## Talks

### On the Discrepancy of Random Matrices with Many Columns

**Abstract**

The discrepancy of a red-blue vertex coloring of a hypergraph is the maximum imbalance between the number of red and blue vertices in any edge. The discrepancy of a hypergraph is the least discrepancy of any red-blue coloring. A major open problem is the Beck-Fíala conjecture, which asserts that the discrepancy of every t-regular hypergraph is on the order of the square root of t. I will discuss some recent joint work with Michael Saks on an application of Fourier analysis to the discrepancy of random regular hypergraphs.

### Efficient algorithms for tensor scaling, quantum marginals, and moment polytopes

**Abstract**

In many cases, a group action on a manifold can be associated with a map
called a moment map. Surprisingly, the image of the manifold under this
map is always a convex polytope, known as its moment polytope. Moment
polytope membership can often be expressed as an optimization problem
which is convex only in the Abelian case. Whether moment polytope
membership can be decided in polynomial time is a fundamental open
problem with deep consequences in algebraic complexity theory and elsewhere.
For example, the map sending a d-tensor to its tuple of marginals is a
moment map for the group GL(n)^d acting on tensors by scaling each
factor. I'll discuss a recent joint work in which we obtain an extremely
simple algorithm for tensor scaling, i.e. approximately scaling tensors
to arbitrary prescribed marginals (whenever possible).
Improving the run-time of our tensor scaling algorithm to polylog(\eps)
would result in a polynomial time algorithm for moment polytope
membership in this special case, and would have applications in
polynomial identity testing, computing Brascamp-Lieb constants, and
optimization problems related to tensor rank, to name a few.

### Operator Scaling with Specified Marginals

STOC (2018),

[slides]

**Abstract**

A characterization theorem and algorithm to test, given positive semidefinite matrices P and Q and a completely positive map T, whether it is possible to pre- and post- compose T with matrix similarities so that T is both trace-preserving and maps P to Q. This generalizes the (r,c)-scaling problem, which asks, given a nonnegative matrix and vectors r and c, if the matrix can be pre- and post- multiplied by diagonals so that its row sums become r and the column sums become c.

### Mechanism Design

Rutgers Graduate Student Pizza Seminar (2016)

**Abstract**

How to fairly distribute goods and services among people who may not be honest about how much they value said goods and services.

### Area Paradoxes

Rutgers Graduate Student Pizza Seminar (2015)

**Abstract**

Besikovitch's construction of sets in the unit cube with volume 0.1 and arbitrarily small surface area (first few steps at right).

## Contact

### W. Cole Franks

Hill Center, Room 606

Department of Mathematics

Rutgers, The State University Of New Jersey

110 Frelinghuysen Rd.

Piscataway, NJ 08854-8019

email: wcf17 at math dot rutgers dot edu