contrary to what people believe, modern computers are sometimes surprisingly bad at computations. integer division is a particular example which computers are agonizingly bad at. we will develop a little bit of the theory of continued fractions and see how these seemingly "only for pure mathematicians" - things can be used for dramatic speed-up of divisions and other types of computations with similar nature.

the effectiveness of the cdcl algorithm for sat is complicated to understand, and so far one of the most successful tools for its analysis has been proof complexity. cdcl is easily seen to be limited by the resolution proof system, and furthermore can be thought of as being equivalent to resolution, in the sense that cdcl can reproduce a given resolution proof with only a polynomial overhead.

but the question of the power of cdcl with respect to resolution is far from closed. to begin with, the previous equivalence is subject to assumptions, only some of which are reasonable. in addition, in a setting where algorithms are expected to run in near-linear time, a polynomial overhead is too coarse a measure.

in this talk we will discuss how exactly cdcl and resolution are equivalent, what breaks when we try to make the assumptions more realistic, and how much of an overhead cdcl needs in order to simulate resolution.

a basic question in model theory is whether a theory admits any kind of quantifier reduction. one form of quantifier reduction is called model completeness, and broadly refers to when arbitrary formulas can be "replaced" by existential formulas.
prior to the negative resolution of the connes embedding problem (cep), a result of goldbring, hart, and sinclair showed that a positive solution to cep would imply that there is no ii$_1$ factor with a theory which is model-complete. in this talk, we discuss work on the question of quantifier reduction for tracial von neumann algebras. in particular, we prove that no ii$_1$ factor has a theory that is model complete by using hastings' quantum expanders and a weaker assumption than cep. this is joint work with ilijas farah and david jekel.

this paper studies how to learn parameters in diagonal gaussian mixture models. the problem can be formulated as computing incomplete symmetric tensor decompositions. we use generating polynomials to compute incomplete symmetric tensor decompositions and approximations. then the tensor approximation method is used to learn diagonal gaussian mixture models. we also do the stability analysis. when the first and third order moments are sufficiently accurate, we show that the obtained parameters for the gaussian mixture models are also highly accurate. numerical experiments are also provided.

for integers $s,t \geq 2$, the ramsey number $r(s,t)$ denotes the minimum $n$ such that every $n$-vertex graph contains a clique of order $s$ or an independent set of order $t$. we prove that \[ r(4,t) = \omega\bigl(\frac{t^3}{\log^4 \! t}\bigr) \quad \quad \mbox{ as }t \rightarrow \infty\] which determines $r(4,t)$ up to a factor of order $\log^2 \! t$, and solves a  conjecture of erdős.


this is a joint work with sam mattheus (accepted in the annals of mathematics).

a countable discrete group is exact if it has a free action on cantor space which is measure-hyperfinite, that is, for every borel probability measure on cantor space, there is a conull set on which the orbit equivalence relation is hyperfinite. for an exact group, it is known that the generic action on cantor space is measure-hyperfinite, and it is open as to whether the generic action is hyperfinite; an exact group for which the generic action is not hyperfinite would resolve a long-standing open conjecture about whether measure-hyperfiniteness and hyperfiniteness are equivalent. we show that for any countable discrete group with finite asymptotic dimension, its generic action on cantor space is hyperfinite. this is joint work with sumun iyer. 

random interface growth is all around us: tumors, bacterial colonies, infections, and propagating flame fronts. the kpz equation is a stochastic pde central to a class of random growth phenomena. in this talk, i will explain how to combine tools from probability, partial differential equations, and integrable systems to understand the behavior of the kpz equation when it exhibits unusual growth.

the fargues-fontaine curve associated to an algebraically closed nonarchimedean field of characteristic $p$ is a fundamental geometric object in $p$-adic hodge theory. via the tilting equivalence it is related to the galois theory of finite extensions of q_p; it also occurs in fargues's program to geometrize the local langlands correspondence for such fields.

recently, peter dillery and alex youcis have proposed using a related object, the "affine cone" over the aforementioned curve, to incorporate some recent insights of kaletha into fargues's program. i will summarize what we do and do not yet know, particularly about vector bundles on this and some related spaces (all joint work in progress with dillery and youcis).

come venture into number theory in this spooky post halloween talk, where i plan on talking about some objects that are (at least tangentially) related to number theory. which objects will show up? maybe elliptic curves, maybe p-adic numbers, maybe lie groups. it's a bit of a mystery, so come to the talk to find out! in order to keep the talk from being too scary, i'll try to keep the prerequisite knowledge to a minimum. 

 it was conjectured by milnor in 1968 that the fundamental group of a complete manifold with nonnegative ricci curvature is finitely generated.  in this talk we will discuss a counterexample, which provides an example m^7 with ric>= 0 such that \pi_1(m)=q/z is infinitely generated.

there are several new points behind the result.  the first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake.  the ability to build such a fractal structure will rely on a very twisted gluing mechanism.  thus the other new point is a careful analysis of the mapping class group \pi_0diff(s^3\times s^3) and its relationship to ricci curvature.  in particular, a key point will be to show that the action of \pi_0diff(s^3\times s^3) on the standard metric g_{s^3\times s^3} lives in a path connected component of the space of metrics with ric>0.