Mathematics Department - Colloquium - Spring 2016

Colloquium - Spring 2016



Mathematics Department Colloquia take place on Friday afternoons from 4:00-5:00PM in the Hill Center, Room 705, on Busch Campus. Also, due to recent construction on Route 18, most on-line maps and driving instructions are out of date. Here are updated driving directions. If you need information on public transportation , you may want to check the New Jersey Transit page for information on fares and schedules for the Northeast Corridor Line. Taxis are available at the New Brunswick train station (fare about $7) and can take you to and from the Hill Center (Victory Cabs, (732) 545-6666). The Rutgers Campus Bus System provides free inter-campus transportation, with the A and H buses taking passengers between Busch Campus and College Avenue, with the A providing a faster ride from College Avenue and the H providing a faster ride from the Hill Center : please visit their website for bus schedules and maps, including real-time tracking of campus buses.

Unfortunately, colloquium cancellations do occur from time to time. Please feel free to call our department (732)-445-3921 before embarking on your journey.

Colloquium participants and hosts may wish to also consult the Rutgers University academic calendar, as well as its calendars of religious holidays and of weather emergencies and university closings.

Organizer(s)

Vladimir Retakh, Nathaniel Shar

Archive

Website

http://www.math.rutgers.edu/~nbs48/colloquium




Past Talks


Friday, April 29th

Bill Duke, University of California, Los Angeles

"Geometric invariants for real quadratic fields"

Time: 4:00 PM
Location: Hill 705
Abstract: I will describe some joint work with A. Toth and O. Imamoglu on a new geometric invariant, a certain bordered Riemann surface, associated to an ideal class of a real quadratic field.

This surface has the usual modular closed geodesic as its boundary and its area is determined by the length of an associated backward continued fraction. We study its distribution properties on average over a genus. This complements in a natural way the distribution of the closed geodesics themselves. In the process we give an extension of the Katok-Sarnak formula relating certain periods of Maass differentials to Weyl-type sums for the surfaces.


Friday, April 22nd

Bernd Sturmfels , University of California, Berkeley

"Eigenvectors of Tensors"

Time: 4:00 PM
Location: Hill 705
Abstract: Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalisation. The spectral theory of tensors was pioneered by Lim and Qi around 2005. It has numerous applications, and ties in closely with optimization and dynamical systems.

We present an introduction that emphasizes algebraic and geometric aspects.


Friday, April 15th

Vladimir Markovic , Caltech

"Harmonic maps and heat flows on hyperbolic spaces"

Time: 4:00 PM
Location: Hill 705
Abstract: We prove that any quasi-conformal map of the (n-1)-dimensional sphere, when n > 2, can be extended to a smooth quasi-isometry F of the n-dimensional hyperbolic space such that the heat flow starting with F converges to a quasi-isometric harmonic map. This implies the Schoen-Li-Wang conjecture that every quasi-conformal map of the (n-1)-sphere can be extended to a harmonic quasi-isometry when n > 2. We also prove the corresponding conjecture when n = 2 (which was the original Schoen Conjecture), but this proof does not involve heat flows.


Friday, April 8th

Sergey Fomin, University of Michigan

"Computing without subtracting (and/or dividing)"

Time: 4:00 PM
Location: Hill 705
Abstract: Algebraic complexity of a rational function can be defined as the minimal number of arithmetic operations required to compute it. Can restricting the set of allowed arithmetic operations dramatically increase the complexity of a given function (assuming it is still computable in the restricted model)? In particular, what can happen if we disallow subtraction and/or division?

This is joint work with D. Grigoriev and G. Koshevoy.


Friday, April 1st

Pavel Etingof , Massachusetts Institute of Technology

"Cherednik algebras and torus knots"

Time: 4:00 PM
Location: Hill 705
Abstract: The Cherednik algebra B(c,n), generated by symmetric polynomials and the quantum Calogero-Moser Hamiltonian, appears in many areas of mathematics. It depends on two parameters - the coupling constant c and number of variables n. I will talk about representations of this algebra, and in particular about a mysterious isomorphism between the representations of B(m/n,n) and B(n/m,m) of minimal functional dimension. We explain the symmetry between m and n by showing that the characters of these representations can be expressed in terms of the colored HOMFLY polynomial of the torus knot T(m/d,n/d), where d=GCD(m,n).

