# 세미나 및 콜로퀴엄

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In this talk we shall try to give a survey of Voevodsky's smash nilpotence conjecture. Considerable time will be spent on explaining the statement of the conjecture, and a result of Kimura which is very fundamental. After this we shall see some examples where the conjecture holds, the work of Kahn-Sebastian, Sebastian, Laterveer, Vial and others.

We discuss the number of proper colorings of hypercubes

given q colors. When q=2, it is easy to see that there are only 2

possible colorings. However, it is already highly nontrivial to figure

out the number of colorings when q=3. Since Galvin (2002) proved the

asymptotics of the number of 3-colorings, the rest cases remained open

so far. In this talk, I will introduce a recent work on the number of

4-colorings, mainly focusing on how entropy can be used in counting.

This is joint work with Jeff Kahn.

The conjecture of Prasanna-Venkatesh predicts that the rational cohomology ring of an arithmetic manifold has special endomorphisms of a motivic origin. Their motivic nature allows one to test the conjecture indirectly through various regulator maps, but the source of these endomorphisms remains mysterious. We propose that the discrete Hodge star operator supplies the predicted endomorphisms in the case when the arithmetic manifold in question is uniformized by the upper half-space. In order to justify the proposal, we analyze how two different rational structures in cohomology interact with the discrete and geometric Hodge star operators, and show that the difference between them is measured by a special value of an $L$-function. Numerical examples will be given to illustrate the result.

One can simulate harmonic analysis on a Riemannian manifold by replacing the de Rham complex with the cochain complex of a triangulated manifold, following the ideas that go back at least to A. Whitney and D. Sullivan, and more recently to S. Wilson. As a result, one obtains the discrete Hodge star operator acting on simplicial cochains, which is an analogue of the usual Hodge star operator acting on differential forms. We will show that the discrete Hodge star operator is a topological invariant a 3-manifold; its action on the cohomology is determined by the underlying manifold together with its orientation, independently of the choice of a triangulation. Like the usual geometric Hodge star operator, it commutes with correspondences, and it is compatible with the Poincare duality. Furthermore, it induces a canonical positive definite bilinear pairing on the singular cohomology.

Toric orbifolds are topological generalization of projective toric varieties. We introduce some sufficient conditions on the combinatorial data associated to a toric orbifold to ensure an invariant CW-structure of the toric orbifold. In this talk I will discuss 3 different equivariant cohomology theories of toric orbifolds. This is a joint work with V. Uma.

In this presentation, we shall analyze random processes exhibiting metastable /tunneling behaviors among several metastable valleys. Such behaviors can be described by a Markov chain after a suitable rescaling. We will focus on three models: random walks in a potential field, condensing zero-range processes, and metastable diffusion processes.

Fock and Goncharov (2006) introduced the notion of positive framed PGL(n,R) representations. In this talk we exhibit framed PGL(3,R) representations of the 3-holed sphere group that are "negative" in a certain sense. If we require the boundary holonomies be all quasi-unipotent, then the boundary-embedded and transversal representations in the corresponding relative character variety form an open subset. These examples may be called "relatively Anosov" and properly include the Pappus representations studied by R. Schwartz (1993). If we further restrict to a certain real 1-dimensional subvariety consisting of representations with 2-fold symmetry, then we obtain a PGL(3,R) analogue of the Goldman-Parker conjecture (solved by R. Schwartz in 2001) on the ideal triangle reflection groups in PU(2,1). Joint work in progress with Sungwoon Kim.

The main interest of this talk is the behavior of Selmer groups of families of twists of elliptic curves. Mazur and Rubin show that there are infinitely many quadratic twists of arbitrary 2-Selmer ranks, under the some conditions on the given elliptic curve.

In this talk, I will introduce a cubic analogue of this result. A naive cubic analogue of the result of Mazur--Rubin does not hold, since all elliptic curves in our settings have ``constant 3-Selmer parity''. I will explain why this problem happens, and how we can manage it.

Formal orbifolds are normal varieties $X$ over perfect fields with a branch data $P$ which encodes compatible system of finite Galois extensions of function fields of formal neighbourhoods of points of $X$. I will introduce these objects and demonstrate how these objects can be used to study (wild) ramification theory in an organised way. In particular I will define etale site, fundamental group, etc. of formal orbifolds. I will discuss a reasonable formulation of Lefschetz theorem for fundamental group of quasi-projective varieties over fields of positive characteristic in the language of formal orbifolds. Time permitting some partial results in this direction will also be stated.

The non-symplectic index of an algebraic K3 surface is the order of the image of the representation of the automorphism group of the K3 surface on the global two forms. If the base field is the complex field, the non-symplectic index is finite and its Euler phi value is at most 20. In this talk, we will see if the base field is of odd characteristic and the K3 surface is of finite height, we have a similar result through a lifting argument. Also we calculate the non-symplectic index of all supersingular K3 surfaces over a field of characteristic at least 5 using the crystalline Torelli theorem.