Building on previous work of Rozansky and Willis, we generalise Rasmussen's s-invariant to links in connected sums of $S^1 \times S^2$. Such an invariant can be computed by approximating the Khovanov-Lee complex of a link in $\#^r S^1 \times S^2$ with that of appropriate links in $S^3$. We use the approximation result to compute the s-invariant of a family of links in $S^3$ which seems otherwise inaccessible, and use this computation to deduce an adjunction inequality for null-homologous surfaces in a (punctured) connected sum of $\bar{CP^2}$. This inequality has several consequences: first, the s-invariant of a knot in the three-sphere does not increase under the operation of adding a null-homologous full twist. Second, the s-invariant cannot be used to distinguish $S^4$ from homotopy 4-spheres obtained by Gluck twist on $S^4$. We also prove a connected sum formula for the s-invariant, improving a previous result of Beliakova and Wehrli. We define two s-invariants for links in $\#^r S^1 \times S^2$. One of them gives a lower bound to the slice genus in $\natural^r S^1 \times B^3$ and the other one to the slice genus in $\natural^r D^2 \times S^2$ . Lastly, we give a combinatorial proof of the slice Bennequin inequality in $\#^r S^1 \times S^2$.

In this talk, we describe self-generating pointwise lower bounds for solutions of the non-cutoff Boltzmann equation, which models the evolution of the particle density of a diffuse gas. These lower bounds imply that vacuum regions in the initial data are filled instantaneously, and also lead to key coercivity estimates for the collision operator. As an application, we can remove the assumptions of mass bounded below and entropy bounded above, from the known criteria for smoothness and continuation of solutions. The proof strategy also applies to the Landau equation, and we will compare this (deterministic) proof with our prior (probabilistic) proof of lower bounds for the Landau equation. This talk is based on joint work with Chris Henderson and Andrei Tarfulea.

Complex networks are said to exhibit four key properties: large scale, evolving over time, small world properties, and power law degree distribution. The Preferential Attachment Model (Barab\'asi--Albert, 1999) and the ACL Preferential Attachment Model (Aiello, Chung, Lu, 2001) for random networks, evolve over time and rely on the structure of the graph at the previous time step. Further models of complex networks include: the Iterated Local Transitivity Model (Bonato, Hadi, Horn, Pralat, Wang, 2011) and the Iterated Local Anti-Transitivity Model (Bonato, Infeld, Pokhrel, Pralat, 2017). In this talk, we will define and discuss the Iterated Local Model. This is a generalization of the ILT and ILAT models, where at each time step edges are added deterministically according to the structure of the graph at the previous time step.

In this talk, we will first introduce several versions of existence and uniqueness theorems for stochastic differential equations(SDEs). Then we will focus on a special type of SDEs, Ito diffusions. We will give a detailed proof of the strong Markov property of Ito diffusions. This talk can be viewed as an extension of math 286 and material is drawn mostly from Chapter 10 of Chung and Williams' $Introduction$ $to$ $Stochastic$ $Integration$.

Though promised to solve intractable problems, quantum computers are much noisier than classical computers. To make quantum computers practical, efficient error correction schemes have to be developed. In the first half of the talk, I will first introduce the theory of Quantum Error Correction (QEC) and current proposals of using QEC to achieve fault-tolerant quantum computing, then I will talk about a new scheme that can greatly lower the cost of QEC. In the second half, I will talk about a promising near-term qubit encoding called the GKP qubit that has error-correcting abilities and a protocol to prepare such qubits.

The Gaussian multiplicative chaos is a relatively new universal object in probability that has many interesting geometric properties.The characteristic polynomial of many classes of random matrices is, in many cases conjecturally, one class of finite approximation to these random measures. Great progress has been made on showing the random matrices from specific ``circular ensembles" converge to the GMC.
Likewise, some progress has been made for unitarily-invariant random matrices. We show some new partial progress in showing the ``Gaussian-beta ensemble'' has a GMC limit. This we do by using the representation of its characteristic polynomial as an entry in a product of independent random two-by-two matrices. For a point $z$ in the complex plane, at which the transfer matrix is to be evaluated, this product of transfer matrices splits into three independent factors, each of which can be understood as a different dynamical system in the complex plane.
This is joint work with Gaultier Lambert.

Central leaves in the special fiber of Shimura varieties are the loci where the isomorphism class of the universal $p$-divisible group remains constant. The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a PEL type Shimura variety is dense in the central leaf containing it. This conjecture is proved for Hilbert modular varieties by C.-F. Yu, and for Siegel modular varieties by Chai and Oort. In this talk I give an overview of the conjecture and present my work that generalizes Chai and Oort's strategy to irreducible components of certain Newton strata on Shimura varieties of PEL type.

A smooth manifold is a topological structure that admits some standard
calculus constructions. A vector bundle over a manifold is a smooth choice
of a vector space at every point of the manifold. One motivation for such
an object is that the differential of a map between two manifolds is a map
between their tangent bundles. At first glance, vector bundles seem as
though they should be Cartesian products of the manifold with a vector
space. In general, this is false, and certain ``characteristic classes''
are the explicit obstructions. In this talk, we will construct these
classes and discuss some theorems explaining why they are of interest to
many mathematicians.