The talk is based on my joint work with E. Gorsky and I. Losev.


Friday, March 25th

Avi Wigderson, IAS, Princeton

"The singularity of symbolic matrices"

Time: 4:00 PM
Location: Hill 705
Abstract: The main object of study of this talk are matrices whose entries are linear forms in a set of formal variables (over some field). The main problem is determining if a given such matrix is invertible or singular (over the appropriate field of rational functions).

As it happens, this problem has a dual life; when the underlying variables commute, and when they do not. Most of the talk will be devoted to explaining (some of) the many origins, motivations and interrelations of these two problems, in computational complexity, non-commutative algebra, (commutative) invariant theory, quantum information theory, optimization and more.

I will describe the state-of-art on the complexity of these problems. For the non-commutative version, where even decidability took decades to establish, we have recently found (with Garg, Gurvits and Olivera) a deterministic polynomial time algorithm (over the rationals). Strangely perhaps, for the commutative version, where decidability is nearly trivial, the best known deterministic algorithm requires exponential time. A probabilistic polynomial time algorithm is known, and making it deterministic is major open problem.


Friday, March 4th

Doron Levy, Maryland University

" Modeling Group Dynamics in Phototaxis"

Time: 4:00 PM
Location: Hill 705
Abstract: Microbes live in environments that are often limiting for growth. They have evolved sophisticated mechanisms to sense changes in environmental parameters such as light and nutrients, after which they swim or crawl into optimal conditions. This phenomenon is known as "chemotaxis" or "phototaxis". Using time-lapse video microscopy we have monitored the movement of phototactic bacteria, i.e., bacteria that move towards light. These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important.

In this talk we will survey our recent results on mathematical models for phototaxis. We will start with a stochastic model, an interacting particle system, and a system of PDEs. Our main theorem establishes the system of PDEs as the limit dynamics of the particle system. We will then present another approach in which we develop particle, kinetic, and fluid models for phototaxis. We will conclude with describing our recent work on modeling selective local interactions with memory.


Friday, February 26th

Laura DeMarco, Northwestern University

" Complex dynamics and elliptic curves"

Time: 4:00 PM
Location: Hill 705
Abstract: In this talk, I will explain some connections between recent research in dynamical systems and the classical theory of elliptic curves and rational points. I will begin with the theorem of Mordell and Weil from the 1920s, presented from a dynamical point of view. I will continue by describing a dynamical/geometric proof of a result of Masser and Zannier about torsion points on elliptic curves and "unlikely intersections". Finally, I aim to explain the role of dynamical stability and bifurcations in finiteness statements.


Friday, February 19th

Ian Agol , University of California, Berkeley

"Veering triangulations and pseudo-Anosov flows"

Time: 4:00 PM
Location: Hill 705
Abstract: We will discuss veering triangulations associated to pseudo-Anosov mapping tori, and how they arise dynamically. We will survey some of the results obtained regarding these triangulations. Then we will discuss a new construction of these triangulations associated to certain pseudo-Anosov flows, which is joint work with Francois Gueritaud.


Friday, February 5th

Jozsef Balogh , University of Illinois at Urbana-Champaign

"On some applications of counting independent sets in hypergraphs"

Time: 4:00 PM
Location: Hill 705
Abstract: Recently, Balogh-Morris-Samotij and Saxton-Thomason developed a method of (approximately) counting independent sets in hypergraphs. This technique, now known as the "Container Method", has already had many applications in extremal and probabilistic combinatorics, additive number theory and discrete geometry. For example it provides approaches to proving classical extremal results (e.g. the theorems of Szemeredi and Turan) in a random setting, and to asymptotic counting of discrete structures such as maximal triangle-free graphs and sum-free sets, and sets without k-term arithmetic progressions. I will give an overview of the area and sketch some sample applications of the technique.


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