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Series: Algebra Seminar

The classical statement that there are 27 lines on every smooth cubic surface in $\mathbb{P}^3$ fails to hold under tropicalization: a tropical cubic surface in $\mathbb{TP}^3$ often contains infinitely many tropical lines. This pathology can be corrected by reembedding the cubic surface in $\mathbb{P}^{44}$ via the anticanonical bundle.

Under this tropicalization, the 27 classical lines become an arrangement of metric trees in the boundary of the tropical cubic surface in $\mathbb{TP}^{44}$. A remarkable fact is that this arrangement completely determines the combinatorial structure of the corresponding tropical cubic surface. In this talk, we will describe their metric and topological type as we move along the moduli space of tropical cubic surfaces. Time permitting, we will discuss the matroid that emerges from their tropical convex hull.

This is joint work with Anand Deopurkar.

Monday, April 1, 2019 - 12:45 ,
Location: Skiles 006 ,
Ahmad Issa ,
University of Texas, Austin ,
Organizer: Jennifer Hom

A link in the 3-sphere is doubly slice if it is the cross-section of an unknotted 2-sphere in the 4-sphere. The double branched cover of a doubly slice link is a 3-manifold which embeds in the 4-sphere. For doubly slice Montesinos links, this produces embeddings of Seifert fibered spaces in S^4. In this pre-talk, I'll discuss a construction and an obstruction to being doubly slice.

Series: CDSNS Colloquium

One of the characteristics observed in real networks is that, as a network's topology evolves so does the network's ability to perform various complex tasks. To explain this, it has also been observed that as a network grows certain subnetworks begin to specialize the function(s) they perform. We introduce a model of network growth based on this notion of specialization and show that as a network is specialized its topology becomes increasingly modular, hierarchical, and sparser, each of which are properties observed in real networks. This model is also highly flexible in that a network can be specialized over any subset of its components. By selecting these components in various ways we find that a network's topology acquires some of the most well-known properties of real networks including the small-world property, disassortativity, power-law like degree distributions and clustering coefficients. This growth model also maintains the basic spectral properties of a network, i.e. the eigenvalues and eigenvectors associated with the network's adjacency network. This allows us in turn to show that a network maintains certain dynamic properties as the network's topology becomes increasingly complex due to specialization.

Series: Algebra Seminar

This is a two day conference (March 30-31) to be held at Georgia Tech on algebraic geometry and related areas. We will have talks by Sam Payne, Eric Larson, Angelica Cueto, Rohini Ramadas, and Jennifer Balakrishnan. See https://sites.google.com/view/gattaca/home for more information.

Series: Combinatorics Seminar

Consider a uniformly chosen proper coloring with q colors of a domain in Z^d (a graph homomorphism to a clique). We show that when the dimension is much higher than the number of colors, the model admits a staggered long-range order, in which one bipartite class of the domain is predominantly colored by half of the q colors and the other bipartite class by the other half. In the q=3 case, this was previously shown by Galvin-Kahn-Randall-Sorkin and independently by Peled. The result further extends to homomorphisms to other graphs (covering for instance the cases of the hard-core model and the Widom-Rowlinson model), allowing also vertex and edge weights (positive temperature models). Joint work with Ron Peled.

Series: Stochastics Seminar

I will talk about the structure of large square random matrices with centered i.i.d. heavy-tailed entries (only two finite moments are assumed). In our previous work with R. Vershynin we have shown that the operator norm of such matrix A can be reduced to the optimal sqrt(n)-order with high probability by zeroing out a small submatrix of A, but did not describe the structure of this "bad" submatrix, nor provide a constructive way to find it. Now we can give a very simple description of this small "bad" subset: it is enough to zero out a small fraction of the rows and columns of A with largest L2 norms to bring its operator norm to the almost optimal sqrt(loglog(n)*n)-order, under additional assumption that the entries of A are symmetrically distributed. As a corollary, one can also obtain a constructive procedure to find a small submatrix of A that one can zero out to achieve the same regularization.

