As a noncommutative fractal geometer, I look for new expressions of the geometry of a fractal through the lens of noncommutative geometry.  At the quantum scale, the wave function of a particle, but not its path in space, can be studied.  Riemannian methods often rely on smooth paths to encode the geometry of a space.  Noncommutative geometry generalizes analysis on manifolds by replacing this requirement with operator algebraic data.  These same "point-free" techniques can also be used to study the geometry of spaces like fractals.  Recently, Michel Lapidus, Frédéric Latrémoliére, and I identified conditions under which differential structures defined on fractal curves can be realized as a metric limit of differential structures on their approximating finite graphs.  Currently, I am using some of the same tools from that project to understand noncommutative discrete structures.  Progress in noncommutative geometry has produced a rich dictionary of quantum analogues of classical spaces.  The addition of noncommutative discrete structure to this dictionary would enlarge its potential to yield insights about both noncommutative sets and classically pathological sets like fractals.  Time permitting, other works in progress, such as on classification of $C^*$-algebras on fractals, may be discussed. 

Interior-point methods are some of the most effective and widely used methods to finding local minimizers of large-scale non-convex optimization problems. In this talk, we introduce three different mechanisms for ensuring global convergence to second-order local minimizers from arbitrary feasible starting points by solving a sequence of trust-region subproblems defined by quadratic models of a shifted primal-dual penalty-barrier merit function. Each of these methods begins by solving the trust-region subproblem to form a new trial point, and proceeds to refine the trial iterate until a sufficient-decrease condition is met. We suggest two different definitions of the trust region, and provide numerical results comparing each of the different approaches.

In this talk, I will discuss characterizations of stationary solutions of the 2D Euler equations in two different directions under different assumptions: rigidity (is every stationary solution radial?) and flexibility (do there exist non-radial stationary solutions?). The proofs will have a calculus of variations' flavor, a new observation that finite energy, stationary solutions with simply-connected vorticity have compactly supported velocity, and an application of the Nash-Moser iteration procedure. Based on joint work with Jaemin Park, Jia Shi and Yao Yao.

Hilbert’s Nullstellensatz is a fundamental result in commutative algebra which is the starting point for classical algebraic geometry.

In this talk, I will discuss joint work with Robert Burklund and Tomer Schlank on a chromatic version of Hilbert’s Nullstellensatz in which Lubin-Tate theories play the role of algebraically closed fields.I will then sample some applications of our results to chromatic support, redshift, and orientation theory for $E_\infty$ rings.

Golden ratio primal-dual algorithm (GRPDA) is a new variant of the classical Arrow-Hurwicz method for solving structured convex-concave saddle point problem. In this talk, we present GRPDAs with adaptive linesearch, which potentially allows much larger stepsizes, and hence, could significantly accelerate the convergence speed. We show global iterate convergence as well as O($\frac{1}{N}$) ergodic convergence rate results, measured by the function value gap and constraint violations of an equivalent optimization problem. When one of the component functions is strongly convex, faster O($\frac{1}{N^2}$) ergodic convergence rate can be established. In addition, linear convergence can be established when subdifferential operators of the component functions are strongly metric subregular. Our preliminary numerical results show our algorithms perform much better than other state-of-art
 comparison algorithms. 

A powerful tool to study the geometry of Radon measures is the decomposability bundle, which I introduced with Alberti in [On the differentiability of Lipschitz functions with respect to measures in the Euclidean space, GAFA, 2016]. This is a map which, roughly speaking, captures at almost every point the tangential directions to the Lipschitz curves along which the measure can be disintegrated. In this talk I will discuss some recent applications of this flexible tool, including a characterization of rectifiable measures as those measures for which Lipschitz functions admit a Lusin type approximation with functions of class ${C^1}$, the converse of Pansu's theorem on the differentiability of Lipschitz functions between Carnot groups, and a characterization of Federer-Fleming flat chains with finite mass.

In this talk, we will consider the angular component of multipliers of repelling cycles of a hyperbolic rational map in one complex variable. Oh-Winter have shown that these angles of multipliers uniformly distribute in the circle (-$\pi$, $\pi$]. Motivated by the sector problem in number theory, we show that for a fixed $K \gg 1$, almost all intervals of length $\frac{2 \pi}{K}$ in (-$\pi$, $\pi$] contains a multiplier angle with the property that the norm of the multiplier is bounded above by a polynomial in K. This is joint work with Hongming Nie.

Given a smooth projective family $f : X \to S$ defined over the ring of integers of a number field, the Shafarevich problem is to describe those fibres of f which have everywhere good reduction. This can be interpreted as asking for the dimension of the Zariski closure of the set of integral points of $S$, and is ultimately a difficult diophantine question about which little is known as soon as the dimension of $S$ is at least 2. Recently, Lawrence and Venkatesh gave a general strategy for addressing such problems which requires as input lower bounds on the monodromy of f over essentially arbitrary closed subvarieties of $S$. In this talk we review their ideas, and describe recent work which gives a fully effective method for computing these lower bounds. This gives a fully effective strategy for solving Shafarevich-type problems for essentially arbitrary families $f$.

Using a combination of methods from applied mathematics and nonlinear dynamics, we present a constructive way to give a discrete time dynamical rule that accurately forecasts the voltage across a neuron cell membrane. This is the only quantity required to build a biological network of realistic neurons. The construction uses simulated 'data' or observed biophysical data alone to develop the dynamical map. We call this data driven forecasting (DDF). The method is described in detail at first using 'data' from simple neuron models and then using observed neurobiological data from laboratory experiments. It provides accurate forecasting of observed quantities in each setting. 

In an example where a detailed Hodgkin-Huxley (HH) model was developed using data assimilation for observed laboratory observations the DDF neuron runs an order of magnitude faster than the HH version in forecasting the important neuron voltage time course. As the computation required for a network of N nodes will be faster by about a factor of 10N using DDF neurons, this will permit building and analyzing the very large networks desired to address realistic biological questions using elements determined via the biophysics of the component neurons. 

If time permits, we will describe how one may use the DDF idea to substantially reduce the geophysical computations required for regional numerical weather forecasting.

The Tower variant of the Number Field Sieve (TNFS) is known to be asymptotically the most efficient algorithm to solve the discrete logarithm problem in finite fields of medium characteristics, when the extension degree is composite. A major obstacle to an efficient implementation of TNFS is the collection of algebraic relations, as it happens in dimensions greater than 2. This requires the construction of new sieving algorithms which remain efficient as the dimension grows. In the work I will present, we overcome this difficulty by considering a lattice enumeration algorithm which we adapt to this specific context. We also consider a new sieving area, a high-dimensional sphere, whereas previous sieving algorithms for the classical NFS considered an orthotope. Our new sieving technique leads to a much smaller running time, despite the larger dimension of the search space, and even when considering a larger target, as demonstrated by a record computation we performed in a 521-bit finite field $GF(p^6)$. The target finite field is of the same form as finite fields used in recent zero-knowledge proofs in some blockchains. This is the first reported implementation of TNFS.

As anyone at UCSD can tell you, I really like making dumb jokes. Unfortunately, I can end up spending so much time on my jokes that I don't end up doing any mathematics. My solution to this problem has been to write math papers which are based on jokes. Somehow I've managed to do this 3 times. In this talk I'll briefly discuss these joke papers. No prior knowledge of jokes or any sense of humor will be assumed. Current UCSD students, prospective students, and anyone else who isn't on my thesis committee is welcome to attend.