Seminars and Colloquia by Series

Exponential varieties

Series
Algebra Seminar
Time
Monday, November 9, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005 or 006
Speaker
Caroline Uhler – MIT
Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, generalizing those of toric varieties and their moment maps. Another special class, including Gaussian graphical models, are varieties of inverses of symmetric matrices satisfying linear constraints. We develop a general theory of exponential varieties, with focus on those defined by hyperbolic polynomials. Joint work with Mateusz Michałek, Bernd Sturmfels, and Piotr Zwiernik.

Analytic methods in graph theory

Series
Joint School of Mathematics and ACO Colloquium
Time
Friday, November 6, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel Kral – University of Warwick

Please Note: Refreshments will be served in the atrium after the talk.

The theory of combinatorial limits provides analytic ways of representing large discrete objects. The theory has opened new links between analysis, combinatorics, computer science, group theory and probability theory. In this talk, we will focus on limits of dense graphs and their applications in extremal combinatorics. We will present a general framework for constructing graph limits corresponding to solutions of extremal graph theory problems, which led to constructing counterexamples to several conjectures concerning graph limits. At the end, we will discuss limits of sparse graphs and possible directions to unify the existing approaches related to dense and sparse graphs.

Folkman Numbers

Series
ACO Student Seminar
Time
Friday, November 6, 2015 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Troy Retter – Emory University
For an integer k, the Folkman number f(k) is the least integer n for which there exists a graph G on n vertices that does not contain a clique of size k and has the property that every two coloring of E(G) yields a monochromatic clique of size of size k. That is, it is the least number of vertices in a K_{k+1}-free graph that is Ramsey to K_k. A recent result of Rodl, Rucinski, and Schacht gives an upper bound on the Folkman numbers f(k) which is exponential in k. A fundamental tool in their proof is a theorem of Saxton and Thomason on hypergraph containers. This talk will give a brief history of the Folkman numbers, introduce the hypergraph container theorem, and sketch the proof of the Rodl, Rucinski, and Schacht result. Recent work with Hiep Han, Vojtech Rodl, and Mathias Schacht on two related problems concerning cycles in graphs and arithmetic progressions in subset of the integers will also be presented.

Ergodic Measures for shifts with eventually constant complexity growth

Series
CDSNS Colloquium
Time
Friday, November 6, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jon Fickenscher – Princeton University –
We will consider (sub)shifts with complexity such that the difference from n to n+1 is constant for all large n. The shifts that arise naturally from interval exchange transformations belong to this class. An interval exchange transformation on d intervals has at most d/2 ergodic probability measures. We look to establish the correct bound for shifts with constant complexity growth. To this end, we give our current bound and discuss further improvements when more assumptions are allowed. This is ongoing work with Michael Damron.

More on Logarithmic sums of convex bodies

Series
Stochastics Seminar
Time
Thursday, November 5, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christos Saraoglou – Kent State University
We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesgue measure in dimension n would imply the log-BMI and, therefore, the B-conjecture for any even log-concave measure in dimension n. As a consequence, we prove the log-BMI and the B-conjecture for any even log-concave measure, in the plane. Moreover, we prove that the log-BMI reduces to the following: For each dimension n, there is a density f_n, which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density f_n. As byproduct of our methods, we study possible log-concavity of the function t -> |(K+_p\cdot e^tL)^{\circ}|, where p\geq 1 and K, L are symmetric convex bodies, which we are able to prove in some instances and as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.

Math Problems in Gene Regulation

Series
School of Mathematics Colloquium
Time
Thursday, November 5, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Caroline Uhler – MIT –
Although the genetic information in each cell within an organism is identical, gene expression varies widely between different cell types. The quest to understand this phenomenon has led to many interesting mathematics problems. First, I will present a new method for learning gene regulatory networks. It overcomes the limitations of existing algorithms for learning directed graphs and is based on algebraic, geometric and combinatorial arguments. Second, I will analyze the hypothesis that the differential gene expression is related to the spatial organization of chromosomes. I will describe a bi-level optimization formulation to find minimal overlap configurations of ellipsoids and model chromosome arrangements. Analyzing the resulting ellipsoid configurations has important implications for the reprogramming of cells during development. Any knowledge of biology which is needed for the talk will be introduced during the lecture.

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