- You are here:
- GT Home
- Home
- News & Events

Series: Geometry Topology Seminar

We introduce a construction of an immersed surface for a null-homologous braid in an annulus open book decomposition. This is hinted by the so called Bennequin surface for a braid in R^3. By resolving the singularities of the immersed surface, we obtain an embedded Seifert surface for the braid. Then we compute a self-linking number formula using this embedded surface and observe that the Bennequin inequality is satisfied if and only the contact structure is tight. We also prove that our self-linking formula is invariant (changes by 2) under a positive (negative) braid stabilization which preserves (changes) the transverse knot class.

Friday, February 27, 2009 - 15:05 ,
Location: Skiles 269 ,
Igor Belegradek ,
Ga Tech ,
Organizer: John Etnyre

Comparison geometry studies Riemannian manifolds with a given curvature bound. This minicourse is an introduction to volume comparison (as developed by Bishop and Gromov), which is fundamental in understanding manifolds with a lower bound on Ricci curvature. Prerequisites are very modest: we only need basics of Riemannian geometry, and fluency with fundamental groups and metric spaces. In the third (2 hour) lecture I shall prove volume and Laplacian comparison theorems.

Series: Combinatorics Seminar

We show that for all \el an \epsilon>0 there is a constant c=c(\ell,\epsilon)>0 such that every \ell-coloring of the triples of an N-element set contains a subset S of size c\sqrt{\log N} such that at least 1-\epsilon fraction of the triples of S have the same color. This result is tight up to the constant c and answers an open question of Erd\H{o}s and Hajnal from 1989 on discrepancy in hypergraphs. For \ell \geq 4 colors, it is known that there is an \ell-coloring of the triples of an N-element set whose largest monochromatic subset has cardinality only \Theta(\log \log N). Thus, our result demonstrates that the maximum almost monochromatic subset that an \ell-coloring of the triples must contain is much larger than the corresponding monochromatic subset. This is in striking contrast with graphs, where these two quantities have the same order of magnitude. To prove our result, we obtain a new upper bound on the \ell-color Ramsey numbers of complete multipartite 3-uniform hypergraphs, which answers another open question of Erd\H{o}s and Hajnal. (This is joint work with D. Conlon and J. Fox.)

Series: Other Talks

Series: Probability Working Seminar

This is a continuation of last week's seminar. The talk is based on a paper by Kuksin, Pyatnickiy, and Shirikyan. In this paper, the convergence to a stationary distribution is established by partial coupling. Here, only finitely many coordinates in the (infinite-dimensional) phase space participate in the coupling while the dynamics takes care of the other coordinates.

Series: SIAM Student Seminar

This talk will follow Peter Lax on the linear algebraic fact of the index of Fredholm operators such as the product formula and stability, all of which are totally elementary.

Series: Stochastics Seminar

Last week we saw combinatorial reconstruction. This time we are going to explain a new approach to Scenery Reconstruction. This new approach could allow us to prove that being able to distinguish sceneries implies reconstructability.

Series: School of Mathematics Colloquium

The study of partition identities has a long history going back to Euler, with applications ranging from Analysis to Number Theory, from Enumerative Combina- torics to Probability. Partition bijections is a combinatorial approach which often gives the shortest and the most elegant proofs of these identities. These bijections are then often used to generalize the identities, find "hidden symmetries", etc. In the talk I will present a modern approach to partition bijections based on the geometry of random partitions and complexity ideas.

Series: ACO Student Seminar

In this talk, I will introduce the class of logconcave and s-concave functions, illustrate their properties, and explain their connections to convex geometry. I will present a simple and unified approach for proving probabilistic inequalities for logconcave and s-concave densities on the real line. Lastly I will use these techniques to prove two important theorems in convex geometry: Grunbaum's theorem, every halfspace cut through the centroid of a convex body contains at least a 1/e volume fraction of the body, and the Milman-Pajor inequality, a convex body in R^n is sandwiched between its inertial ellipsoid and a factor n scaling of it. Joint work with Santosh Vempala.

Series: Research Horizons Seminar

I'll give a brief introduction to the to Quantum Statistical Mechanics in the case of systems of Fermions (e.g. electrons) and try to show that a lot of the mathematical problems can be framed in term of counting (Feynman) graphs or estimating large determinants.