equations of tropical varieties,” J.H.Giansiracusa and I introduced a scheme-theoretic framework for tropicalization and tropical geometry. In this talk I’ll discuss recent developments in this program. Specifically, we introduce a canonical embedding of any scheme in an F1-scheme (in essence, a non-finite type toric variety) such that the corresponding tropicalization is the inverse limit of all tropicalizations and its T-points form the space underlying Berkovich’s analytification. This is related to Payne’s topological inverse limit result.
(in 5 days)
Colloquium: Department
Extremal Problems In Combinatorics
Mathias Schacht, University of Hamburg
Contact: Steve Batterson, sb@mathcs.emory.edu
Venue: MSC E208