## Seminars and Colloquia by Series

### Introduction to metric and comparison geometry

Series
Geometry Topology Working Seminar
Time
Friday, February 27, 2009 - 15:05 for 2.5 hours
Location
Skiles 269
Speaker
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.

### Large almost monochromatic subsets in hypergraphs

Series
Combinatorics Seminar
Time
Friday, February 27, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Benny SudakovUCLA
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.)

### Introduction to metric and comparison geometry

Series
Other Talks
Time
Friday, February 27, 2009 - 15:00 for 2 hours
Location
Skiles 269
Speaker
Igor BelegradekSchool of Mathematics, Georgia Tech
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.

### Coupling in ergodic problems for Stochastic Navier--Stokes. Part II

Series
Probability Working Seminar
Time
Friday, February 27, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 268
Speaker
Sergio AlmadaSchool of Mathematics, Georgia Tech
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.

### Fredholm operators

Series
SIAM Student Seminar
Time
Friday, February 27, 2009 - 12:30 for 2 hours
Location
Skiles 269
Speaker
Weizhe ZhangSchool of Mathematics, Georgia Tech
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.

### Scenery reconstruction part II

Series
Stochastics Seminar
Time
Thursday, February 26, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Henri MatzingerSchool of Mathematics, Georgia Tech
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.

### Geometry and complexity of partition bijections

Series
School of Mathematics Colloquium
Time
Thursday, February 26, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Igor PakUniversity of Minnesota
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.

### The Geometry of Logconcave Functions

Series
ACO Student Seminar
Time
Wednesday, February 25, 2009 - 13:30 for 2 hours
Location
Skiles 269
Speaker
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.

### Quantum Statistical Mechanics, graphs and determinants

Series
Research Horizons Seminar
Time
Wednesday, February 25, 2009 - 12:00 for 2 hours
Location
Skiles 255
Speaker
Federico BonettoSchool of Mathematics, Georgia Tech
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.

### Nonlinear effect of copy number variation on gene expression

Series
Mathematical Biology Seminar
Time
Wednesday, February 25, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Yuriy MileykoSchool of Biology, Georgia Tech
The expression dynamics of interacting genes depends on the topology of the regulatory network, the quantitative nature of feedbacks and interactions between DNA, RNA and proteins, and the biochemical state of the intracellular and surrounding environment. In this talk we show that dynamics of a gene regulatory network can also depend sensitively on the copy number of genes and promoters. Genetic regulatory networks include an overrepresentation of subgraphs commonly known as network motifs. We consider positive feedback, bistable feedback, and toggle switch motifs and show that variation in gene copy number can cause a sequence of saddle-node bifurcations in the corresponding differential equations models, which leads to multiple orders of magnitude change in gene expression. A similar analysis of a 3-gene motif with successive inhibition (the repressilator'') reveals that changes in gene copy number can also cause a Hopf bifurcation, thus leading to a qualitative switch in system behavior among oscillatory and equilibrium dynamics. Importantly, we show that these bifurcations exist over a wide range of parameter values, thus reinforcing our claim that copy number is a key control parameter in the expression dynamics of regulatory networks.