Colloquia — Spring 2012
Friday, March 9, 2012
| Title |
TBA |
| Speaker |
Shanjian Tang
Fudan University
China |
| Time |
3:00pm-4:00pm |
| Place |
PHY 130 |
| Sponsor |
Yuncheng You |
| Abstract |
TBA |
Monday, February 13, 2012
| Title |
An introduction to handlebody-knot theory |
| Speaker |
Prof. Atsushi Ishii
Tsukuba University
Japan |
| Time |
3:05pm-3:55pm |
| Place |
PHY 108 |
| Sponsor |
Masahiko Saito |
| Abstract |
A handlebody-knot is a handlebody embedded in the \(3\)-sphere. A handlebody-link is a disjoint union of handlebodies embedded in the \(3\)-sphere. A handlebody-knot is a \(1\)-component handlebody-link. Two handlebody-links are equivalent if one can be transformed into the other by an isotopy of the \(3\)-sphere. I will explain how two handlebody-links can be distinguished. We may decompose handlebody-links to distinguish them, or we may use invariants. This talk is an introduction to handlebody-knot theory. |
Friday, February 10, 2012
| Title |
Some Combinatoric and Comedic Consequences of the Proof of the Pizza Conjecture |
| Speaker |
Rick Mabry*
Louisiana State University in Shreveport
Shreveport, LA |
| Time |
4:00pm-5:00pm |
| Place |
PHY 130 |
| Sponsor |
A. G. Kartsatos |
| Abstract |
The recent proof of the “Pizza Conjecture” by Deiermann and Mabry was accomplished by wishful thinking. Unable to solve the original problem, the would-be solvers generalized the problem in a fairly extreme way and hoped for the best. The resulting problem was ultimately solved by reducing a nice, continuous (calculus) problem to a gruesome, discrete (combinatorial) one. The solution made some news, and, in a comedy of errors, cheesy comments flooded the internet in the aftermath. This talk will highlight some mathematical, gastronomical, and comical slices of what ensued, and look at a more recent pizza theorem and its combinatorial solution. |
| Title |
Sparse Ramsey Hosts |
| Speaker |
Kevin Milans
University of South Carolina
Columbia, SC |
| Time |
3:00pm-4:00pm |
| Place |
PHY 130 |
| Sponsor |
Brendan Nagle |
| Abstract |
In Ramsey Theory, we study conditions under which every partition of a large structure yields a part with additional structure. For example, Van der Waerden's theorem states that every \(s\)-coloring of the integers contains arbitrarily long monochromatic arithmetic progressions, and the Hales--Jewett Theorem guarantees that every game of tic-tac-toe in high dimensions has a winner. Ramsey's Theorem implies that for any target graph \(G\), every \(s\)-coloring of the edges of some sufficiently large host graph contains a monochromatic copy of \(G\). In Ramsey's Theorem, the host graph is dense (in fact complete). We explore conditions under which the host graph can be sparse and still force a monochromatic copy of \(G\).
We write \(H \stackrel{s}{\to} G\) if every \(s\)-edge-coloring of \(H\) contains a monochromatic copy of \(G\). The \(s\)-color Ramsey number of \(G\) is the minimum \(k\) such that some \(k\)-vertex graph \(H\) satisfies \(H \stackrel{s}{\to} G\). The degree Ramsey number of \(G\) is the minimum \(k\) such that some graph \(H\) with maximum degree \(k\) satisfies \(H \stackrel{s}{\to} G\). Chvátal, Rödl, Szemerédi, and Trotter proved that the Ramsey number of bounded-degree graphs grows only linearly, sharply contrasting the exponential growth that generally occurs when the bounded-degree assumption is dropped. We are interested in the analogous degree Ramsey question: is the \(s\)-color degree Ramsey number of \(G\) bounded by some function of \(s\) and the maximum degree of \(G\)? We resolve this question in the affirmative when \(G\) is restricted to a family of graphs that have a global tree structure; this family includes all outerplanar graphs. We also investigate the behavior of the \(s\)-color degree Ramsey number as \(s\) grows. This talk includes results from three separate projects that are joint with P. Horn, T. Jiang, B. Kinnersley, V. Rödl, and D. West. |
Wednesday, February 8, 2012
| Title |
Automata generating free products of groups of order \(2\) |
| Speaker |
Dmytro Savchuk
SUNY Binghamton
Binghamton, NY |
| Time |
3:00pm-4:00pm |
| Place |
PHY 118 |
| Sponsor |
Nataša Jonoska |
| Abstract |
We construct a family of automata with \(n\) states, \(n>3\), acting on a rooted binary tree that generate the free products of cyclic groups of order \(2\). Groups generated by automata is a fascinating class of groups that includes counterexamples to several famous conjectures in group theory. I will start from discussing the definition and main properties of these groups. Then I will give a short exposition of the history of the question, explain the construction and main ideas behind the proof, which involve the notion of a dual automaton.
