William Brian

Speaker: Dr. Luo Ye from Xiamen University

Date and Time: Friday, February 11, 2022, 9:00-10:00 via Zoom. Please contact Will Brian to obtain the Zoom link.

Title: Tropical convexity analysis and some applications

Abstract:  The tropical semiring is an idempotent semiring where  the usual operations of addition and multiplication are replaced by operations of minimum/maximum and addition respectively. Tropical geometry is a theory of geometry over the tropical semiring which has rich combinatorial features and can be described as a degenerated version of algebraic geometry over the field of complex numbers under Maslov dequantization or over a non-archimedean field under the valuation map. The  features of “linear combinations” in tropical geometry can be captured by the notion of tropical convexity.   In this talk, I will introduce a general theory of tropical convexity analysis based on the so-called “B-pseudonorms” on tropical projective spaces, and show some subsequent results, e.g., a tropical version of Mazur’s Theorem on closed tropical convex hulls and a fixed point theorem for tropical projections. Two applications will also be presented. The first is to establish a connection between tropical projections and  reduced divisors on (metric) graphs, and the second is to construct  min-max-plus neural networks, a new type of artificial neural networks.

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Speaker: Dr. Yuexiao Dong from Temple University

Date and Time: Friday, February 4, 2022, 11:00-12:00 via Zoom. Please contact Will Brian to obtain the Zoom link.

Title: Testing the linear mean and constant variance conditions in sufficient dimension reduction

Abstract:  Sufficient dimension reduction (SDR) methods characterize the relationship between the response and the predictors through a few linear combinations of the predictors. Sliced inverse regression and sliced average variance estimation are among the most popular SDR methods as they do not involve multi-dimensional smoothing and are easy to implement. However, these inverse regression-based methods require the linear conditional mean (LCM) and(or) the constant conditional variance (CCV) assumption. We propose novel tests to check the validity of the LCM and the CCV conditions through the martingale difference divergence. Extensive simulation studies and a real data application are performed to demonstrate the effectiveness of our proposed tests.

Speaker: Dr. Eshita Mazumdar from Ahmedabad University.

Date and Time: Friday, January 28, 2022, 11:00-12:00 via Zoom. Please contact Will Brian to obtain the Zoom link.

Title: Zero-sum problems

Abstract:  Zero-sum problems are basically combinatorial in nature. It deals with the condition which ensures that a given sequence over a finite group has a zero-sum subsequence with some prescribed property. There are many invariants associated with zero-sum problems. One of such invariants is the Davenport Constant.  The original motivation for introducing the Davenport Constant was to study the problem of non-unique factorization domain over number fields. The precise value of this group invariant for any finite abelian group is still unknown. In this talk I will discuss an extremal problem related to Weighted Davenport Constant and introduce several exciting combinatorial results for finite abelian groups. Also, characteristics of these constants on restricted sequences will be discussed. If time permits I will talk about an ongoing project where we introduced a new group invariant which is a natural generalization of the Davenport Constant.

Speaker: Dr. Steven Clontz, from the University of South Alabama

Date and Time: Friday, January 21, 2022, 12:00-1:00 via Zoom. Please contact Will Brian to obtain the Zoom link.

Title: Games Topologists Play

Abstract: Several ideas from topology and set theory may be characterized by considering two-player infinite-length games. During each round n ∈ {0,1,2, . . . }, suppose Player 1 makes a move a_n (perhaps choosing an open cover of a given regular space), followed by Player 2 making a move b_n (perhaps choosing a finite subcollection from 1’s chosen cover); the winner of such a game is determined by the sequence of moves ( a₀,b₀,a₁,b₁, . . . ) (perhaps Player 2 wins if their choices form a cover).

The topological game specified above is known as Menger’s game, and Player 2 has an unbeatable strategy that only uses information limited to the round number and the most recent move of Player 1 in this game if and only if the given regular space is σ-compact. In this talk, we will explore various results of this flavor found in the literature, including an interesting game-theoretic proof appropriate for undergraduates that the real numbers are uncountable.