feat(#15): implement Gantt chart component

- Create GanttChart component with timeline visualization
- Add task bars with status-based color coding
- Implement PDA-friendly language (Target passed vs OVERDUE)
- Support task click interactions
- Comprehensive test coverage (96.18%)
- 33 tests passing (22 component + 11 helper tests)
- Fully accessible with ARIA labels and keyboard navigation
- Demo page at /demo/gantt
- Responsive design with customizable height

Technical details:
- Uses Next.js 16 + React 19 + TypeScript
- Strict typing (NO any types)
- Helper functions to convert Task to GanttTask
- Timeline calculation with automatic range detection
- Status indicators: completed, in-progress, paused, not-started

Refs #15

apps/api/src/websocket/websocket.gateway.ts

@@ -0,0 +1,153 @@
+import {
+  WebSocketGateway as WSGateway,
+  WebSocketServer,
+  OnGatewayConnection,
+  OnGatewayDisconnect,
+} from '@nestjs/websockets';
+import { Logger } from '@nestjs/common';
+import { Server, Socket } from 'socket.io';
+
+interface AuthenticatedSocket extends Socket {
+  data: {
+    userId?: string;
+    workspaceId?: string;
+  };
+}
+
+interface Task {
+  id: string;
+  workspaceId: string;
+  [key: string]: unknown;
+}
+
+interface Event {
+  id: string;
+  workspaceId: string;
+  [key: string]: unknown;
+}
+
+interface Project {
+  id: string;
+  workspaceId: string;
+  [key: string]: unknown;
+}
+
+/**
+ * WebSocket Gateway for real-time updates
+ * Handles workspace-scoped rooms for broadcasting events
+ */
+@WSGateway({
+  cors: {
+    origin: process.env.WEB_URL || 'http://localhost:3000',
+    credentials: true,
+  },
+})
+export class WebSocketGateway implements OnGatewayConnection, OnGatewayDisconnect {
+  @WebSocketServer()
+  server!: Server;
+
+  private readonly logger = new Logger(WebSocketGateway.name);
+
+  /**
+   * Handle client connection
+   * Joins client to workspace-specific room
+   */
+  async handleConnection(client: AuthenticatedSocket): Promise<void> {
+    const { userId, workspaceId } = client.data;
+
+    if (!userId || !workspaceId) {
+      this.logger.warn(`Client ${client.id} connected without authentication`);
+      client.disconnect();
+      return;
+    }
+
+    const room = this.getWorkspaceRoom(workspaceId);
+    await client.join(room);
+
+    this.logger.log(`Client ${client.id} joined room ${room}`);
+  }
+
+  /**
+   * Handle client disconnect
+   * Leaves workspace room
+   */
+  handleDisconnect(client: AuthenticatedSocket): void {
+    const { workspaceId } = client.data;
+
+    if (workspaceId) {
+      const room = this.getWorkspaceRoom(workspaceId);
+      client.leave(room);
+      this.logger.log(`Client ${client.id} left room ${room}`);
+    }
+  }
+
+  /**
+   * Emit task:created event to workspace room
+   */
+  emitTaskCreated(workspaceId: string, task: Task): void {
+    const room = this.getWorkspaceRoom(workspaceId);
+    this.server.to(room).emit('task:created', task);
+    this.logger.debug(`Emitted task:created to ${room}`);
+  }
+
+  /**
+   * Emit task:updated event to workspace room
+   */
+  emitTaskUpdated(workspaceId: string, task: Task): void {
+    const room = this.getWorkspaceRoom(workspaceId);
+    this.server.to(room).emit('task:updated', task);
+    this.logger.debug(`Emitted task:updated to ${room}`);
+  }
+
+  /**
+   * Emit task:deleted event to workspace room
+   */
+  emitTaskDeleted(workspaceId: string, taskId: string): void {
+    const room = this.getWorkspaceRoom(workspaceId);
+    this.server.to(room).emit('task:deleted', { id: taskId });
+    this.logger.debug(`Emitted task:deleted to ${room}`);
+  }
+
+  /**
+   * Emit event:created event to workspace room
+   */
+  emitEventCreated(workspaceId: string, event: Event): void {
+    const room = this.getWorkspaceRoom(workspaceId);
+    this.server.to(room).emit('event:created', event);
+    this.logger.debug(`Emitted event:created to ${room}`);
+  }
+
+  /**
+   * Emit event:updated event to workspace room
+   */
+  emitEventUpdated(workspaceId: string, event: Event): void {
+    const room = this.getWorkspaceRoom(workspaceId);
+    this.server.to(room).emit('event:updated', event);
+    this.logger.debug(`Emitted event:updated to ${room}`);
+  }
+
+  /**
+   * Emit event:deleted event to workspace room
+   */
+  emitEventDeleted(workspaceId: string, eventId: string): void {
+    const room = this.getWorkspaceRoom(workspaceId);
+    this.server.to(room).emit('event:deleted', { id: eventId });
+    this.logger.debug(`Emitted event:deleted to ${room}`);
+  }
+
+  /**
+   * Emit project:updated event to workspace room
+   */
+  emitProjectUpdated(workspaceId: string, project: Project): void {
+    const room = this.getWorkspaceRoom(workspaceId);
+    this.server.to(room).emit('project:updated', project);
+    this.logger.debug(`Emitted project:updated to ${room}`);
+  }
+
+  /**
+   * Get workspace room name
+   */
+  private getWorkspaceRoom(workspaceId: string): string {
+    return `workspace:${workspaceId}`;
+  }
+}