Im am planning to discuss some details of the proof, the main component of which is the development of techniques that extend constructive regularization approaches known for the Bernoulli matrices (from the works of Feige and Ofek, and Le, Levina and Vershynin) to the considerably broader class of heavy-tailed random matrices.

Series: Graph Theory Seminar

Reed

and Wood and independently Norine, Seymour, Thomas, and Wollan showed

that for each $t$ there is $c(t)$ such that every graph on $n$ vertices

with no $K_t$ minor has

at most $c(t)n$ cliques. Wood asked in 2007 if

$c(t)<c^t$ for some absolute constant $c$. This problem was recently

solved by Lee and Oum. In this paper, we determine the exponential

constant. We prove that every graph on $n$ vertices

with no $K_t$ minor has at most $3^{2t/3+o(t)}n$ cliques. This bound is tight for $n \geq 4t/3$.

We use the similiar idea to give an upper bound on the number of cliques in

an $n$-vertex graph with no $K_t$-subdivsion. Easy computation will

give an upper

bound of $2^{3t+o(t)}n$; a more careful examination gives an upper bound

of $2^{1.48t+o(t)}n$. We conjecture that the optimal exponential

constant is $3^{2/3}$ as in the case of minors.

This is a joint work with Jacob Fox.

Series: School of Mathematics Colloquium

The Remez inequality for polynomials quantifies the way the maximum of a polynomial over an interval is controlled by its maximum over a subset of positive measure. The coefficient in the inequality depends on the degree of the polynomial; the result also holds in higher dimensions. We give a version of the Remez inequality for solutions of second order linear elliptic PDEs and their gradients. In this context, the degree of a polynomial is replaced by the Almgren frequency of a solution. We discuss other results on quantitative unique continuation for solutions of elliptic PDEs and their gradients and give some applications for the estimates of eigenfunctions for the Laplace-Beltrami operator. The talk is based on a joint work with A. Logunov.

Series: Stochastics Seminar

One of the most famous methods for solving large-scale over-determined linear systems is Kaczmarz algorithm, which iteratively projects the previous approximation x_k onto the solution spaces of the next equation in the system. An elegant proof of the exponential convergence of this method using correct randomization of the process is due to Strohmer and Vershynin (2009). Many extensions and generalizations of the method were proposed since then, including the works of Needell, Tropp, Ward, Srebro, Tan and many others. An interesting unifying view on a number of iterative solvers (including several versions of the Kaczmarz algorithm) was proposed by Gower and Richtarik in 2016. The main idea of their sketch-and-project framework is the following: one can observe that the random selection of a row (or a row block) can be represented as a sketch, that is, left multiplication by a random vector (or a matrix), thereby pre-processing every iteration of the method, which is represented by a projection onto the image of the sketch.

I will give an overview of some of these methods, and talk about the role that random matrix theory plays in the showing their convergence. I will also discuss our new results with Deanna Needell on the block Gaussian sketch and project method.

Series: High Dimensional Seminar

One of the most famous methods for solving large-scale over-determined linear systems is Kaczmarz algorithm, which iteratively projects the previous approximation x_k onto the solution spaces of the next equation in the system. An elegant proof of the exponential convergence of this method using correct randomization of the process is due to Strohmer and Vershynin (2009). Many extensions and generalizations of the method were proposed since then, including the works of Needell, Tropp, Ward, Srebro, Tan and many others. An interesting unifying view on a number of iterative solvers (including several versions of the Kaczmarz algorithm) was proposed by Gower and Richtarik in 2016. The main idea of their sketch-and-project framework is the following: one can observe that the random selection of a row (or a row block) can be represented as a sketch, that is, left multiplication by a random vector (or a matrix), thereby pre-processing every iteration of the method, which is represented by a projection onto the image of the sketch.

I will give an overview of some of these methods, and talk about the role that random matrix theory plays in the showing their convergence. I will also discuss our new results with Deanna Needell on the block Gaussian sketch and project method.