Georgia Institute of Technology College of Sciences

The colloquium will be the second lecture of the TRIAD Distinguished Lecture Series by Prof. Sara van de Geer. For further information please see http://math.gatech.edu/events/triad-distinguished-lecture-series-sara-van-de-geer-0.

We consider a class of nonlinear, degenerate drift-diffusion equations in R^d. By a scaling argument, it is widely believed that solutions are uniformly Holder continuous given L^p-bound on the drift vector field for p>d. We show the loss of such regularity in finite time for p≤d, by a series of examples with divergence free vector fields. We use a barriers argument.

The concentration of Lipschitz functions around their expectation is a classical topic and continues to be very active. In these talks, we will discuss some recent progress in detail, including: A tight log-Sobolev inequality for isotropic logconcave densities A unified and improved large deviation inequality for convex bodies An extension of the above to Lipschitz functions (generalizing the Euclidean squared distance)The main technique of proof is a simple iteration (equivalently, a Martingale process) that gradually transforms any density into one with a Gaussian factor, for which isoperimetric inequalities are considerably easier to establish. (Warning: the talk will involve elementary calculus on the board, sometimes at an excruciatingly slow pace). Joint work with Yin Tat Lee.

I will discuss some free probability inequalities on the circle which can be seen in two different ways, one is via random matrix approximation, and another one by itself. I will show what I believe to be the key of these new forms, namely the fact that the circle acts on itself. For instance the Poincare inequality has a certain form which reflects this aspect. I will also briefly show how a transportation inequality can be discussed and how the standard Wasserstein distance can be modified to introduce this interesting phenomena. I will end the talk with a conjecture and some supporting evidence in the classical world of functional inequalities.

This is the second lecture of the series on h-principle. We will introduce jet bundle and it's various properties. This played a big role in the devloping modern geometry and topology. And using this we will prove Whitney embedding theorem. Only basic knowledge of calculus is required.

We will talk about discrete versions of the Bethe-Sommerfeld conjecture. Namely, we study the spectra of multi-dimensional periodic Schrödinger operators on various discrete lattices with sufficiently small potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we characterize those energies at which gaps may open, and we give sharp arithmetic criteria on the periods that ensure no gaps open. We will also provide examples that open the maximal number of gaps and estimate the scaling behavior of the gap lengths as the coupling constant goes to zero. This talk is based on a joint work with Svetlana Jitomirskaya and another work with Jake Fillman.

We will go through the proof of the hypergraph container result of Balogh, Morris, and Samotij. We will also discuss some applications of this container result.

The three-dimensional Maxwell-Pauli-Coulomb (MPC) equations are a system of nonlinear, coupled partial differential equations describing the time evolution of a single electron interacting with its self-generated electromagnetic field and a static (infinitly heavy) nucleus of atomic number Z. The time local (and, hence, global) well-posedness of the MPC equations for any initial data is an open problem, even when Z = 0. In this talk we present some progress towards understanding the well-posedness of the MPC equations and, in particular, how the existence of solutions depends on the stability of the one-electron atom. Our main result is that time global finite-energy weak solutions to the MPC equations exist provided Z is less than a critical charge. This is an oral comprehensive exam. All are welcome to attend.

One way to analyze a (finitely generated) module over a ring is to consider its minimal free resolution and look at its Betti table. The Betti table would be obtained by algebraic computations in general, but in case of the ideal (consists of relations) is generated by monomial quadratics, we can obtain Betti numbers (which are entries of the Betti table) by looking at the corresponding graphs and its associated simplicial complex. In this talk, we will introduce the Stanley-Reisner ideal which is the ideal generated by monomial quadratics and Hochster’s formula. Also, we will deal with some theorems and corollaries which are related to our topic.

The seminar will be the third lecture of the TRIAD Distinguished Lecture Series by Prof. Sara van de Geer. For further information please see http://math.gatech.edu/events/triad-distinguished-lecture-series-sara-van-de-geer-0

We give a complete description of the coefficients of the characteristic polynomial $\chi_H(\lambda)$ of a ($k$-uniform) hypergraph $H$, defined by the hyperdeterminant $\det(\mathcal{A} - \lambda \mathcal{I})$, where $\mathcal{A}$ is of the adjacency tensor/hypermatrix of $H$, and the hyperdeterminant is defined in terms of resultants of homogeneous systems associated to its argument. The co-degree $k$ coefficients can be obtained by an explicit formula yielding a linear combination of subgraph counts in $H$ of certain ``Veblen hypergraphs''. This generalizes the Harary-Sachs Theorem for graphs, provides hints of a Leibniz-type formula for symmetric hyperdeterminants, and can be used in concert with computational algebraic methods to obtain the full characteristic polynomial of many new hypergraphs, even when the degrees of these polynomials is enormous. Joint work with Greg Clark of USC.

In this talk we will discuss the paper of McGehee titled "The stable manifold theorem via an isolating block," in which a proof of the theorem is made using only elementary topology of Euclidean spaces and elementary linear algebra.