T-Duality and Homological Mirror Symmetry of Toric Varieties

Let $X_\Sigma$ be a complete toric variety. The coherent-constructible correspondence $\kappa$ of \cite{FLTZ} equates \Perf_T(X_\Sigma) with a subcategory Sh_{cc}(M_\bR;\LS) of constructible sheaves on a vector space M_\bR. The microlocalization equivalence $\mu$ of \cite{NZ,N} relates these sheaves to a subcategory Fuk(T^*M_\bR;\LS) of the Fukaya category of the cotangent T^*M_\bR. When X_\Si is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, DCoh_T(X_\Si)\cong DFuk(T^*M_\bR;\LS), which is an equivalence of triangulated tensor categories. The nonequivariant coherent-constructible correspondence $\bar{\kappa}$ of \cite{T} embeds \Perf(X_\Si) into a subcategory Sh_c(T_\bR^\vee;\bar{\Lambda}_\Si) of constructible sheaves on a compact torus T_\bR^\vee. When X_\Si is nonsingular, the composition of $\bar{\kappa}$ and microlocalization yields a version of homological mirror symmetry, DCoh(X_\Sigma)\hookrightarrow DFuk(T^*T_\bR;\bar{\Lambda}_\Si), which is a full embedding of triangulated tensor categories. When X_\Si is nonsingular and projective, the composition $\tau=\mu\circ \kappa$ is compatible with T-duality, in the following sense. An equivariant ample line bundle \cL has a hermitian metric invariant under the real torus, whose connection defines a family of flat line bundles over the real torus orbits. This data produces a T-dual Lagrangian brane $\mathbb L$ on the universal cover T^*M_\bR of the dual real torus fibration. We prove \mathbb L\cong \tau(\cL) in Fuk(T^*M_\bR;\LS). Thus, equivariant homological mirror symmetry is determined by T-duality.Comment: 34 pages, 2 figures. The previous version of this paper has now been broken into two parts. The other part is available at arXiv:1007.005

A categorification of Morelli's theorem

We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli's description of the K-theory of a smooth projective toric variety. Specifically, let $X$ be a proper toric variety of dimension $n$ and let M_\bR = \mathrm{Lie}(T_\bR^\vee)\cong \bR^n be the Lie algebra of the compact dual (real) torus T_\bR^\vee\cong U(1)^n. Then there is a corresponding conical Lagrangian \Lambda \subset T^*M_\bR and an equivalence of triangulated dg categories \Perf_T(X) \cong \Sh_{cc}(M_\bR;\Lambda), where \Perf_T(X) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on $X$ and \Sh_{cc}(M_\bR;\Lambda) is the triangulated dg category of complex of sheaves on M_\bR with compactly supported, constructible cohomology whose singular support lies in $\Lambda$. This equivalence is monoidal---it intertwines the tensor product of coherent sheaves on $X$ with the convolution product of constructible sheaves on M_\bR.Comment: 20 pages. This is a strengthened version of the first half of arXiv:0811.1228v3, with new results; the second half becomes arXiv:0811.1228v

Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type {\rm II}\sb 1 factors and M_n(\cc)) and symmetric gauge norms on $L^\infty[0,1]$ and \cc^n. As the first application, we obtain that the class of unitarily invariant norms on a type {\rm II}\sb 1 factor coincides with the class of symmetric gauge norms on $L^\infty[0,1]$ and von Neumann's classical result \cite{vN} on unitarily invariant norms on M_n(\cc). As the second application, Ky Fan's dominance theorem \cite{Fan} is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative $L^p$-theory (e.g., non-commutative H$\ddot{\text{o}}$lder's inequality, duality and reflexivity of non-commutative $L^p$-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of \NN(\M), the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor \M. We obtain all extreme points of \NN(M_2(\cc)) and many extreme points of \NN(M_n(\cc)) ($n\geq 3$). For a type {\rm II}\sb 1 factor \M, we prove that if $t$ ($0\leq t\leq 1$) is a rational number then the Ky Fan $t$-th norm is an extreme point of \NN(\M).Comment: 48 pages, final version, to appear in J. Funct. Ana

Navigating the Nuclear Landscape: Insights into Nuclear Fuel Design

In this interview, Yixing Sung, a veteran nuclear engineer at Westinghouse Electric Corporation, discusses their career in nuclear fuel design, touching on key moments in the history of nuclear power. Born in China, Yixing came to the U.S. in the 1980s to study nuclear engineering during a time when the field's popularity waned due to incidents like the Three Mile Island accident. They describe their fascination with nuclear power’s efficiency despite public concerns, discussing the challenges of safety, cost, and nuclear waste. Yixing reflects on the impact of global nuclear disasters such as Chernobyl and Fukushima, noting how these events shaped public perception, research funding, and industry adaptation. Additionally, they highlight Westinghouse's role in the global nuclear landscape, particularly in Asia, and discuss the influence of geopolitics on nuclear energy development. Despite setbacks, Yixing remains committed to nuclear power as a solution for the world’s growing energy needs

Investigation of Training Algorithms for Hidden Markov Models Applied to Automatic Speech Recognition

The work presented in this thesis focuses on simulating a speech recognizer which is trained by different people with different speaking styles and investigates how sensitive the training and recognition processes are to the variations in the training data. There are four main parts to this work. The first involves an experiment of weighting methods for training with multiple observation sequences. The second involves the testing of different initial parameters. The third part includes the first experiment involving training with multiple observation sequences. The model\u27s sensitivity to variations in training data was evaluated by comparing the cases of different values of variation. The final part varied the observation vectors with the variation restricted to only one of the eight positions in the sequence. The experiment was repeated for each of eight positions in the observation sequence, and the effect on recognition was evaluated

Phoneme Weighting and Energy-Based Weighting for Speaker Recognition

This dissertation focuses on determining specific vowel phonemes which work best for speaker identification and speaker verification, and also developing new algorithms to improve speaker identification accuracy. Results from the first part of our research indicate that the vowels /i/, /E/ and /u/ were the ones having the highest recognition scores for both the Gaussian mixture model (GMM) and vector quantization (VQ) methods (at most one classification error). For VQ, /i/, /I/, /e/, /E/ and /@/ had no classification errors. Persons speaking /E/, /o/ and /u/ have been verified well by both GMM and VQ methods in our experiments. For VQ, the verification results are consistent with the identification results since the same five phonemes performed the best and had less than one verification error. After determining several ideal vowel phonemes, we developed new algorithms for improved speaker identification accuracy. Phoneme weighting methods (which performed classification based on the ideal phonemes we found from the previous experiments) and other weighting methods based on energy were used. The energy weighting methods performed better than the phoneme weighting methods in our experiments. The first energy weighting method ignores the speech frames which have relatively small magnitude. Instead of ignoring the frames which have relatively small magnitude, the second method emphasizes speech frames which have relatively large magnitude. The third method and the adjusted third method are a combination of the previous two methods. The error reduction rate was 7.9% after applying the first method relative to a baseline system (which used Mel frequency cepstral coefficients (MFCCs) as feature and VQ as classifier). After applying the second method and the adjusted third method, the error reduction rate was 28.9% relative to a baseline system