Schedule, Notes/Handouts & Videos

Mathematical modeling has been used to study diverse biological topics ranging from protein folding to cell interactions to interacting populations of humans but has only recently been used to study the physiology of the eye. In recent years, computer (in silico) experiments have given researchers invaluable insights and in some cases have re-directed experimental research and theory. In this talk I will give a brief overview of the relevant physiology of the eye as it pertains to Retinitis pigmentosa (RP), a group of inherited degenerative eye diseases that characterized by the premature death of both rod and cone photoreceptors often resulting in total blindness. With mathematics and in silico experiments, we explore the experimentally observed results highlighting the delicate balance between the availability of nutrients and the rates of shedding and renewal of photoreceptors needed for a normal functioning retina. This work provides a framework for future physiological investigations potentially leading to long-term targeted multi-faceted interventions and therapies dependent on the particular stage and subtype of RP under consideration. The mathematics presented will be accessible to an undergraduate math audience and the biology will be at the level of a novice (and with a little help from Dr. Seuss).

In this talk, I will share three pieces of advice regarding how to be "successful" in mathematics (think about the meaning of the word "success"). I will then discuss the effect MSRI-UP has had in my career. We will then discuss some open problems regarding roots of "peak polynomials".

Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, thus obstructing the use of greatest common divisors. Suppose f is any nonzero univariate polynomial of degree d in Z/p^nZ[x]. For the case n=2, we prove a new efficient algorithm to count the roots in Z/p^2Z within time polynomial in d + log p. The key idea is to partition the roots via their image under reduction modulo p, and then carefully use Hensel's Lemma. This builds upon recent work of Cheng, Gao, Rojas, and Wan.

Canetti, Friedlander, et al. (2002) studied the randomness of powers over finite fields and along the way derived an analogue of Descartes’™ rule over the finite field F_q with q elements: They showed that the number of roots of any univariate t-nomial, with exponents {0,a_2,...,a_t} and the differences a_i-a_j all relatively prime to q-1, is O(q^{(t-2)/(t-1)}). The correct optimal bounds remain a mystery for prime fields, even in the case of polynomials with three terms. Following the work of Kelley (2016), we seek to prove the conjecture that the number of roots in F_p of a trinomial with a linear middle term is always O(log p). We expand current evidence by using a supercomputer to determine the  number of roots of these trinomials for 139,571 < p 191,491. We also prove that the search can be restricted to trinomials with a middle linear term when p-1 has less than three distinct prime factors.

One way to prove the algorithmic hardness of a function is to consider its circuit complexity: the minimal size of a circuit computing the function. Recent work in circuit complexity has revealed that it is enough to restrict to depth-4 circuits, inspiring the study of certain structured polynomials called sum-of-products-of-sparse (SPS) polynomials. In particular, showing sufficiently good upper bounds for the number of real roots of SPS polynomials would imply the hardness of the permanent. If f and g are univariate polynomials with real coefficients and t terms, then does fg-1 have a number of real roots that grows at most linearly in t? We consider the first non-trivial approach towards such a bound via A-discriminants, leading to interesting questions involving sparse polynomial systems.

Given a univariate trinomial f in R[x], we analyze the Archimedean Newton polytope of f and the corresponding lower binomials. The roots of these lower binomials conjecturally provide high quality approximations of the roots of f. We implement Smale's alpha-criterion to analyze whether our approximations converge quickly under Newton iteration. We know that under certain conditions every root of a lower binomial is an approximate root of a trinomial. We expect to determine when at least one root of a lower binomial is an approximate root. Moreover, for roots that are not approximate, we examine when Newton's method yields approximate roots.

A fundamental problem in many applications is determining the real solutions of a polynomial. In recent years, the A-discriminant variety of Gelfand, Kapranov and Zelevinsky has proved to be a valuable tool. Its complement over the real numbers defines regions of coefficient values called chambers. We implement an algorithm in MATLAB that draws A-discriminant curves for families of bivariate pentanomials where the topology of the underlying zero set is constant. Our program graphs the discriminant curve with all of its chambers and automatically computes the topology of all smooth zero set for the given families of bivariate pentanomials. We hope automated topology computation for high degree polynomials will prove useful for the algebraic geometry community.

Let f be a polynomial of degree d with exactly n+4 monomial terms in R[x_1,…,x_n]. We show that one can efficiently compute an explicit polyhedral complex with the same isotopy type as the positive zero set of f. In particular, the complexity of our construction is polynomial in u + log d with high probability. Along the way, we derive and implement an algorithm that, given an n-variate (n+4)-nomial f, outputs a plot of the reduced A-discriminant contour in R^3.