In this meeting we plan to discuss the topics for the semester, decide on the near future plans. My suggestions starts from Patterson-Sullivan theory, and several topics in manifolds of negative curvature. I will give some overview of some results that I propose to study in the seminar.

We will discus the direction of the seminar for this semester

Given a polynomial x^n+a_1x^{n-1}+ . . . + a_n, what is the simplest formula for the roots in terms of the coefficients a_1, . . . a_n? Following Abel, we can no longer take “simplest” to mean in radicals, but we could ask for a solution using only 1 or 2 or d-variable functions. Hilbert conjectured that for degrees 6,7 and 8, we need 2,3 and 4 variable functions respectively. In a too little known paper, he then sketched how the 27 lines on a cubic surface should give a 4-variable solution of the general degree 9. In this talk, I’ll review the geometry of solving polynomials, explain Hilbert’s idea, and then extend his geometric methods to get best-to-date upper bounds on the number of variables needed to solve a general degree n polynomial.

There are still completely open fundamental questions about polynomials in one variable. One example is Hilbert's 13th Problem, a conjecture going back long before Hilbert. Indeed, the invention of algebraic topology grew out of an effort to understand how the roots of a polynomial depend on the coefficients. The goal of this talk is to explain part of the circle of ideas surrounding these questions. Along the way, we will encounter some beautiful classical objects - the space of monic, degree d square-free polynomials, algebraic functions, lines on cubic surfaces, level structures on Jacobians, braid groups, Galois groups, and configuration spaces - all intimately related to each other, all with mysteries still to reveal.

A group $G$ is a knit product of subgroups $H$ and $K$ if $G=HK$ and $H\cap K=\{1\}$, or equivalently, if the group operation $G\times G\to G$ restricts to a bijection $H\times K\to G$. We will discuss knit products in the context of large-scale geometry.

Sparse partial least squares (SPLS) is widely used in applied sciences as a method that performs dimension reduction and variable selection simultaneously in linear regression. Several implementations of SPLS have been derived, among which the SPLS proposed in Chun and Keleş (2010) is very popular and highly cited. However, for all of these implementations, the theoretical properties of SPLS are largely unknown. In this paper, we propose a new version of SPLS, called the envelope-based SPLS, using a connection between envelope models and partial least squares (PLS). We establish the consistency, oracle property and asymptotic normality of the envelope-based SPLS estimator. The large-sample scenario and high-dimensional scenario are both considered. We also develop the envelope-based SPLS estimators under the context of generalized linear models, and discuss its theoretical properties including consistency, oracle property and asymptotic distribution. Numerical experiments and examples show that the envelope-based SPLS estimator has better variable selection and prediction performance over the existing SPLS estimators.

Mirror Symmetry provides a link between symplectic and algebraic geometry through a duality in string theory. In particular, it asserts a link from the symplectic geometry of a space M to the algebraic geometry of its mirror space W. One way we see this is now known as classical mirror symmetry: the Gromov-Witten or enumerative theory of a symplectic space is encapsulated by the Hodge theory / periods of the mirror algebraic space. In the 90s this was articulated just for Calabi-Yau varieties, but it has expanded even further to Fano varieties; however, the mirror space is now not an algebraic variety but a mildly non-commutative object known as a Landau-Ginzburg model. Recently, this notion has been developed even to articulate mirror symmetry between Landau-Ginzburg models. In this talk, we will explain what non-commutative Hodge theory / periods look like for a Landau-Ginzburg model and how they predict phenomena in open enumerative theories for the mirror.

We will introduce a notion of a K\"ahler metric with constant weighted scalar curvature on a compact K\"ahler manifold $X$, depending on a fixed real torus $\T$ in the reduced group of automorphisms of $X$, and two smooth (weight) functions defined on the momentum image of $X$. We will also define a notion of weighted Mabuchi energy adapted to our setting, and of a weighted Futaki invariant of a $\T$-compatible smooth K\"ahler test configuration associated to $(X, \T)$. After that, using the geometric quantization scheme of Donaldson, we will show that if a projective manifold admits in the corresponding Hodge K\"ahler class a K\"ahler metric with constant weighted scalar curvature, then this metric minimizes the weighted Mabuchi energy, which implies a suitable notion of weighted K-semistability. As an application, we describe the K\"ahler classes on a geometrically ruled complex surface of genus greater than 2, which admits conformally K\"ahler Einstein-Maxwell metrics.

In the $n$-dimensional hypercube graph, one can easily choose half of the vertices such that they induce an empty graph. However, having even just one more vertex would cause the induced subgraph to contain a vertex of degree at least $\sqrt{n}$. This result is best possible, and improves a logarithmic lower bound shown by Chung, Furedi, Graham and Seymour in 1988. In this talk we will discuss a very short algebraic proof of it. As a direct corollary of this purely combinatorial result, the sensitivity and degree of every boolean function are polynomially related. This solves an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.

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When 1 -> H -> G -> Q -> 1 is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an ``ending lamination'' on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(F_n), one can identify the resultant quotient space with a certain R-tree in the boundary of Culler-Vogtmann's Outer space.

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