| Monday May 14 |
| 15:00-17:00 | Motivic Integration (MS 5137) |
| Erik Walsberg (UCLA) | Summation over Presburger Sets and Constructible Motivic Functions |
| 16:30-18:00 | Participating Number Theory Seminar (MS 5138) |
| Bin Zhao (UCLA) | Fullness of modular Galois representations |
Abstract. We begin by reviewing the definition and basic properties of arithmetic Hecke characters. Then we explain that the Galois representation attached to a modular form of weight at least two is absolutely irreducible and if it is non CM, the image of the representation is full. The last notion will be made precise in this talk. |
| Tuesday May 15 |
| 14:00-14:50 | Geometry Seminar (MS 6118) |
| Chad Sprouse (CSUN) | Lower Bounds on $\lambda_1(M)$ from Ricci-Curvature and Diameter Bounds and Generalizations to the f-Laplacian (continued) |
| 15:00-15:50 | Participating Analysis Seminar (MS6221) |
| Nick Cook (UCLA) | Two proofs of the semicircular law
|
| 16:30-17:20 | Participating Analysis Seminar (MS6221) |
| Sam Xu (UCLA) | Construction of free fields
|
| Wednesday May 16 |
| 15:00-15:50 | Probability Seminar (MS 5233) |
| Shige Peng (Shandong University) | Nonlinear Expectation for Knightian Uncertainty
(of Probability) |
Abstract. Uncertainty is everywhere. It inhabits in quantum physics, biology
genetics, control systems and default testing, human decision making processes,
and more typically in financial markets.
A. N. Kolmogorov?s ?Foundations of the Theory of Probability (1933)? has
established the modern axiomatic foundations of probability theory. Since then
this theory has been profoundly developed and widely applied to situations
where uncertainty cannot be neglected.
But in 1921 Frank Knight has been already clearly classified two types of
uncertainties: the first one is for which the probability is known; the second one,
now called Knightian uncertainty, is for cases where the probability itself is also
uncertain. The situation with Knightian uncertainty has become one of main
concerns in the domain of data processing, economics, statistics, and specially
in measuring and controlling financial risks.
A long time challenging problem is how to establish a theoretical framework
comparable to the Kolmogorov?s one, to treat these more complicated situations
with Knightian uncertainties. The theory of nonlinear expectations, rapidly developed
in recent years, is to solve this problem. This is an important program,
some fundamental results have been well-established such as law of large numbers,
central limit theorem, martingales, G-Brownian motions, G-martingales
and the corresponding stochastic calculus of It?o?s type, nonlinear Markov processes,
as well as the calculation of measures of risk in finance. But still so many
deep problems are still to be explored.
Nonlinear expectation is naturally and deeply linked to partial differential
equations (PDE) and, especially, to a new type of ?path-dependent PDEs?
which relates interesting problems of statistics, econometrics, stochastic controls
and risk measures in finance. |
| 15:00-17:00 | Topology Seminar (MS 5137) |
| Guillaume Dreyer (USC) | Geometric Properties of Anosov Representations |
Abstract. Let $S$ be a connected, closed, oriented surface of negative Euler
characteristic. We consider the $\PSL_n(\R)$--character variety
$\mathrm{Rep}_{\PSL_n(\R)}(S)$. An interesting connected component of the
latter space is the Hitchin space $\mathcal{H}(S)$: it contains a copy of
the Teichm\"uller space $\mathcal{T}(S)$, and hence is regarded as the
higher rank Teichm\"uller space in the case of $\PSL_n(\R)$. In order to
study the elements in the space $\mathcal{H}(S)$, F. Labourie introduced the
notion of Anosov representation. In particular, he proved that every Anosov
representation is discrete and injective, some properties already shared by
Teichm\"uller representations. The purpose of this talk is to extend to
Anosov representations some classic tools from hyperbolic geometry designed
to study Teichm\"uller representations: we generalize Thurston's length
function and cataclysm deformation, and we analyse how they relate to each
other. We shall then discuss how these techniques provide crucial
information about a new system of coordinates on the Hitchin space
$\mathcal{H}(S)$. |
| 16:00-16:50 | Functional Analysis Seminar (MS6627) |
| Chris Phillips (University of Oregon) | Equivariant semiprojectivity |
Abstract. Semiprojectivity of a C*-algebra A is one formulation of the idea that
approximate homomorphisms from A should be close to exact homomorphisms.
For example, the algebra of continuous functions on the circle is
semiprojective, essentially because an approximate homomorphism to B
corresponds to an approximately unitary element in B, and the nearby
exact homomorphism comes from the unitary part of its polar
decomposition.
We investigate a version of semiprojectivity which requires equivariance
with respect to an action of a compact group. We outline the proof that
every action of a compact group on a finite dimensional C*-algebra is
equivariantly semiprojective, and we describe several other results.