This is a joint result with Yaroslav Vorobets of Texas A&M University. |
Monday, February 6, 2012
| Title |
Cyclic Sieving and Cluster Multicomplexes |
| Speaker |
Brendon Rhoades
University of Southern California
Los Angeles, CA |
| Time |
3:05pm-3:55pm |
| Place |
PHY 108 |
| Sponsor |
Brian Curtin |
| Abstract |
Let \(X\) be a finite set, \(C=\langle c\rangle\) be a finite cyclic group acting on \(X\), and \(X(q)\in N[q]\) be a polynomial with nonnegative integer coefficients. Following Reiner, Stanton, and White, we say that the triple \((X,C,X(q))\) exhibits the *cyclic sieving phenomenon* if for any integer \(d>0\), the number of fixed points of \(c^d\) is equal to \(X(\zeta^d)\), where \(\zeta\) is a primitive \(|C|^{\mathrm{th}}\) root of unity. We explain how one can use representation theory to prove instances of the cyclic sieving phenomenon involving the action of tropical Coxeter elements on (complexes closely related to) cluster complexes. The representation theory involves cluster monomial bases of geometric realizations of finite type cluster algebras. |
Friday, February 3, 2012
| Title |
Searching for Structure in Graph Theory: Chromatic Index and Immersion |
| Speaker |
Jessica McDonald
Simon Fraser University
Burnaby, British Columbia
Canada |
| Time |
3:00pm-4:00pm |
| Place |
PHY 130 |
| Sponsor |
Brendan Nagle |
| Abstract |
Searching for structure is a fundamental theme in graph theory. The celebrated Goldberg-Seymour Conjecture is an example of this — it asserts that all multigraphs with “high” chromatic index contain a “dense” subgraph. In this talk we discuss edge-colouring in this context, and use the method of Tashkinov trees to gain new insights. Namely, we extend a classical characterization result of Vizing, and prove an approximation bound towards the Goldberg-Seymour Conjecture. We also consider the important containment relation of immersion. In particular, motivated by the Graph Minors Project of Robertson and Seymour and by Hadwiger's Conjecture, we explore conditions under which graphs and digraphs contain clique immersions. The results we obtain are in analogue to the clique subdivision theorem of Bollobas-Thomason and Komlos-Szemeredi. |
Wednesday, February 1, 2012
| Title |
Duality and equivalence of graphs in surfaces |
| Speaker |
Iain Moffatt
University of South Alabama
Mobile, AL |
| Time |
3:00pm-4:00pm |
| Place |
PHY 108 |
| Sponsor |
Masahiko Saito |
| Abstract |
This talk revolves around two fundamental constructions in graph theory: duals and medial graphs. There are a host of well-known relations between duals and medial graphs of graphs drawn in the plane. (This can also be thought of in terms of knot diagrams and their graphs.) By considering these relations we will be led to the working principle that duality and equality of plane graphs are equivalent concepts. It is then natural to ask what happens when we change our notion of equality. In this talk we will see how isomorphism of abstract graphs corresponds to an extension of duality called twisted duality, and how twisted duality extends the fundamental relations between duals and medial graphs from graphs in the plane to graphs in other surfaces. We will then go on to see how this group action leads to a deeper understanding of the properties of, and relationships among, various graph polynomials, including the chromatic polynomial, the Penrose polynomial, and topological Tutte polynomials. |
Friday, January 20, 2012
| Title |
Freak Waves in the Ocean |
| Speaker |
Victor L'vov
Weizmann Institute of Science
Rehovot, Israel |
| Time |
3:00pm-4:00pm |
| Place |
PHY 130 |
| Sponsor |
Arcadii Grinshpan |
| Abstract |
Ships are disappearing all over the world’s oceans at a rate of about one every week. These drownings often happen in mysterious circumstances. With little evidence researchers usually put the blame on human errors or poor maintenance. But an alarming series of drownings and near drownings including world class vessels has pushed the search for better reasons than the regular ones: freak (or rogue, monster, killing) waves.
A freak wave in the ocean is a catastrophic event when energy and momentum of the wave field spontaneously concentrate in a localized area of space generating of short wave train consisting of several waves with energy and momentum density in order of magnitude exceeding the background level. Freak waves could be disastrous for ships, drilling platforms, lighthouses and other coastal constructions.
I will present observations of freak waves, their effect on ships and discuss possible mechanisms of their creation and evolutions. |
* Rick Mabry received his Ph.D. from USF in 1985 under the direction of Professor A. G. Kartsatos.↵