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Friday, February 23, 2018 - 10:00 ,
Location: Skiles 006 ,
Tim Duff ,
Georgia Tech ,
tduff3@gatech.edu ,
Organizer: Kisun Lee

TBA

Friday, February 23, 2018 - 10:00 ,
Location: Skiles 006 ,
Tim Duff ,
Georgia Tech ,
tduff3@gatech.edu ,
Organizer: Kisun Lee

TBA

Series: School of Mathematics Colloquium

A distinct covering system of congruences is a finite collection of arithmetic progressions $$a_i \bmod m_i, \qquad 1 < m_1 < m_2 < \cdots < m_k.$$Erdős asked whether the least modulus of a distinct covering system of congruences can be arbitrarily large. I will discuss my proof that minimum modulus is at most $10^{16}$, and recent joint work with Pace Nielsen, in which it is proven that every distinct covering system of congruences has a modulus divisible by either 2 or 3.

Wednesday, February 21, 2018 - 14:00 ,
Location: Skiles 006 ,
Kevin Kodrek ,
GaTech ,
Organizer: Anubhav Mukherjee

There are a number of ways to define the braid group. The traditional definition involves equivalence classes of braids, but it can also be defined in terms of mapping class groups, in terms of configuration spaces, or purely algebraically with an explicit presentation. My goal is to give an informal overview of this group and some of its subgroups, comparing and contrasting the various incarnations along the way.

Series: Analysis Seminar

I will speak how to ``dualize'' certain martingale estimates related to the dyadic square function to obtain estimates on the Hamming and vice versa. As an application of this duality approach, I will illustrate how to dualize an estimate of Davis to improve a result of Naor--Schechtman on the real line. If time allows we will consider one more example where an improvement of Beckner's estimate will be given.

Series: PDE Seminar

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - i.e. periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. To overcome these problems, we first reduce the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme that requires very weak Melnikov non-resonance conditions (which lose derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments. This is a joint work with P. Baldi, M. Berti and R. Montalto.

Series: Geometry Topology Seminar

Now that the geometrization conjecture has been proven, and the virtual Haken conjecture has been proven, what is left in
3-manifold topology? One remaining topic is the computational complexity of geometric topology problems. How difficult is it to
distinguish the unknot? Or 3-manifolds from each other? The right approach to these questions is not just to consider quantitative
complexity, i.e., how much work they take for a computer; but also qualitative complexity, whether there are efficient algorithms with
one or another kind of help. I will discuss various results on this theme, such as that knottedness and unknottedness are both in NP; and
I will discuss high-dimensional questions for context.

Series: Geometry Topology Seminar

Although the Alexander polynomial does not satisfy an unoriented skein relation, Manolescu (2007) showed that there exists an unoriented skein exact triangle for knot Floer homology. In this talk, we will describe some developments in this direction since then, including a combinatorial proof using grid homology and extensions to the Petkova-Vertesi tangle Floer homology (joint work with Ina Petkova) and Zarev's bordered sutured Floer homology (joint work with Shea Vela-Vick).

Series: CDSNS Colloquium

Using techniques from local bifurcation theory, we prove the existence of various types of temporally periodic solutions for damped wave equations, in higher dimensions. The emphasis is on understanding the role of external bifurcation parameters and symmetry, in generating the periodic motion. The work presented is joint with Brian Pigott

Series: Combinatorics Seminar

A tight k-uniform \ell-cycle, denoted by TC_\ell^k, is a k-uniform hypergraph whose vertex set is v_0, ..., v_{\ell-1}, and the edges are all the k-tuples {v_i, v_{i+1}, \cdots, v_{i+k-1}}, with subscripts modulo \ell. Motivated by a classic result in graph theory that every n-vertex cycle-free graph has at most n-1 edges, Sos and, independently, Verstraete asked whether for every integer k, a k-uniform n-vertex hypergraph without any tight k-uniform cycles has at most \binom{n-1}{k-1} edges. In this talk I will present a construction giving negative answer to this question, and discuss some related problems. Joint work with Jie Ma.