An important area of study in the classification of II_1 factors is to investigate the dependence of a group von Neumann algebra L(G) relative to the group G. Popa’s deformation/rigidity theory has provided novel insights into this question over the past 20 years. 

In this talk, we demonstrate how one can import group cohomological information into the von Neumann algebra framework to unravel the structure of a large family of von Neumann algebras.

The truncated Brown-Peterson spectra admit actions by the cyclic group of order 2 via complex conjugation. Their fixed point spectra are higher height analogues of real K-theory. We describe how to use Tate square methods along with the slice spectral sequence to compute their mod 2 homology. This is joint work in progress with Mike Hill and Doug Ravenel.

The balancing of items — or discrepancy — arises naturally in Computer Science and Discrete Mathematics. Here we consider n vectors in n-space, all coordinate +1 or −1. We create a signed sum of the vectors, with the goal that this signed sum be as small as possible, Here we use the max (or ${L^∞}$) norm, though many variants are possible.

We create a game with Paul (Erdos) selecting the vectors and Carole (find the anagram!) choosing to add or subtract. This becomes four (two TIMES two) different problems. The vectors (Paul) can be chosen randomly or adversarially, equivalently average case and worst case analysis for Carole. The choice of signed sum (Carole) can be done on-line or off-line.

All four variants are interesting and are at least partially solved. We emphasize the random (Paul) on-line (Carole) case, joint work with Nikhil Bansal.

Since there is NOTHING special about March 2nd, in particular, it is no famous person's birthday, we will get back on track with serious, graduate-level mathematics. I will present the definition of a von Neumann algebra which is the main object I study. We will go through some common constructions and see their relationships to concepts in group theory. The talk will DEFINITELY NOT incorporate anything frivolous such as rhyme, meter, or visual media.
 

Stochastic majorization-minimization (SMM) is an online extension of the classical principle of majorization-minimization, which consists of sampling i.i.d. data points from a fixed data distribution and minimizing a recursively defined majorizing surrogate of an objective function. In this paper, we introduce stochastic block majorization-minimization, where the surrogates can now be only block multi-convex and a single block is optimized at a time within a diminishing radius. Relaxing the standard strong convexity requirements for surrogates in SMM, our framework gives wider applicability including online CANDECOMP/PARAFAC (CP) dictionary learning and yields greater computational efficiency especially when the problem dimension is large. We provide an extensive convergence analysis on the proposed algorithm, which we derive under possibly dependent data streams, relaxing the standard i.i.d. assumption on data samples. We show that the proposed algorithm converges almost surely to the set of stationary points of a nonconvex objective under constraints at a rate $O(  \frac{  (\log n)^{1+\epsilon}  }{  n^{1/2}  }  )$ for the empirical loss function and $O(  \frac{  (\log n)^{1+\epsilon}  }{  n^{1/4}  }  )$ for the expected loss function, where n denotes the number of data samples processed. Under some additional assumption, the latter convergence rate can be improved to $O(  \frac{  (\log n)^{1+\epsilon}  }{  n^{1/2}  }  )$. Our results provide first convergence rate bounds for various online matrix and tensor decomposition algorithms under a general Markovian data setting.

Introduced by Lehmer in 1933, the classical Mahler measure of a complex rational function $P$ in one or more variables is given by integrating $\log|P(x_1, \ldots, x_n)|$ over the unit torus. Lehmer asked whether the Mahler measures of integer polynomials, when nonzero, must be bounded away from zero, a question that remains open to this day. In this talk we generalize Mahler measure by associating it with a discrete dynamical system $f: \mathbb{C} \to \mathbb{C}$, replacing the unit torus by the $n$-fold Cartesian product of the Julia set of $f$ and integrating with respect to the equilibrium measure on the Julia set. We then characterize those two-variable integer polynomials with dynamical Mahler measure zero, conditional on a dynamical version of Lehmer's conjecture.
 

