Mathematics

Phylogenomics: Inverting Random Trees

Sebastien Roch (University of Wisconsin-Madison)

Abstract: Phylogenomic analysis, in particular the estimation of species phylogenies from genome-scale data, is a common step in modern evolutionary studies. This estimation is complicated by the fact that genes evolve under biological processes that produce discordant trees. Such processes include horizontal gene transfer (HGT), incomplete lineage sorting (ILS), and gene duplication and loss (GDL), all of which can be modeled using specialized random tree distributions. I will survey some recent results regarding the analysis of these probabilistic models. Specifically their identifiability, or "invertibility," will be discussed as well as the asymptotic properties of species tree estimation methods (time permitting). Based on joint works with Max Bacharach, Brandon Legried, Erin Molloy, Elchanan Mossel, Allan Sly, Tandy Warnow, Shuqi Yu.

Exponential concentration of overlap, free energy, and replica symmetry breaking for inhomogeneous Sherrington-Kirkpatrick Spin glass

Qiang Wu (UIUC)

Abstract: Multi-species Sherrington-Kirkpatrick Spin glass model, as an inhomogeneous generalization of classical SK model, was first introduced by Barra etal in 2015, later Panchenko set up the Parisi formula to compute limiting free energy at all temperatures. However, all those results can only hold when the disorder variance matrix is positive semi-definite. For the indefinite MSK model, nearly nothing rigorous is known, physicists conjectured that this model has some intrinsic difference with the positive definite case. In this talk, we will discuss some fluctuation results for the general MSK model in replica symmetric regime. First, A unified argument for the exponential concentration of overlap will be presented, and this concentration result further enables one to prove a CLT of free energy. Besides that, we also introduce a new species-wise cavity approach to study the fluctuation of overlap vectors, and this approach does not require positive definite assumption. The fluctuation results also suggest the phase boundary (known as AT line) of replica symmetry and replica symmetry breaking for general MSK model, we will conclude with an explicit form of conjectured AT equation.

Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

Radek Adamczak (University of Warsaw)

Abstract: I will present recent results concerning the equivalence between the modified log-Sobolev inequality and a family of Beckner type inequalities with constants uniformly separated from zero. Next I will discuss moment estimates which can be derived from such inequalities, generalizing previous results due to Aida and Stroock, based on a stronger log-Sobolev inequality due to Federbush and Gross. If time permits I will present examples to moment estimates for certain Cauchy type measures, for invariant measures of Glauber dynamics and on the Poisson path space. Based on joint work with B. Polaczyk and M. Strzelecki.

Random Graph Matching with Improved Noise Robustness

Konstantin Tikhomirov (Georgia Institute of Technology)

Abstract: Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields such as computer vision and biology. In this work we will discuss a new algorithm for exact matching of correlated Erdos-Renyi graphs. Based on joint work with Cheng Mao and Mark Rudelson.

To Be Announced

Firas Rassoul-Agha (University of Utah)

To Be Announced

Wenqing Hu (Missouri University of Science and Technology)

To Be Announced

Jeremy Hoskins (University of Chicago)