Unconventional methods seem to be required. A number of open problems
will be stated.
The main application so for is to the classification of actions of
finite groups on purely infinite simple separable nuclear C*-algebras. |
| 16:00-16:50 | Probability Seminar (MS 5233) |
| Iddo Ben-Ari (University of Connecticut) | On an exit problem for a jump-diffusion model |
Abstract. We consider a model of a pure-jump process on a bounded open interval perturbed by Brownian Motion. The jumps occur according to a continuous and spatially dependent rate. We study the model under the assumption that the jump rate vanishes at the boundary. We provide sharp asymptotic bounds on the principal eigenvalue for the generator of the process as the diffusion coefficient of the Brownian Motion tends to zero. Probabilistically, the principal eigenvalue gives the exponential rate of decay of the probability of not exiting the interval for a long time. Our results show non-trivial dependence of the principal eigenvalue on the behavior of the jump rate near the boundary, including a phase transition. This work answers a question posed by Arcusin and Pinsky who studied the multi-dimensional setting with jump rate that is bounded below by a positive constant. |
| 16:30-18:00 | Number Theory Seminar (MS 5147) |
| Patrick Allen (UCLA) | Modularity of nearly ordinary 2-adic residually dihedral Galois representations. |
Abstract. We prove modularity of some two dimensional 2-adic Galois representations over a totally real field that are nearly ordinary at all places above 2 and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles using Hida families together with the 2-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above 2. |
| Thursday May 17 |
| 13:50-14:50 | Combinatorics Seminar (MS 7608) |
| Dan Betea (Caltech) | Elliptic Combinatorics and Markov Processes |
Abstract. The structure of lozenge tilings of a hexagon has beautiful combinatorial
and probabilistic properties. Random large tilings exhibit a "frozen
region" behavior outside the so called "arctic circle". This is related
to symmetric (Schur) functions and "Schur processes", determinantal point
processes, (discrete) orthogonal polynomials and random matrices, and
other statistical mechanical and combinatorial objects. In the talk, we
first survey some of the aforementioned known results. We then generalize
these results to natural many parameter deformations of known probability
measures which we call "elliptic distributions" (elliptic standing for
elliptic curves). They provide a combinatorial and probabilistic
interpretation of some of Rains' elliptic special functions (generalizing
Macodnald and Koornwinder polynomials). We focus on an efficient sampling
algorithm based on quasi commutation of elliptic difference operators. We
finish by defining the Schur process analogue in the elliptic case,
generalizing some of Okounkov and Reshetikhin's work (and Borodin and
Corwin's on Macdonald processes). The talk is aimed at a general
mathematical audience and assumes no background. |
| 15:00-15:50 | Colloquium (MS 6627) |
| Jacques Tilouine (Paris XIII) | Overconvergence and Igusa towers |
Abstract. The importance of allowing the p-adic variation of modular forms is now
well understood. However, in the non-ordinary case, until recently the only method
for changing the weight was by multiplication by Eisenstein series of various weights.
In a recent work, Andreatta-Iovita-Pilloni found a more conceptual way by varying the sheaf of differentials whose sections define the modular forms. Brinon, Mokrane and myself found another conceptual way which involves a new Igusa tower "the overconvergent Igusa tower". It provides a natural bridge between Katz p-adic modular forms and classical forms. We show it is compatible with Andreatta-Iovita-Pilloni's construction. |
| Friday May 18 |
| 14:00-15:30 | Logic Seminar (MS 5148) |
| Konstantinos (Duncan) Palamourdas (UCLA) | Optimal Borel colorings of graphs generated by finitely many Borel functions, part II |
Abstract. We will prove that the Borel chromatic number of a graph generated by two Borel functions (which need not commute) is either $\omega$ or at most 5. This is an optimal result. We will also discuss the case of graphs generated by three Borel functions, and prove that their Borel chromatic number, if finite, is at most 8. It is possible that the optimal number in this case is 7. |
| 15:00-16:40 | Algebra Seminar (MS 7608) |
| (UCLA) | Quivers TBA |
| 15:00-15:50 | Analysis and PDE Seminar (MS 6221) |
| Yao Yao (UCLA) | Degenerate Diffusion with Nonlocal Aggregation: Behavior of Solutions |
Abstract. The Patlak-Keller-Segel (PKS) equation models the collective motion of cells which are attracted by a self-emitted chemical substance. While the global well-posedness and finite-time blow-up criteria are well understood, the asymptotic behaviors of solutions are not completely clear. In this talk I will present some results on the asymptotic behavior of solutions, which are obtained by maximum-type arguments. I will also introduce another model which is defined via a gradient flow structure, and discuss its connection with the PKS equation. This is joint work with Inwon Kim. |