Fully connected networks are roughly described by two structural parameters: a depth L and a width n. It is well known that, with some important caveats on the scale at initialization, in the regime of fixed L and the limit of infinite n, neural networks at the start of training are a free (i.e. Gaussian) field and that network optimization is kernel regression for the so-called neural tangent kernel (NTK). This is a striking and insightful simplification of infinitely overparameterized networks. However, in this particular infinite width limit neural networks cannot learn data-dependent features, which is perhaps their most important empirical feature. To understand feature learning one must therefore study networks at finite width. In this talk I will do just that. I will report on recent work joint with Dan Roberts and Sho Yaida (done at a physics level of rigor) and some more mathematical ongoing work which allows one to compute, perturbatively in 1/n and recursively in L, all correlation functions of the neural network function (and its derivatives) at initialization. An important upshot is the emergence of L/n, instead of simply L, as the effective network depth. This cut-off parameter provably measures the extent of feature learning and the distance at initialization to the large n free theory.

I will outline a proof that the Ising model has a continuous phase transition on any nonamenable Cayley graph. This will involve some neat probabilistic applications of ergodic-theoretic machinery such as factors of IID and the spectral theory of group actions. I will aim to make the talk accessible to a broad community.

Inherent fluctuations may play an important role in biochemical and biophysical systems when the system involves some species with low copy numbers. This talk will present the recent work on the stochastic modeling of enzyme-catalyzed reactions in biology.

In the first part of the talk, I will introduce a multiscale approximation method that helps reduce network complexity using various scales in species numbers and reaction rate constants. I will apply the multiscale approximation method to simple enzyme kinetics and derive quasi-steady-state approximations. In the second part of the talk, I will show another example for glucose metabolism where we see different-sized enzyme complexes. We hypothesized that the size of multienzyme complexes is related to their functional roles. We will see two models: one using a system of ordinary differential equations and the other using the Langevin dynamics.

Often, a goal of mathematics is to classify objects of a particular type.  In algebraic geometry, the objects are usually of some geometric interest: manifolds, varieties, vector bundles, etc; and after fixing several discrete invariants (like the dimension of the object), we try to classify all the objects with those invariants. This leads to a notion of moduli space, i.e. a space parametrizing all of these objects. We will do several examples and mention both the usefulness and difficulty of these problems!  No background in algebraic geometry is required.

We apply results of Garsia-Haiman-Tesler on Macdonald polynomials to the problem of computation of integrals of tautological classes over the Hilbert schemes of surfaces, studied by Marian-Oprea-Pandharipande. Using localization, these results allow us to find new functional equations for the generating series of integrals. The MOP paper considers two kinds of integrals: the so-called Chern integrals resp. Verlinde integrals. The answer to the problem is encoded in series A1, A2, A3, A4, A5 resp. B1, B2, B3, B4. All the series except A4, A5, B3, B4 were computed in MOP and a conjecture motivated by mathematical physics was formulated relating A4 to B3 and A5 to B4. It was also conjectured that A4, A5, B3, B4 are algebraic functions. Solving our functional equations we prove the former conjecture and obtain explicit formulas for A4 and B3, thus proving a part of the latter conjecture. We also give a conjectural formula for A5 and B4. This is a joint work with Lothar Göttsche

We apply results of Garsia-Haiman-Tesler on Macdonald polynomials to the problem of computation of integrals of tautological classes over the Hilbert schemes of surfaces, studied by Marian-Oprea-Pandharipande. Using localization, these results allow us to find new functional equations for the generating series of integrals. MOP paper considers two kind of integrals: the so-called Chern integrals resp. Verlinde integrals. The answer to the problem is encoded in series A1, A2, A3, A4, A5 resp. B1, B2, B3, B4. All the series except A4, A5, B3, B4 were computed in MOP and a conjecture motivated by mathematical physics was formulated relating A4 to B3 and A5 to B4. It was also conjectured that A4, A5, B3, B4 are algebraic functions. Solving our functional equations we prove the former conjecture and obtain explicit formulas for A4 and B3, thus proving a part of the latter conjecture. We also give a conjectural formula for A5 and B4. This is a joint work with Lothar Göttsche.