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Courcelle’s Theorem is an influential metatheorem published in 1990. It tells us that a property of graph can be tested in polynomial time, as long as the property can expressed in the monadic secondorder logic of graphs, and as long as the input is restricted to a class of graphs with bounded treewidth. There are several properties that are NPcomplete in general, but which can be expressed in monadic logic (3colourability, Hamiltonicity…), so Courcelle’s Theorem implies that these difficult properties can be tested in polynomial time when the structural complexity of the input is limited.
Matroids can be considered as a special class of hypergraphs. Any finite set of vectors over a field leads to a matroid, and such a matroid is said to be representable over that field. Hlineny produced a matroid analogue of Courcelle’s Theorem for input classes with bounded branchwidth that are representable over a finite field.
We have now identified the structural properties of hypergraph classes that allow a proof of Hliněný’s Theorem to go through. This means that we are able to extend his theorem to several other natural classes of matroids.
This talk will contain an introduction to matroids, monadic logic, and treeautomata.
This is joint work with Daryl Funk, Mike Newman, and Geoff Whittle.
What is the largest subset of $\mathbb Z_{2^n}$ that doesn’t contain a projective dcube? In the Boolean lattice, Sperner’s, Erdos’s, Kleitman’s and Samotij’s theorems state that families that do not contain many chains must have a very specific layered structure. We show that if instead of
Z
n
2
we work in
Z
2
n
, analogous statements hold if one replaces the word kchain by projective cube of dimension
2
k
−
1
. The largest dcubefree subset of
Z
2
n
, if d is not a power of two, exhibits a much more interesting behaviour.
This is joint work with Jason Long.
An important family of incidence problems are discrete analogs of deep questions in geometric measure theory. Perhaps the most famous example of this is the finite field Kakeya conjecture, proved by Dvir in 2008. Dvir’s proof introduced the polynomial method to incidence geometry, which led to the solution to many longstanding problems in the area.
I will talk about a generalization of the Kakeya conjecture posed by Ellenberg, Oberlin, and Tao. A
(
k
,
m
)
Furstenberg set S in
F
n
q
has the property that, parallel to every affine
k
plane V, there is a kplane W such that

W
∩
S

>
m
. Using sophisticated ideas from algebraic geometry, Ellenberg and Erman showed that if S is a
(
k
,
m
)
Furstenberg set, then

S

>
c
m
n
/
k
, for a constant c depending on n and k. In recent joint work with Manik Dhar and Zeev Dvir, we give simpler proofs of stronger bounds. For example, if
m
>
2
n
+
7
q
, then

S

=
(
1
−
o
(
1
)
)
m
q
n
−
k
, which is tight up to the
o
(
1
)
term.
In many applications of machine learning, interpretable or explainable models for binary classification, such as decision trees or decision lists, are preferred over potentially more accurate but less interpretable models such as random forests or support vector machines. In this talk, we consider boolean decision rule sets (equivalent to boolean functions in disjunctive normal form) as interpretable models for binary classification. We define the complexity of a rule set to be the number of rules (clauses) plus the number of conditions (literals) across all clauses, and assume that simpler or less complex models are more interpretable. We discuss an integer programming formulation for such models that trades off classification accuracy against rule simplicity, and obtain highquality classifiers of this type using column generation techniques. Compared to some recent alternatives, our algorithm dominates the accuracysimplicity tradeoff in 8 out of 16 datasets, and also produced the winning entry in the 2018 FICO explainable machine learning challenge. When compared to rule learning methods designed for accuracy, our algorithm sometimes finds significantly simpler solutions that are no less accurate.
자연과학동(E61) Room 1401
확률론
남경식 (UC Berkeley)
Elliptic and hypoelliptic diffusions with singular drift
자연과학동(E61) Room 1401
확률론
It has been known that stochastic differential equations with nondegenerate diffusion admit a unique solution for subcritical drifts. In this talk, we extend this in two different directions: the critical drift case and the degenerate diffusion case. I will briefly introduce a hypoelliptic theory, and then explain how to obtain probabilistic results on stochastic differential equations.
A classical problem in mechanics is the following: given a set of forces applied at some points in space, is there any discrete network with all the wires under tension that supports such a system of forces? In this talk I will present some recent results on the topic, obtained in collaboration with Graeme Milton, Pierre Seppecher and Guy Bouchitte. In usual wire or cable networks, such as in a bridge or bicycle wheel, one distributes the forces by adjusting the tension in the wires. Here our discrete networks provide an alternative way of distributing the forces through the geometry of the network. In particular the network can be chosen so it is uniloadable, i.e. supports only one set of forces at the terminal nodes. Such uniloadable networks are relevant as the tension in one element determines the tension in the joint wires and, by extension, uniquely determines the stress in the entire network.