SCMA30045 Colloquium | 論文研討 || 2026 Spring

We will learn ...

This course provides an overview of some fundamental areas in mathematics, including calculus, linear algebra, discrete mathematics, and probability. We will explore key definitions from these subjects and learn how to prove or disprove whether an example satisfies a given definition. We will then advance to more proof writing by exploring properties of numbers, sets, functions, and relations. By the end of the course, you will know how to explore and manipulate new mathematical objects.

You need to do ...

HW0: Tell me your email before September 12 to get extra 2pt — this is a required work. Important information will be announced through email.

A few tips for learning mathematics ...

Mistakes Make You Smarter: Everyone learns through experiences and mistakes. For each new concept you learn, generate as many examples as possible to train your brain to distinguish between right and wrong.

Ask Questions: Beyond knowledge, mathematics is fundamentally about logic. Question everything you encounter—why it is defined this way, why an assumption is required, why a proof needs a particular step, and so on.

Think Carefully: Sound arguments should hold true in any circumstance. Verify the examples you generate to ensure they align with your argument.

Help Each Other: Learning together can make the process easier. Teaching others is also an effective way to reinforce your own understanding.

Course Info

Tentative Schedule

• February 24
  13:20課程介紹
• February 26
  11:10李志光教授, College of William and Mary
  Linear preservers of matrix pairs with some extremal norm properties [ slides ]

  Abstract

  We present some recent results on linear maps $T$ on matrix pairs $(A,B)$ preserving some extremal norm properties such as

  1. $\|T(A) + T(B)\| = \|T(A)\| + \|T(B)\|$ whenever $\|A+B\| = \|A|\ + \|B\|$, or
  2. $\|T(A)T(B)\| = \|T(A)\| \|T(B)\|$ whenever $\|AB\| = \|A|| \|B\|$.

  Related results and open problems will be mentioned.

• March 3
  13:30台積電智能製造中心 ★ 2026 校園講座：智能製造．智造未來 ★ (to 15:00)
  Details

  
• March 10
  13:20呂秉澤教授, 國立中正大學
  Geometry-Aware Iterative and Direct Methods for Unitary Quantum Channel Reconstruction

  Abstract

  Unitary quantum channels play a central role in modeling coherent quantum dynamics and quantum circuits. Reconstructing such channels from finite input–output data is inherently challenging, particularly in the presence of noise, and is closely related to nonconvex optimization on matrix manifolds.

  In this work, we develop a unified geometry-aware framework for unitary quantum channel reconstruction by formulating the problem as a constrained optimization task on the Stiefel manifold. For noisy measurement data, we propose an iterative algorithm based on polar decomposition that embeds the unitary constraint directly into the update rule. We prove that the resulting sequence monotonically decreases the objective function and converges to a critical point on the manifold.

  In the noise-free setting, we further introduce a direct reconstruction methodology. We show that the global minimizers of the objective function form an equivalence class of unitary matrices, differing only by a global phase factor, and establish conditions under which the underlying quantum channel can be recovered exactly. Leveraging spectral properties of non-degenerate quantum states, we derive a reconstruction procedure that significantly reduces the effective search dimension and requires only a minimal number of quantum observables.

  Together, the proposed iterative and direct methods provide a theoretically rigorous and computationally efficient approach for approximating or exactly recovering unitary quantum channels from limited data.

• March 17
  13:20王新博教授, 國立臺灣大學
  TBA

  Abstract

  TBA

• March 24
  13:20尤釋賢教授, 中央研究院
  TBA

  Abstract

  TBA

• March 31
  13:20Professor Swee Hong Chan, Rutgers University
  Spanning trees and continued fractions

  Abstract

  Consider the set of positive integers representing the number of spanning trees in simple graphs with n vertices. How quickly can this set grow as a function of n? In this talk, we discuss a proof of the exponential growth of this set, which resolves an open problem of Sedlacek from 1966. The proof uses a connection with continued fractions and advances towards Zaremba’s conjecture in number theory. This is joint work with Alex Kontorovich and Igor Pak. This talk is intended for general audience.

• April 7
  13:20性別平等課程宣導
• April 14
  13:20陳柏廷醫師, 臺大醫院
  TBA

  Abstract

  TBA

• April 21
  13:20吳德琪教授, 中央研究院
  Long-Time Asymptotics for the Kadomtsev–Petviashvili II Equation

  Abstract

  The Kadomtsev-Petviashvili II (KPII) equation is one of the few physically relevant integrable systems in more than one spatial dimension. In this talk, we present an overview of the inverse scattering theory and the stationary phase method, and explain how these tools are used to derive the long-time asymptotic behavior of solutions.

• April 28
  13:20黃楓台​處長, 國家太空中心
  TBA

  Abstract

  TBA

  14:20黃楓南教授, 國立中央大學
  TBA

  Abstract

  TBA

• May 5
  13:20機率演講
  TBA

  Abstract

  TBA

• May 12
  13:20李宜霖博士, 國立臺灣師範大學
  TBA

  Abstract

  TBA

• May 19
  無安排演講
• May 26
  13:20張昌祐教授
  , 中央研究院 TBA

  Abstract

  TBA

• June 2
  13:20TBA
  TBA

  Abstract

  TBA

• June 9
  期末考週，無安排演講

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